Differential Geometry of Curves and Surfaces

To my students in MATH 4230/5230, Fall 2026—
may these notes help you see geometry in motion.

Preface

These notes accompany MATH 4230/5230, Differential Geometry of Curves and Surfaces, at the University of Colorado Boulder in Fall 2026. They are organized lecture by lecture and follow the course’s progression from parametrized curves in Euclidean space to the local and global geometry of surfaces.

The primary text is Kristopher Tapp’s Differential Geometry of Curves and Surfaces [Tap16]. Andrew Browder’s Mathematical Analysis: An Introduction [Bro96] serves as a supplementary reference for analytic background, especially for students enrolled in MATH 5230.

The aim of the notes is not to replace either the textbook or the classroom. Instead, each lecture records the principal definitions, examples, statements, and calculations that organize the day’s discussion. Proofs may occasionally be abbreviated in class and completed here; conversely, some geometric pictures and informal explanations will remain most natural at the board.

Organization of the notes

Each lecture begins with the assigned reading, the related homework, and a short list of goals. Definitions, examples, and results are numbered within that lecture. The index is meant to make the notes useful as a reference near the end of the semester, while the bibliography records the sources used in preparing them.

These notes will be revised throughout the semester. The date on the title page identifies the current working edition.

Course Information

Course

MATH 4230/5230, Differential Geometry of Curves and Surfaces.

Semester

Fall 2026.

Instructor

Sebastian Casalaina-Martin.

Class meetings

Monday, Wednesday, and Friday, 11:15 a.m.–12:05 p.m., in ECCR 151.

Primary text

Kristopher Tapp, Differential Geometry of Curves and Surfaces [Tap16].

Supplementary analysis reference

Andrew Browder, Mathematical Analysis: An Introduction [Bro96].

Scope

The course studies the geometry of plane and space curves and regular surfaces in $R^{3}$ . Central topics include speed and arc length, curvature and torsion, the Frenet frame, the first and second fundamental forms, Gaussian and mean curvature, geodesics, parallel transport, and the Gauss–Bonnet theorem.

How to use these notes

Read the assigned section of Tapp before class. During class, use these notes to organize definitions and calculations rather than as a substitute for working them out. After class, complete the assigned exercises and add your own examples, diagrams, and corrections in the margins.

Conventions and Notation

Unless stated otherwise, all maps are smooth. An interval means a connected subset of $R$ , usually open unless endpoints are explicitly included. A parametrized curve is generally denoted

γ : I \to R^{n},

with velocity $γ^{'} (t)$ and acceleration $γ^{″} (t)$ . The standard Euclidean inner product and norm are written

⟨ u, v ⟩ and ‖ u ‖ = \sqrt{⟨ u, u ⟩} .

Beginning in Lecture 4 we follow the notation of Tapp’s curve chapter: names of vectors and vector-valued functions are boldface. Thus $𝜸$ is a curve, $𝐯$ and $𝐚$ are its velocity and acceleration, and the lowercase bold symbols $𝐭$ and $𝐧$ denote the unit tangent and unit normal vectors. The curvature function is $κ$ ; the vector $𝐧 (t)$ is defined only when $κ (t) \neq 0$ .

For plane curves, $R_{𝜃}$ denotes counterclockwise rotation through angle $𝜃$ , and $R_{90} = R_{π / 2}$ . The signed curvature is $κ_{s}$ , an angle function for the unit tangent is $𝜃$ , and $rot (𝜸)$ denotes the rotation index of a closed plane curve.

For the global theory of plane curves, the unit circle is

S^{1} = {𝐮 \in R^{2} : | 𝐮 | = 1} .

If $f : [a, b] \to S^{1}$ is continuous with $f (a) = f (b)$ , then $\deg (f)$ denotes its degree. For a simple closed plane curve with trace $C$ and a point $𝐩 \notin C$ ,

W_{C} (𝐩) = \deg (t ⟼ \frac{𝜸 (t) - 𝐩}{| 𝜸 (t) - 𝐩 |})

is the winding number of $C$ about $𝐩$ . The bounded and unbounded components of $R^{2} - C$ are called the interior and exterior. For a piecewise-regular plane curve, $α_{i}$ denotes the signed turning angle at the $i$ th corner.

For space curves in $R^{3}$ , $𝐛 = 𝐭 \times 𝐧$ is the unit binormal vector, and ${𝐭, 𝐧, 𝐛}$ is the Frenet frame. The torsion function is denoted by $τ$ , using Tapp’s sign convention

τ = - \frac{⟨ 𝐛^{'}, 𝐧 ⟩}{| 𝐯 |} .

The osculating plane is spanned by $𝐭$ and $𝐧$ , with $𝐛$ as its unit normal.

For rigid motions, $O (n)$ denotes the orthogonal group. If $A \in O (n)$ , then $L_{A} (𝐱) = A 𝐱$ , while $T_{𝐪} (𝐱) = 𝐱 + 𝐪$ denotes translation by $𝐪$ . Every rigid motion has the form $T_{𝐪} \circ L_{A}$ ; it is proper when $\det A = 1$ and improper when $\det A = - 1$ .

For differential forms on an open set $U \subseteq R^{n}$ , $Ω^{k} (U)$ denotes the vector space of smooth $k$ -forms. The coordinate one-forms are $d x_{1}, \dots, d x_{n}$ , and for an ordered index set $I = (i_{1} < \dots < i_{k})$ ,

d x_{I} = d x_{i_{1}} \land \dots \land d x_{i_{k}} .

The symbol $d$ also denotes the exterior derivative. A vector field $𝐅 = (F_{1}, \dots, F_{n})$ corresponds, through the Euclidean inner product, to the one-form

ω_{𝐅} = ∑_{i = 1}^{n} F_{i} d x_{i} .

Its line integral along a curve $𝜸$ is written

\int_{𝜸} ω_{𝐅} = \int_{𝜸} 𝐅 \cdot d 𝜸 .

For a function $f$ , the notation $𝐷𝑓$ denotes its derivative as a linear map. Later, when covariant differentiation is introduced, its notation will be stated explicitly to avoid confusing it with the ordinary Euclidean derivative.

The end of a proof is marked by $□$ . Definitions, examples, and results are numbered by lecture; for example, Proposition 4.2 is the second numbered item in Lecture 4.

Preface
Course Information
Conventions and Notation
1Parametrized Curves: Regularity, Speed, and Arc Length
1.1Review
1.2Parametrized curves
1.3Velocity, regularity, and speed
1.4Arc length
1.5Summary
1.1.1Number systems and Euclidean space
1.1.2Intervals
1.1.3Smooth functions and maps
2Inner Products, Projections, and Orthonormal Bases
2.1The Euclidean inner product
2.2Orthogonal projection
2.3Differentiating inner products
2.4A geometric application: straight lines are shortest (not covered in class)
2.5Summary
2.2.1Review: Components and projections (not covered in class)
2.2.2Review: Orthonormal sets and bases
3Acceleration and Reparametrization
3.1Acceleration
3.2Trace and reparametrization
3.3More on reparameterization (not covered in class)
3.4Reparametrization by arc length (Covered on Friday August 28)
3.5Closed curves (not covered in class)
3.6Summary
3.1.1Tangential and perpendicular acceleration
4Curvature
4.1The curvature function
4.2Unit tangent and unit normal vectors
4.3Curvature of a graph at a critical point (not covered in class)
4.4A worked example (not covered in class)
4.5Further topics
4.6Summary
4.3.1Computational formula
5Plane Curves
5.1Rotation matrices
5.2Signed curvature
5.3Angle functions
5.4Rotation index
5.5Summary
5.2.1The unit-speed case
6Space Curves: Frenet Frame, Torsion, and Frenet Equations
6.1The cross product
6.2The Frenet frame
6.3The Frenet equations
6.4Some further topics
6.5Summary
6.1.1Rotations and the cross product
6.4.1Zero torsion
6.4.2Product rule
6.4.3A curvature formula in R 3
7Rigid Motions and Fundamental Theorems
7.1Rigid motions of Euclidean space
7.2Geometric quantities under rigid motion
7.3The fundamental theorems of curves
7.4A short overview of curvature formulas
7.5Summary
7.3.1The plane-curve construction
8Hopf’s Umlaufsatz and the Jordan Curve Theorem
8.1Degree and the unit tangent map
8.2Hopf’s Umlaufsatz
8.3The Jordan curve theorem
8.4A first look at the piecewise-regular formula
8.5Summary
9Differential Forms, Vector Fields, and Line Integrals
9.1Coordinate forms and the wedge product
9.2Smooth differential forms
9.3The exterior derivative
9.4Vector fields and one-forms
9.5Line integrals
9.6Summary
9.1.1Wedge products of forms
Bibliography
Index

Lecture 1Parametrized Curves: Regularity, Speed, and Arc Length

Friday, August 21, 2026

Reading. [Tap16, §1.1, pp. 2–8].

After class. Tapp, Exercises 1.5–1.10.

Goals. Review intervals and smooth maps, describe curves as smooth maps, distinguish regular from singular parametrizations, compute speed, and define arc length.

1.1 Review

1.1.1 Number systems and Euclidean space

Recall the familiar inclusions

N \subseteq Z \subseteq Q \subseteq R,

where $N = {1, 2, 3, \dots}$ are the natural numbers, $Z = {\dots, - 1, 0, 1, 2, \dots}$ are the integers, $Q = {a / b : a, b \in Z, b \neq 0}$ are the rational numbers, and $R$ is the set of real numbers (all the numbers on the number line).

For each positive integer $n$ , Euclidean $n$ -space is

R^{n} = {(x_{1}, \dots, x_{n}) : x_{i} \in R} .

In particular, $R^{1} = R$ . For $x = (x_{1}, \dots, x_{n}) \in R^{n}$ , the Euclidean norm is

‖ x ‖ = \sqrt{x_{1}^{2} + \dots + x_{n}^{2}},

and the distance between $x, y \in R^{n}$ is $‖ x - y ‖$ .

1.1.2 Intervals

An interval is a subset $I \subseteq R$ with the property that, if $a, b \in I$ and $a < c < b$ , then $c \in I$ . A bounded interval with endpoints $a < b$ has one of the four forms

(a, b), (a, b], [a, b), [a, b] .

We also use $- \infty$ and $\infty$ to describe unbounded intervals, for example $(a, \infty)$ , $[a, \infty)$ , and $(- \infty, b]$ . The symbols $\pm \infty$ are never included as endpoints.

1.1.3 Smooth functions and maps

Let $I \subseteq R$ be an interval. A function $f : I \to R$ is smooth if it is infinitely differentiable at every $x \in I$ ; i.e., for all $x \in I$ and all integers $k \geq 0$ , we have $f^{(k)} (x)$ exists. For intervals that include one or both of their endpoints, then at these endpoints, we mean that the derivatives exist as one sided limits; for instance, if $I = [a, b)$ , then to say that $f^{'} (a)$ exists means that $\lim_{h \to 0^{+}} \frac{f (a + h) - f (a)}{h}$ exists. We write

C^{\infty} (I) = {f : I \to R : f is smooth} .

In particular, $C^{\infty} (R)$ denotes the smooth real-valued functions on $R$ . A map

f = (f_{1}, \dots, f_{n}) : I \to R^{n}

f (t) = (f_{1} (t), \dots, f_{n} (t))

is smooth when each coordinate function $f_{i} : I \to R$ is smooth.

Remark 1.1 (Smoothness at a boundary point). The one-sided definition of smoothness above agrees with the extension definition of smoothness used in [Tap16, Definition/Exercise 1.2]. That is to say, $f : I \to R^{n}$ is smooth if and only if there exists an open interval $Î \subseteq R$ with $I \subseteq Î$ and a smooth map $\hat{f} : Î \to R^{n}$ such that for all $x \in I$ and all integers $k \geq 0$ , one has $f^{(k)} (x) = {\hat{f}}^{(k)} (x)$ . (Note that $f^{(k)} (x)$ is a one-sided limit if $x$ is an endpoint of $I$ , whereas ${\hat{f}}^{(k)} (x)$ is a two-sided limit at such a point.) For instance, if $f : [a, b) \to R^{n}$ is smooth, then there is a real number $𝜖 > 0$ and a smooth map $\hat{f} : (a - 𝜖, b) \to R^{n}$ such that for all $x \in [a, b)$ and all integers $k \geq 0$ , one has $f^{(k)} (x) = {\hat{f}}^{(k)} (x)$ . A similar statement holds for intervals $(a, b]$ , and $[a, b]$ . We will use this extension fact without proof.

Exercise 1.2 (For students in MATH 5230). Let $I$ be an interval containing $0$ in its interior, and consider the linear map

𝒯 : C^{\infty} (I) \to R [[x]], 𝒯 (f) = ∑_{k = 0}^{\infty} \frac{f^{(k)} (0)}{k!} x^{k},

which sends a smooth function to its formal Taylor series at $0$ . Borel’s lemma says that $𝒯$ is surjective. Look up the statement of Borel’s lemma; no proof is required. Then use that to show that our definition of smoothness agrees with the definition in Tapp; i.e., prove that the remark above is true. For instance, if $I = [0, b)$ , then let $g (x) \in C^{\infty} (R)$ be any function such that $𝒯 (g) = \sum_{k = 0}^{\infty} \frac{f^{(k)} (0)}{k!} x^{k}$ ; then define $\hat{f} (x) = f (x)$ for $x \in I = [0, b)$ and $\hat{f} (x) = g (x)$ for $x \in (- 𝜖, 0]$ . Also describe $\ker (𝒯)$ : what does it mean for a smooth function to have zero Taylor series at $0$ ? Is the kernel finite dimensional?

Remark 1.3 (Analysis background for MATH 5230). Graduate students should review [Bro96, Chapter 8, especially §§ 8.1–8.2 and 8.4] and learn the definition of the differential of a map $f : R^{n} \to R^{m}$ , its interpretation as a linear approximation, its matrix representation, and the chain rule. They should also work several relevant exercises from § 8.6.

1.2 Parametrized curves

A geometric curve is a path traced through space. A parametrization records both the points of the path and the manner in which the path is traversed. Thus two maps can have the same image while moving along it at different speeds, in different directions.

The basic object in this course is therefore a map rather than merely a subset of Euclidean space.

Definition 1.4 (Parametrized curve). A smooth parametrized curve in $R^{n}$ is a smooth map

γ : I \to R^{n},

where $I \subseteq R$ is an interval.

Example 1.5 (Lines). Fix $p, v \in R^{n}$ with $v \neq 0$ . The map

γ (t) = p + 𝑡𝑣

parametrizes the line through $p$ in direction $v$ . Its velocity is constant:

γ^{'} (t) = 𝑣.

Example 1.6 (A circle). For $r > 0$ , the map

γ (t) = (r \cos t, r \sin t), 0 \leq t \leq 2 π,

traverses the circle of radius $r$ once counterclockwise.

A line through $p$ in direction $v$ .

A circle traversed counterclockwise.

Figure 1.1: Two basic parametrized curves.

Example 1.7 (The graph of a function). Let $f : I \to R$ be smooth. The curve

γ (t) = (t, f (t))

parametrizes the graph $y = f (x)$ . Its derivative is

γ^{'} (t) = (1, f^{'} (t)) .

Figure 1.2: The graph of a smooth function can be viewed as a parametrized curve.

1.3 Velocity, regularity, and speed

Definition 1.8 (Velocity and regularity). The velocity of $γ$ at time $t$ is $γ^{'} (t)$ . The curve is regular at $t$ if $γ^{'} (t) \neq 0$ , and it is regular if it is regular at every point of its domain.

Regularity says that the parametrization never comes instantaneously to a complete stop. It is a condition on the parametrization, not only on its image.

Definition 1.9 (Speed). The speed of a parametrized curve $γ$ is the function

v_{γ} (t) = ‖ γ^{'} (t) ‖ .

Remark 1.10. A parametrized curve is regular if and only if its speed $‖ γ^{'} (t) ‖$ is never zero. This is the definition used in Tapp.

Example 1.11 (Speed of the circle). For $γ (t) = (r \cos t, r \sin t)$ ,

γ^{'} (t) = (- r \sin t, r \cos t) and hence ‖ γ^{'} (t) ‖ = 𝑟.

Thus the standard angular parametrization has constant speed $r$ .

Example 1.12 (A nonregular parametrization). Consider

γ (t) = (t^{2}, t^{3}), t \in 𝑅.

Then

γ^{'} (t) = (2 t, 3 t^{2}),

so $γ^{'} (0) = 0$ . The curve is regular away from $t = 0$ , but it is not regular at $0$ . Its image is the semicubical cusp $y^{2} = x^{3}$ , with $x \geq 0$ .

Figure 1.3: A semicubical cusp: the image is a curve, but the parametrization is not regular at

t = 0

1.4 Arc length

Definition 1.13 (Arc length). Let $γ : [a, b] \to R^{n}$ be a smooth parametrized curve. Its arc length is

L (γ) = \int_{a}^{b} ‖ γ^{'} (t) ‖ d 𝑡.

The formula expresses a familiar physical principle: distance traveled is the integral of speed over time.

Figure 1.4: A curve from

γ (a)

γ (b)

, with instantaneous velocity vector

γ^{'} (t)

Example 1.14 (Length of a line segment). If $γ (t) = p + t (q - p)$ for $0 \leq t \leq 1$ , then

L (γ) = \int_{0}^{1} ‖ q - p ‖ d t = ‖ q - p ‖ .

Example 1.15 (Circumference of a circle). For the circle above,

L (γ) = \int_{0}^{2 π} r d t = 2 𝜋𝑟.

1.5 Summary

The number systems $N$ , $Z$ , $Q$ , and $R$ , and Euclidean space $R^{n}$ ;
intervals in $R$ , including open, closed, half-open, and unbounded intervals;
smooth functions on intervals, including smoothness at boundary points;
smooth maps $I \to R^{n}$ and derivatives of vector-valued functions;
parametrized curves $γ : I \to R^{n}$ ;
examples of parametrized curves, including lines, circles, and graphs of functions;
velocity $γ^{'} (t)$ , speed $∥ γ^{'} (t) ∥$ , and regular curves;
examples of nonregular curves, including the cusp $γ (t) = (t^{2}, t^{3})$ ;
arc length and the formula
$L (γ) = \int_{a}^{b} ∥ γ^{'} (t) ∥ 𝑑𝑡.$

Lecture 2Inner Products, Projections, and Orthonormal Bases

Monday, August 24, 2026

Reading. [Tap16, §1.2, pp. 9–15].

After class. Tapp, Exercises 1.15–1.20.

Goals. Use the Euclidean inner product to measure angles and orthogonality, decompose vectors into parallel and perpendicular parts, compute projections onto subspaces using orthonormal bases, and apply these ideas to vector-valued functions and curves.

2.1 The Euclidean inner product

The norm introduced in Lecture 1 measures the length of a vector. The inner product contains additional information: it measures how much one vector points in the direction of another.

Definition 2.1 (Euclidean inner product). For

x = (x_{1}, \dots, x_{n}), y = (y_{1}, \dots, y_{n}) \in R^{n},

the Euclidean inner product of $x$ and $y$ is

⟨ x, y ⟩ = x_{1} y_{1} + \dots + x_{n} y_{n} .

It is also commonly called the dot product.

Proposition 2.2 (Basic properties). For all $x, y, z \in R^{n}$ and $a, b \in R$ ,

\begin{array}{l} ⟨ 𝑎𝑥 + 𝑏𝑦, z ⟩ & = a ⟨ x, z ⟩ + b ⟨ y, z ⟩, \\ ⟨ x, 𝑎𝑦 + 𝑏𝑧 ⟩ & = a ⟨ x, y ⟩ + b ⟨ x, z ⟩, \\ ⟨ x, y ⟩ & = ⟨ y, x ⟩, \\ ⟨ x, x ⟩ & \geq 0, and ⟨ x, x ⟩ = 0 if and only if x = 0 . \end{array}

Proof. Each identity follows directly by expanding the coordinate formula. For the final assertion,

⟨ x, x ⟩ = x_{1}^{2} + \dots + x_{n}^{2},

which is nonnegative and vanishes exactly when every coordinate of $x$ vanishes. □

Remark 2.3. The first two conditions say that $⟨ -, - ⟩$ is bilinear, the next that it is symmetric, and the third that it is positive definite.

Theorem 2.4 (Cauchy–Schwarz inequality). For all $x, y \in R^{n}$ ,

| ⟨ x, y ⟩ | \leq ‖ x ‖ ‖ y ‖ .

Equality holds precisely when $x$ and $y$ are linearly dependent.

Remark 2.5. The proof of the inequality is Tapp, Exercise 1.15. It should be proved algebraically rather than by appealing to the angle formula below, since that formula is justified using the Schwarz inequality.

Proof. If $x$ and $y$ are linearly dependent, then one can easily show that one has equality above. So assume that $x$ and $y$ are not linearly dependent. Then for $t \in R$ , consider

0 < ⟨ x + 𝑡𝑦, x + 𝑡𝑦 ⟩ = ⟨ x, x ⟩ + 2 ⟨ x, y ⟩ t + ⟨ y, y ⟩ t^{2},

where the strict inequality holds since $x$ and $y$ are not linearly dependent. A real quadratic polynomial $a z^{2} + 𝑏𝑧 + c$ has no zeros if and only if $b^{2} - 4 𝑎𝑐 < 0$ ; so therefore, we have

4 {⟨ x, y ⟩}^{2} - 4 ⟨ y, y ⟩ ⟨ x, x ⟩ < 0 .

The result then follows from some algebra. □

Note that the theorem above implies that

- 1 \leq \frac{| ⟨ x, y ⟩ |}{‖ x ‖ ‖ y ‖} \leq 1

Definition 2.6 (Angle, orthogonality, and parallelism). If $x, y \in R^{n}$ are nonzero, their angle is the unique $𝜃 \in [0, π]$ satisfying

\cos 𝜃 = \frac{⟨ x, y ⟩}{‖ x ‖ ‖ y ‖} .

Equivalently,

⟨ x, y ⟩ = ‖ x ‖ ‖ y ‖ \cos 𝜃.

The vectors $x$ and $y$ are orthogonal if $⟨ x, y ⟩ = 0$ , and they are parallel if one is a scalar multiple of the other.

Thus nonzero vectors are orthogonal exactly when their angle is $π / 2$ . The sign of $⟨ x, y ⟩$ records whether the angle is acute, right, or obtuse.

Example 2.7. Let $x = (1, 2, - 1)$ and $y = (2, 0, 2)$ . Then

⟨ x, y ⟩ = 2 + 0 - 2 = 0,

so $x$ and $y$ are orthogonal.

2.2 Orthogonal projection

Our goal is to understand and prove the following lemma:

Lemma 2.8. Let $V \subseteq R^{n}$ be a subspace. There exists a unique linear map

Π_{V} : R^{n} \to R^{n}

with the following properties:

$Im (Π_{V}) = V$ , i.e., the image of the linear map is $V$ ;
$Π_{V} |_{V} = {Id}_{V}$ , i.e., for all $v \in V$ , we have $Π_{V} (v) = v$ ;
$\ker Π_{V} = V^{⊥}$ , i.e., if $Π_{V} (x) = 0$ , then $x \in V^{⊥}$ .

Proof. Exercise. □

Recall that the kernel of a linear map $L$ (denoted $\ker L$ ) is sometimes also referred to as the null space. We will recall the definition of the orthogonal complement later. For those of you who know what an orthonormal basis is (this definition is recalled below also):

Exercise 2.9. Show that if $(u_{1}, \dots, u_{n})$ is an orthonormal basis for $V$ , then for all $x \in R^{n}$ ,

Π_{V} (x) = ∑_{i = 1}^{n} ⟨ x, u_{1} ⟩ u_{1} .

Exercise 2.10 (For students in MATH 5230). Show that there is a unique isomorphism

L : R^{n} \tilde{\to} V \oplus V^{⊥}

such that $L |_{V} = {Id}_{V}$ and $L |_{V^{⊥}} = {Id}_{V^{⊥}}$ .

In the subsections below, we review the terminology above, as well as some of the notation and terminology in Tapp.

2.2.1 Review: Components and projections (not covered in class)

Fix a nonzero vector $y$ . We want to split an arbitrary vector $x$ into one part that points along $y$ and another part perpendicular to $y$ .

Definition 2.11 (Component and projection in a direction). Let $x, y \in R^{n}$ , with $y \neq 0$ . The component of $x$ in the direction of $y$ is the signed scalar

{comp}_{y} (x) = \frac{⟨ x, y ⟩}{‖ y ‖} .

The projection of $x$ in the direction of $y$ is the vector

{proj}_{y} (x) = \frac{⟨ x, y ⟩}{{‖ y ‖}^{2}} 𝑦.

If $y$ is a unit vector, these formulas simplify to

{comp}_{y} (x) = ⟨ x, y ⟩, {proj}_{y} (x) = ⟨ x, y ⟩ 𝑦.

Proposition 2.12 (Parallel–perpendicular decomposition). Let $x, y \in R^{n}$ , with $y \neq 0$ . There is a unique decomposition

x = x^{∥} + x^{⊥},

where $x^{∥}$ is parallel to $y$ and $x^{⊥}$ is orthogonal to $y$ . It is given by

x^{∥} = {proj}_{y} (x), x^{⊥} = x - {proj}_{y} (x) .

Proof. Since $x^{∥}$ is parallel to $y$ , it must have the form $x^{∥} = 𝑐𝑦$ . The condition that $x^{⊥} = x - 𝑐𝑦$ be orthogonal to $y$ gives

0 = ⟨ x - 𝑐𝑦, y ⟩ = ⟨ x, y ⟩ - c {‖ y ‖}^{2} .

Hence

c = \frac{⟨ x, y ⟩}{{‖ y ‖}^{2}},

which proves both existence and uniqueness. □

Example 2.13. Let $x = (3, 1)$ and $y = (1, 1)$ . Since $⟨ x, y ⟩ = 4$ and ${‖ y ‖}^{2} = 2$ ,

x^{∥} = {proj}_{y} (x) = 2 (1, 1) = (2, 2),

and therefore

x^{⊥} = x - x^{∥} = (1, - 1) .

Indeed, $⟨ (1, - 1), (1, 1) ⟩ = 0$ .

Figure 2.1: Projection splits a vector into a part parallel to

y

and a part perpendicular to

y

2.2.2 Review: Orthonormal sets and bases

Definition 2.14 (Orthonormal set). A finite set ${u_{1}, \dots, u_{k}} \subseteq R^{n}$ is orthonormal if

⟨ u_{i}, u_{j} ⟩ = {\begin{matrix} 1, & i = j, \\ 0, & i \neq 𝑗. \end{matrix}

Thus each $u_{i}$ is a unit vector, and distinct members are mutually orthogonal. An orthonormal set that is a basis of a subspace $V$ is an orthonormal basis of $V$ .

Every orthonormal set is linearly independent; this is Tapp, Exercise 1.19. The standard orthonormal basis of $R^{n}$ is

e_{1} = (1, 0, \dots, 0), e_{2} = (0, 1, 0, \dots, 0), \dots, e_{n} = (0, \dots, 0, 1) .

If $x = (x_{1}, \dots, x_{n})$ , then

x = ⟨ x, e_{1} ⟩ e_{1} + \dots + ⟨ x, e_{n} ⟩ e_{n} .

In other words, the usual coordinates of $x$ are precisely its components in the directions of the standard basis vectors.

For a subspace $V \subseteq R^{n}$ , its orthogonal complement is

V^{⊥} = {w \in R^{n} : ⟨ w, v ⟩ = 0 for every v \in V} .

Proposition 2.15 (Projection onto a subspace). Let $V \subseteq R^{n}$ be a subspace with orthonormal basis ${u_{1}, \dots, u_{k}}$ . Every $x \in R^{n}$ has a unique decomposition

x = x_{V} + x_{V^{⊥}},

where $x_{V} \in V$ and $x_{V^{⊥}}$ is orthogonal to every vector in $V$ . The projection of $x$ onto $V$ is

x_{V} = {proj}_{V} (x) = ∑_{i = 1}^{k} ⟨ x, u_{i} ⟩ u_{i},

and

x_{V^{⊥}} = x - {proj}_{V} (x) .

Proof. Write a prospective vector in $V$ as

x_{V} = a_{1} u_{1} + \dots + a_{k} u_{k} .

The vector $x - x_{V}$ is orthogonal to all of $V$ exactly when it is orthogonal to each basis vector $u_{j}$ . Orthonormality gives

0 = ⟨ x - x_{V}, u_{j} ⟩ = ⟨ x, u_{j} ⟩ - a_{j} .

Thus the only possible coefficients are $a_{j} = ⟨ x, u_{j} ⟩$ , and these coefficients do give the desired decomposition. □

Example 2.16 (Projection onto a plane). In $R^{3}$ , let

u_{1} = \frac{1}{\sqrt{2}} (1, 1, 0), u_{2} = (0, 0, 1),

and let $V = span {u_{1}, u_{2}}$ . The set ${u_{1}, u_{2}}$ is an orthonormal basis of $V$ . For $x = (2, 0, 3)$ ,

\begin{array}{l} {proj}_{V} (x) & = ⟨ x, u_{1} ⟩ u_{1} + ⟨ x, u_{2} ⟩ u_{2} \\ = \sqrt{2} u_{1} + 3 u_{2} \\ = (1, 1, 3) . \end{array}

Hence the perpendicular part is

x - {proj}_{V} (x) = (1, - 1, 0) .

Remark 2.17 (Gram–Schmidt). Every finite-dimensional subspace of $R^{n}$ has an orthonormal basis. Starting from a basis $v_{1}, \dots, v_{k}$ , normalize $v_{1}$ , then subtract from each later vector its projections onto the orthonormal vectors already constructed, and normalize the result. For the first two steps,

u_{1} = \frac{v_{1}}{‖ v_{1} ‖}, w_{2} = v_{2} - ⟨ v_{2}, u_{1} ⟩ u_{1}, u_{2} = \frac{w_{2}}{‖ w_{2} ‖} .

Continuing this process gives the Gram–Schmidt process; its proof is Tapp, Exercise 1.20.

2.3 Differentiating inner products

The inner product also interacts well with differentiation.

Lemma 2.18 (Product rules). Let $x, y : I \to R^{n}$ and $c : I \to R$ be smooth maps. Then

\begin{array}{l} \frac{d}{d t} ⟨ x (t), y (t) ⟩ & = ⟨ x^{'} (t), y (t) ⟩ + ⟨ x (t), y^{'} (t) ⟩, \\ \frac{d}{d t} (c (t) x (t)) & = c^{'} (t) x (t) + c (t) x^{'} (t) . \end{array}

Proof. Differentiate each coordinate and use the ordinary product rule for real-valued functions. □

Proposition 2.19 (Derivatives of orthogonality relations). Let $x, y : I \to R^{n}$ be smooth.

If $‖ x (t) ‖$ is constant, then $⟨ x (t), x^{'} (t) ⟩ = 0$ for every $t \in I$ .
If $⟨ x (t), y (t) ⟩ = 0$ for every $t \in I$ , then
$⟨ x^{'} (t), y (t) ⟩ = - ⟨ x (t), y^{'} (t) ⟩ .$

Proof. For the first statement, differentiate the constant function

⟨ x (t), x (t) ⟩ = {‖ x (t) ‖}^{2} .

The product rule gives

0 = 2 ⟨ x (t), x^{'} (t) ⟩ .

For the second, differentiate the identity $⟨ x (t), y (t) ⟩ = 0$ . □

Example 2.20 (Motion on a sphere). If $x : I \to R^{3}$ satisfies $‖ x (t) ‖ = r$ , then its image lies on the sphere of radius $r$ centered at the origin. The proposition says that $x^{'} (t)$ is orthogonal to the radius vector $x (t)$ . Thus the velocity is tangent to the sphere.

Figure 2.2: If

∥ x (t) ∥ = r

is constant, then the velocity is tangent to the sphere and orthogonal to the radius vector.

Corollary 2.21 (Constant-speed curves). If a curve $γ : I \to R^{n}$ has constant speed, then

⟨ γ^{'} (t), γ^{″} (t) ⟩ = 0

for every $t \in I$ . Thus its velocity and acceleration are orthogonal.

Proof. Apply the first part of the proposition to the vector-valued function $x = γ^{'}$ . □

2.4 A geometric application: straight lines are shortest (not covered in class)

Theorem 2.22 (Length dominates endpoint distance). Let $γ : [a, b] \to R^{n}$ be a smooth curve, with $p = γ (a)$ and $q = γ (b)$ . Then

L (γ) \geq ‖ q - p ‖ .

Thus no curve joining $p$ to $q$ is shorter than the line segment between them.

Proof. The result is immediate if $p = q$ , so assume $p \neq q$ and set

u = \frac{q - p}{‖ q - p ‖} .

This is the unit vector pointing from $p$ toward $q$ . By Cauchy–Schwarz,

⟨ γ^{'} (t), u ⟩ \leq ‖ γ^{'} (t) ‖ ‖ u ‖ = ‖ γ^{'} (t) ‖ .

Therefore

\begin{array}{l} L (γ) & = \int_{a}^{b} ‖ γ^{'} (t) ‖ d t \\ \geq \int_{a}^{b} ⟨ γ^{'} (t), u ⟩ d t \\ = ⟨ γ (b) - γ (a), u ⟩ \\ = ⟨ q - p, \frac{q - p}{‖ q - p ‖} ⟩ \\ = ‖ q - p ‖ . \end{array}

□

Remark 2.23. The proof illustrates a recurring strategy: choose a unit vector or an orthonormal basis adapted to the geometry of the problem, and then use inner products to extract the relevant components.

Figure 2.3: Among smooth curves from

p

q

, the line segment gives the shortest possible length.

2.5 Summary

The inner product encodes length, angle, and orthogonality.
Projection separates a vector into parallel and perpendicular parts.
With an orthonormal basis, coordinates and projections are computed directly by inner products.
Differentiating an identity such as constant norm or orthogonality produces geometric information about derivatives.
These tools will be used in Lecture 3 to separate acceleration into components parallel and perpendicular to velocity.

Lecture 3Acceleration and Reparametrization

Wednesday, August 26, 2026

Reading. [Tap16, §§ 1.3–1.4, pp. 16–23].

After class. Tapp, Exercises 1.26–1.29, 1.31, and 1.33.

Goals. Interpret acceleration geometrically, split it into components parallel and perpendicular to velocity, relate tangential acceleration to the rate of change of speed, define reparametrization and orientation, prove invariance of arc length, construct unit-speed parametrizations, and discuss closed curves.

3.1 Acceleration

Let $γ : I \to R^{n}$ be a regular curve. Following the notation used in physics, we write

v (t) = γ^{'} (t) and a (t) = γ^{″} (t) .

The vector $v (t)$ is the velocity, while $a (t)$ measures the instantaneous change in velocity. Thus acceleration can result from a change in speed, a change in direction, or both.

Definition 3.1 (Acceleration). The acceleration of a smooth parametrized curve $γ$ at time $t$ is

a (t) = γ^{″} (t) .

The physical interpretation is supplied by Newton’s equation

F (t) = 𝑚𝑎 (t) .

For a particle of mass $m$ , the acceleration points in the direction of the net force. A force may pull partly along the direction of motion, thereby changing speed, and partly across the direction of motion, thereby bending the path.

Example 3.2 (Circular motion). For

γ (t) = (r \cos t, r \sin t),

we have

v (t) = (- r \sin t, r \cos t), a (t) = (- r \cos t, - r \sin t) = - γ (t) .

Thus the velocity is tangent to the circle, while the acceleration points toward its center. Moreover,

⟨ a (t), v (t) ⟩ = 0,

so the acceleration changes the direction of motion but not the speed.

Figure 3.1: For circular motion, velocity is tangent to the path and acceleration is center-pointing.

3.1.1 Tangential and perpendicular acceleration

At each time, decompose the acceleration into the part parallel to velocity and the part perpendicular to velocity.

Definition 3.3 (Tangential and perpendicular components). For a regular curve, define

\begin{array}{l} a^{∥} (t) & = {proj}_{v (t)} (a (t)) = \frac{⟨ a (t), v (t) ⟩}{{‖ v (t) ‖}^{2}} v (t), \\ a^{⊥} (t) & = a (t) - a^{∥} (t) . \end{array}

Then

a (t) = a^{∥} (t) + a^{⊥} (t),

where $a^{∥} (t)$ is parallel to $v (t)$ and $a^{⊥} (t)$ is orthogonal to $v (t)$ .

Figure 3.2: Acceleration decomposes into a component parallel to velocity and a component perpendicular to velocity.

The parallel component controls the speed.

Proposition 3.4 (Derivative of speed). Let $γ : I \to R^{n}$ be regular. Then

\frac{d}{d t} ‖ v (t) ‖ = \frac{⟨ a (t), v (t) ⟩}{‖ v (t) ‖} = {comp}_{v (t)} (a (t)) .

Thus the signed scalar component of acceleration in the direction of velocity is exactly the rate of change of speed.

Proof. We start with ${‖ v ‖}^{2} = ⟨ v, v ⟩$ . We then differentiate both sides. On the left we have

\frac{d}{d t} {‖ v ‖}^{2} = 2 ‖ v ‖ \frac{d}{d t} ‖ v ‖ .

On the right we have

\frac{d}{d t} ⟨ v, v ⟩ = ⟨ v^{'}, v ⟩ + ⟨ v, v^{'} ⟩ = 2 ⟨ a, v ⟩ .

The result follows from some algebra. □

Corollary 3.5. For a regular curve:

the speed is increasing when $⟨ a (t), v (t) ⟩ > 0$ and decreasing when $⟨ a (t), v (t) ⟩ < 0$ ;
the speed is constant if and only if $a^{∥} (t) = 0$ for every $t$ , equivalently $a (t) ⊥ v (t)$ for every $t$ .

Example 3.6 (Acceleration along a parabola). Let

γ (t) = (t, t^{2}) .

Then

v (t) = (1, 2 t), a (t) = (0, 2) .

At $t = 1$ ,

⟨ a (1), v (1) ⟩ = 4, {‖ v (1) ‖}^{2} = 5,

\begin{array}{l} a^{∥} (1) & = \frac{4}{5} (1, 2) = (\frac{4}{5}, \frac{8}{5}), \\ a^{⊥} (1) & = (0, 2) - a^{∥} (1) = (- \frac{4}{5}, \frac{2}{5}) . \end{array}

The positive value

\frac{d}{d t} ‖ v (t) ‖ |_{t = 1} = \frac{4}{\sqrt{5}}

confirms that the particle is speeding up at this time.

The perpendicular component changes the direction of motion. Its magnitude is influenced both by how sharply the path bends and by how quickly the path is traversed. Reparametrization will allow us to separate these two effects.

3.2 Trace and reparametrization

Definition 3.7 (Trace). The trace of a parametrized curve $γ : I \to R^{n}$ is the subset

γ (I) = {γ (t) : t \in I} \subseteq R^{n} .

The trace remembers the path but forgets the clock: it does not record where the particle is at a particular time, how fast it moves, or which direction it travels.

Example 3.8 (Same trace, different clocks). Let

γ (s) = (s, s^{2}), - 2 \leq s \leq 2,

and let

ϕ (t) = t + t^{3}, - 1 \leq t \leq 1 .

Since $ϕ^{'} (t) = 1 + 3 t^{2} > 0$ , the map $ϕ : [- 1, 1] \to [- 2, 2]$ is a smooth bijection. The curve

\tilde{γ} (t) = γ (ϕ (t)) = (t + t^{3}, {(t + t^{3})}^{2})

has the same trace as $γ$ , but its speed varies according to the new clock $t$ .

Figure 3.3: A reparametrization keeps the trace but changes the association between parameter values and points on the trace.

Definition 3.9 (Reparametrization). Let $γ : I \to R^{n}$ be a regular curve. A reparametrization of $γ$ is a curve of the form

\tilde{γ} = γ \circ ϕ : \tilde{I} \to R^{n},

where $ϕ : \tilde{I} \to I$ is a smooth bijection with $ϕ^{'} (t) \neq 0$ for every $t \in \tilde{I}$ .

In diagram form we say that the following diagram commutes:

3.3 More on reparameterization (not covered in class)

The derivative condition ensures that the new curve is still regular.

Proposition 3.10 (Velocity and acceleration under reparametrization). If $\tilde{γ} = γ \circ ϕ$ , then

\begin{array}{l} {\tilde{γ}}^{'} (t) & = ϕ^{'} (t) γ^{'} (ϕ (t)), \\ {\tilde{γ}}^{″} (t) & = {(ϕ^{'} (t))}^{2} γ^{″} (ϕ (t)) + ϕ^{″} (t) γ^{'} (ϕ (t)) . \end{array}

Consequently,

‖ {\tilde{γ}}^{'} (t) ‖ = | ϕ^{'} (t) | ‖ γ^{'} (ϕ (t)) ‖ .

Proof. Apply the ordinary chain rule componentwise to obtain the first formula, then differentiate it using the product rule to obtain the second. Taking norms in the first formula gives the speed transformation law. □

Remark 3.11 (What changes and what does not). The term $ϕ^{″} (t) γ^{'} (ϕ (t))$ in the acceleration formula is parallel to velocity. Therefore the perpendicular component satisfies

{\tilde{a}}^{⊥} (t) = {(ϕ^{'} (t))}^{2} a^{⊥} (ϕ (t)) .

Since speed squared transforms by the same factor,

\frac{‖ {\tilde{a}}^{⊥} (t) ‖}{{‖ {\tilde{γ}}^{'} (t) ‖}^{2}} = \frac{‖ a^{⊥} (ϕ (t)) ‖}{{‖ γ^{'} (ϕ (t)) ‖}^{2}} .

This reparametrization-invariant ratio will become the curvature in the next lecture.

Because $\tilde{I}$ is an interval and $ϕ^{'}$ never vanishes, the intermediate value theorem implies that $ϕ^{'}$ has a constant sign.

Definition 3.12 (Orientation). A reparametrization $\tilde{γ} = γ \circ ϕ$ is orientation-preserving if $ϕ^{'} (t) > 0$ everywhere and orientation-reversing if $ϕ^{'} (t) < 0$ everywhere.

The reparametrization in the previous example preserves orientation. By contrast,

\hat{γ} (t) = γ (- t)

reverses orientation.

Proposition 3.13 (Arc length is independent of parametrization). Let

\tilde{γ} = γ \circ ϕ : [c, d] \to R^{n}

be a reparametrization. Set

u_{0} = \min {ϕ (c), ϕ (d)}, u_{1} = \max {ϕ (c), ϕ (d)} .

Then

\int_{c}^{d} ‖ {\tilde{γ}}^{'} (t) ‖ d t = \int_{u_{0}}^{u_{1}} ‖ γ^{'} (u) ‖ d 𝑢.

Thus corresponding portions of the two parametrizations have the same arc length.

Proof. If $ϕ^{'} > 0$ , then the speed transformation formula and the substitution $u = ϕ (t)$ give

\int_{c}^{d} ‖ {\tilde{γ}}^{'} (t) ‖ d t = \int_{c}^{d} ϕ^{'} (t) ‖ γ^{'} (ϕ (t)) ‖ d t = \int_{ϕ (c)}^{ϕ (d)} ‖ γ^{'} (u) ‖ d 𝑢.

If $ϕ^{'} < 0$ , then $| ϕ^{'} | = - ϕ^{'}$ , and the same substitution gives

\int_{c}^{d} ‖ {\tilde{γ}}^{'} (t) ‖ d t = \int_{ϕ (d)}^{ϕ (c)} ‖ γ^{'} (u) ‖ d 𝑢.

These are precisely the asserted formulas in the two cases. □

3.4 Reparametrization by arc length (Covered on Friday August 28)

Unit-speed curves are especially convenient: the parameter difference equals the distance traveled along the curve. Every regular curve admits such a parametrization.

Theorem 3.14 (Arc-length parametrization). Every regular curve can be reparametrized by arc length. Moreover, for such a parameterization, the speed is constant and equal to $1$ .

Proof. For $I = (a, b)$ (for $[a, b)$ , etc., a similar argument works) define the signed arc-length function

s (t) = \int_{a}^{t} ‖ γ^{'} (u) ‖ d 𝑢.

By the fundamental theorem of calculus,

s^{'} (t) = ‖ γ^{'} (t) ‖ > 0 .

(3.1)

Thus $s : I \to J = s (I) = (0, s (b))$ is a smooth increasing bijection with a smooth inverse $ϕ : J \to I$ . Define

\bar{γ} = γ \circ ϕ : J \to R^{n} .

Next, since for all $v \in J$ we have $s (ϕ (v)) = v$ , differentiating with respect to $v$ gives

s^{'} (ϕ (v)) ϕ^{'} (v) = 1 .

Therefore

\begin{array}{l} ‖ {\bar{γ}}^{'} (v) ‖ & = ϕ^{'} (v) ‖ γ^{'} (ϕ (v)) ‖ & 3.10 \\ = ϕ^{'} (v) s^{'} (ϕ (v)) & (3.1) \\ = 1 . \end{array}

From this we can check that $\bar{γ}$ gives a paramerization by arclength. In other words, for all $v \in J$ , we have $\int_{0}^{v} ‖ {\bar{γ}}^{'} (u) ‖ d u = \int_{0}^{v} 1 d u = v$ . □

Remark 3.15. There are some details for you to check in the argument above. First, to prove that $s$ is smooth, you must check that if $γ$ is regular, then $‖ γ^{'} ‖$ is smooth. Then you should review the proof from your calculus class that $s^{'} > 0$ implies that $s$ is strictly increasing (see for instance [Bro96, Cor. 4.23]). Then you should check that $s$ is invertible, and the inverse is differentiable. Then you should check that the inverse of $s$ is smooth.

Example 3.16 (A circle parametrized by arc length). For

γ (t) = (r \cos t, r \sin t), 0 \leq t \leq 2 π,

the speed is $r$ . Taking $t_{0} = 0$ ,

s (t) = 𝑟𝑡, ϕ (s) = \frac{s}{r} .

Hence

\bar{γ} (s) = (r \cos \frac{s}{r}, r \sin \frac{s}{r}), 0 \leq s \leq 2 𝜋𝑟,

and $‖ {\bar{γ}}^{'} (s) ‖ = 1$ .

Remark 3.17. The proof gives an explicit procedure, but the inverse of the arc-length function often cannot be written in elementary closed form. The theorem is consequently more useful in theoretical arguments than as a computational recipe.

3.5 Closed curves (not covered in class)

A loop should meet itself smoothly at its beginning and ending point, not just have the same endpoint twice.

Definition 3.18 (Closed and simple closed curves). A closed curve is a regular curve $γ : [a, b] \to R^{n}$ such that

γ (a) = γ (b)

and every derivative agrees at the endpoints:

γ^{(k)} (a) = γ^{(k)} (b) for every k \geq 1 .

It is simple closed if it is one-to-one on $[a, b)$ .

The standard circle is simple closed. The figure-eight curve

η (t) = (\sin t, \sin 2 t), 0 \leq t \leq 2 π,

closes smoothly but is not simple because it passes through the origin more than once.

Figure 3.4: A simple closed curve and a closed curve with a self-intersection.

Proposition 3.19 (Periodic extension). A regular curve $γ : [a, b] \to R^{n}$ is closed if and only if there is a periodic regular curve $\hat{γ} : R \to R^{n}$ of period $b - a$ whose restriction to $[a, b]$ is $γ$ .

Proof. This is Tapp, Exercise 1.33. The matching of every derivative at the two endpoints is precisely what permits the translated copies of $γ$ to glue together smoothly. □

Remark 3.20 (Reparametrizing a closed curve). For closed curves, changing the clock should also allow a cyclic shift of the starting point. If $\hat{γ}$ is the periodic extension above and $γ_{λ}$ is its restriction to $[a + λ, b + λ]$ , then a closed reparametrization has the form

\tilde{γ} = γ_{λ} \circ 𝜙.

In addition to being a smooth bijection with nonzero derivative, the derivatives of $ϕ$ must match at its two endpoints so that the composite also closes smoothly.

Proposition 3.21. Two simple closed curves have the same trace if and only if each is a reparametrization of the other.

For nonsimple closed curves, the trace does not contain enough information: one trip around a circle and two trips around the same circle have identical traces, but they are not reparametrizations of one another.

3.6 Summary

Acceleration is $a (t) = γ^{″} (t)$ and decomposes as $a = a^{∥} + a^{⊥}$ .
The tangential component controls speed:
$\frac{d}{d t} ‖ v (t) ‖ = \frac{⟨ a (t), v (t) ⟩}{‖ v (t) ‖} .$
A reparametrization changes the clock but preserves the trace.
Under $\tilde{γ} = γ \circ ϕ$ , velocity and acceleration transform by the chain rule, and arc length is unchanged.
Every regular curve admits a unit-speed reparametrization using its arc-length function.
A closed curve has matching position and derivatives at its endpoints; a simple closed curve has no self-intersections apart from the closing point.
The ratio $‖ a^{⊥} ‖ / {‖ v ‖}^{2}$ is unchanged by reparametrization and will lead to curvature in Lecture 4.

Lecture 4Curvature

Friday, August 28, 2026

Reading. [Tap16, § 1.5, pp. 24–31].

After class. HW 1 is due. Begin Tapp, Exercises 1.36, 1.37, 1.38, and 1.43.

Goals. Define curvature in a parametrization-independent way, define the unit tangent and unit normal vectors, and relate curvature to the change in the direction of motion.

Remark 4.1 (Scope). We omit the discussion of osculating planes and osculating circles. The notation, order of presentation, and principal definitions below follow Tapp’s treatment of curvature.

4.1 The curvature function

Let

𝜸 : I \to R^{n}

be a regular curve. Following Tapp, write the velocity and acceleration as

𝐯 (t) = 𝜸^{'} (t), 𝐚 (t) = 𝜸^{″} (t) .

Decompose acceleration into parts parallel and perpendicular to velocity:

𝐚 (t) = 𝐚^{∥} (t) + 𝐚^{⊥} (t) .

The perpendicular part detects turning, but its length also depends on how quickly the curve is traversed. If $\tilde{𝜸} = 𝜸 \circ ϕ$ is a regular reparametrization, then

\tilde{𝐯} (t) = ϕ^{'} (t) 𝐯 (ϕ (t)), {\tilde{𝐚}}^{⊥} (t) = ϕ^{'} {(t)}^{2} 𝐚^{⊥} (ϕ (t)) .

Thus the quotient below is unchanged by reparametrization.

Definition 4.2: Curvature function

Let $𝜸 : I \to R^{n}$ be a regular curve. Its curvature vector $κ : I \to R^{n}$ is

\vec{κ} (t) = \frac{𝐚^{⊥} (t)}{{| 𝐯 (t) |}^{2}} .

Its curvature function $κ : I \to [0, \infty)$ is

κ (t) = \frac{| 𝐚^{⊥} (t) |}{{| 𝐯 (t) |}^{2}} .

The value $κ (t)$ measures how sharply the trace bends near $𝜸 (t)$ . Solving the definition for the perpendicular acceleration gives

| 𝐚^{⊥} (t) | = κ (t) {| 𝐯 (t) |}^{2} .

Thus the perpendicular acceleration increases both with curvature and with the square of the speed.

Proposition 4.3: Unit-speed curves

If $𝜸$ is parametrized by arc length, then

κ (t) = | 𝐚 (t) | .

Proof. For a unit-speed curve, $| 𝐯 | = 1$ and $𝐚 ⊥ 𝐯$ . Hence $𝐚^{⊥} = 𝐚$ , and the result follows directly from the definition. □

Example 4.4 (A circle). For $r > 0$ , consider

𝜸 (t) = (r \cos t, r \sin t) .

Then

𝐯 (t) = (- r \sin t, r \cos t), 𝐚 (t) = (- r \cos t, - r \sin t) .

Since $𝐚 (t) ⊥ 𝐯 (t)$ , one has $𝐚^{⊥} (t) = 𝐚 (t)$ . Therefore

κ (t) = \frac{r}{r^{2}} = \frac{1}{r} .

A circle of radius $r$ has constant curvature $1 / r$ .

4.2 Unit tangent and unit normal vectors

Definition 4.5: Unit tangent and unit normal vectors

Let $𝜸 : I \to R^{n}$ be regular. Define

𝐭 (t) = \frac{𝐯 (t)}{| 𝐯 (t) |}, 𝐧 (t) = \frac{𝐚^{⊥} (t)}{| 𝐚^{⊥} (t) |} .

The vector $𝐭 (t)$ is the unit tangent vector. The vector $𝐧 (t)$ is the unit normal vector, and it is defined only at times for which $κ (t) \neq 0$ .

Whenever the input is clear, we suppress it and write simply $𝐭$ and $𝐧$ . Where $𝐧$ is defined, the set ${𝐭, 𝐧}$ is orthonormal. The path $t \mapsto 𝐭 (t)$ records the direction in which the curve is moving. Its derivative measures how quickly that direction changes.

Proposition 4.6: Curvature and change of direction

For every regular curve,

𝐚 = \underset{𝐚^{∥}}{\underset{⏟}{{| 𝐯 |}^{'} 𝐭}} + \underset{𝐚^{⊥}}{\underset{⏟}{| 𝐯 | 𝐭^{'}}} = | 𝐯 |^{'} 𝐭 + κ | 𝐯 |^{2} 𝐧 .

and

κ (t) = \frac{| 𝐭^{'} (t) |}{| 𝐯 (t) |} .

Proof. Since $𝐯 = | 𝐯 | 𝐭$ , differentiation gives

𝐚 = \underset{𝐚^{∥}}{\underset{⏟}{{| 𝐯 |}^{'} 𝐭}} + \underset{𝐚^{⊥}}{\underset{⏟}{| 𝐯 | 𝐭^{'}}}

The identity $| 𝐭 | = 1$ implies $𝐭^{'} ⊥ 𝐭$ . The first term above is therefore parallel to $𝐯$ , while the second is perpendicular to $𝐯$ . Hence

𝐚^{⊥} = | 𝐯 | 𝐭^{'} .

Substitution into the definition of curvature proves the formula $κ (t) = \frac{| 𝐭^{'} (t) |}{| 𝐯 (t) |}$ . Then, putting this formula for curvature into the right hand side of $𝐚 = \underset{𝐚^{∥}}{\underset{⏟}{{| 𝐯 |}^{'} 𝐭}} + \underset{𝐚^{⊥}}{\underset{⏟}{| 𝐯 | 𝐭^{'}}} = | 𝐯 |^{'} 𝐭 + κ | 𝐯 |^{2} 𝐧$ shows that the right and middle terms are equal. □

At every time for which $κ (t) \neq 0$ , the same calculation gives

𝐧 (t) : = \frac{𝐚^{⊥} (t)}{| 𝐚^{⊥} (t) |} = \frac{𝐭^{'} (t)}{| 𝐭^{'} (t) |}

and then, clearing denominators and using the formula for curvature in the proposition gives,

𝐭^{'} (t) = κ (t) | 𝐯 (t) | 𝐧 (t) .

4.3 Curvature of a graph at a critical point (not covered in class)

Example 4.7. Let $f : I \to R$ be smooth, let $t_{0} \in I$ satisfy $f^{'} (t_{0}) = 0$ , and consider the natural parametrization of its graph,

𝜸 (t) = (t, f (t)) .

At $t_{0}$ ,

𝐯 (t_{0}) = (1, 0), 𝐚 (t_{0}) = (0, f^{″} (t_{0})) .

These vectors are orthogonal, so

κ (t_{0}) = | f^{″} (t_{0}) | .

Thus, at a critical point of a graph, curvature is the absolute value of its concavity.

For the parabola $f (t) = t^{2}$ at $t_{0} = 0$ ,

𝐭 (0) = (1, 0), 𝐧 (0) = (0, 1), κ (0) = 2 .

Figure 4.1: For the parabola at its vertex, the unit tangent is horizontal and the unit normal points toward the side on which the curve bends.

4.3.1 Computational formula

The perpendicular component can be computed directly from velocity and acceleration:

Proposition 4.8 (Computational formula). For a regular curve,

κ (t) = \frac{\sqrt{{‖ γ^{'} (t) ‖}^{2} {‖ γ^{″} (t) ‖}^{2} - {⟨ γ^{'} (t), γ^{″} (t) ⟩}^{2}}}{{‖ γ^{'} (t) ‖}^{3}} .

Proof. Since $a^{∥} = {proj}_{v} (a)$ , orthogonality gives

\begin{array}{l} {‖ a^{⊥} ‖}^{2} & = {‖ a ‖}^{2} - {‖ a^{∥} ‖}^{2} \\ = {‖ a ‖}^{2} - \frac{{⟨ a, v ⟩}^{2}}{{‖ v ‖}^{2}} . \end{array}

Substitute this into $κ = ‖ a^{⊥} ‖ / {‖ v ‖}^{2}$ and simplify. □

4.4 A worked example (not covered in class)

Example 4.9 (A parabola). Consider

γ (t) = (t, t^{2}) .

Then

γ^{'} (t) = (1, 2 t), γ^{″} (t) = (0, 2) .

The computational formula gives

\begin{array}{l} κ (t) & = \frac{\sqrt{(1 + 4 t^{2}) (4) - {(4 t)}^{2}}}{{(1 + 4 t^{2})}^{3 / 2}} \\ = \frac{2}{{(1 + 4 t^{2})}^{3 / 2}} . \end{array}

The curvature is largest at the vertex $t = 0$ , where

𝔱 (0) = (1, 0), N (0) = (0, 1), κ (0) = 2 .

Figure 4.2: At the vertex of the parabola, the unit tangent is horizontal and the principal normal points toward the side on which the curve bends.

4.5 Further topics

Here are a few things to think about going forward. We will return to this later.

Recall that we have

𝐭^{'} = κ | 𝐯 | 𝐧 .

What about $𝐧^{'}$ ? We have

𝐧^{'} = - κ | 𝐯 | 𝐭 + 𝐱

for some $𝐱 \in {(Span 𝐭)}^{⊥}$ . In fact, we know that $⟨ 𝐧^{'}, 𝐧 ⟩ = 0$ , so that $𝐱 \in {(Span (𝐭, 𝐧))}^{⊥}$ .

Write $𝐛 = 𝐱 / ‖ 𝐱 ‖$ , assuming $𝐱$ is non-zero. Write

τ = | 𝐱 | / | 𝐯 | .

Then we have by definition

𝐧^{'} = - κ | 𝐯 | 𝐭 + τ | 𝐯 | 𝐛

for some scalar function $τ (t)$ .

Exercise 4.10. What happens with the above under orientation preserving reparameterization? In other words, if we take $\tilde{γ} = γ \circ ϕ$ , then how are $\tilde{𝐱}$ , $\tilde{𝐛}$ , and $\tilde{τ}$ related to $𝐱$ , $𝐛$ , and $τ$ change?

Solution. If we write $t = ϕ (u)$ , so that $\tilde{γ} (u) = γ (t)$ , then we know that $\tilde{𝐭} (u) = 𝐭 (t)$ , since this is the unit tangent direction at $\tilde{γ} (u) = γ (t)$ . Similarly $\tilde{𝐧} (u) = 𝐧 (t)$ . So on the one hand we have

\frac{d}{d u} (\tilde{𝐧} (u)) = (𝐧 {(ϕ (u))}^{'} = 𝐧^{'} (ϕ (u)) ϕ^{'} (u)

Recall that we have

\tilde{𝐯} (u) = \frac{d}{d u} (\tilde{γ} (u)) = \frac{d}{d u} (γ (ϕ (u))) = (γ^{'} (ϕ (u)) ϕ^{'} (u) = 𝐯 (ϕ (u)) ϕ^{'} (u),

so that

| \tilde{𝐯} (u) | = | 𝐯 (ϕ (u)) | ϕ^{'} (u) .

Since $\tilde{κ} (u) = κ (ϕ (u))$ , we have

\begin{array}{l} \tilde{𝐱} (u) : = {\tilde{𝐧}}^{'} (u) + \tilde{κ} (u) | \tilde{𝐯} (u) | \tilde{𝐭} (u) & = 𝐧^{'} (ϕ (u)) ϕ^{'} (u) + κ (ϕ (u)) | 𝐯 (ϕ (u)) | ϕ^{'} (u) 𝐭 (ϕ (u)) \\ = (𝐧^{'} + κ | 𝐯 | 𝐭) ϕ^{'} (u) = 𝐱 (ϕ (u)) ϕ^{'} (u) . \end{array}

Thus $\tilde{𝐱} (u)$ and $𝐱 (ϕ (u))$ point in the same direction (with the exact scalar above), and so $\tilde{𝐛} (u) = 𝐛 (ϕ (u))$ . With this and the formula $𝐧^{'} = - κ | 𝐯 | 𝐭 + τ | 𝐯 | 𝐛$ defining $τ$ , we can conclude that $\tilde{τ} (u) = τ (ϕ (u))$ . □

Then what about $𝐛^{'}$ ? We have $⟨ 𝐛^{'}, 𝐛 ⟩ = 0$ . We have $⟨ 𝐛, 𝐭 ⟩ = 0$ , so that $⟨ 𝐛^{'}, 𝐭 ⟩ = - ⟨ 𝐛, 𝐭^{'} ⟩ = 0$ (from formula above). We have $⟨ 𝐛^{'}, 𝐧 ⟩ = - ⟨ 𝐛, 𝐧^{'} ⟩ = - τ | 𝐯 |$ . So we have

𝐛^{'} = - τ | 𝐯 | 𝐧 + 𝐲

where $𝐲 \in {(Span (𝐭, 𝐧, 𝐛))}^{⊥}$

4.6 Summary

Curvature is defined by
$κ = \frac{| 𝐚^{⊥} |}{{| 𝐯 |}^{2}} .$
The unit tangent and unit normal are
$𝐭 = \frac{𝐯}{| 𝐯 |}, 𝐧 = \frac{𝐚^{⊥}}{| 𝐚^{⊥} |},$

with $𝐧$ defined only where $κ \neq 0$ .
Curvature is the rate of change of direction divided by speed:
$κ = \frac{| 𝐭^{'} |}{| 𝐯 |} .$
For unit-speed curves, $κ = | 𝐚 |$ .

Lecture 5Plane Curves

Monday, August 31, 2026

Reading. [Tap16, § 1.6, pp. 32–40].

After class. Begin HW 2(b): Tapp, Exercises 1.44–1.49.

Goals. Use counterclockwise rotations in the plane, define signed curvature, construct an angle function for the velocity of a unit-speed plane curve, and define the rotation index of a closed plane curve.

The plane has an extra feature that is not available in higher-dimensional Euclidean space: once an orientation is chosen, it makes sense to distinguish counterclockwise turning from clockwise turning. This gives curvature a sign.

5.1 Rotation matrices

For $𝜃 \in R$ , define the counterclockwise rotation through angle $𝜃$ by

R_{𝜃} = (\begin{matrix} \cos 𝜃 & - \sin 𝜃 \\ \sin 𝜃 & \cos 𝜃 \end{matrix}) .

Thus $R_{𝜃} : R^{2} \to R^{2}$ is a linear map. The rotation through $9 0^{\circ}$ is

R_{90} = R_{π / 2} = (\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}), R_{90} (x, y) = (- y, x) .

For every $𝐰 \in R^{2}$ ,

⟨ 𝐰, R_{90} (𝐰) ⟩ = 0, | R_{90} (𝐰) | = | 𝐰 |, R_{90}^{2} (𝐰) = - 𝐰 .

Consequently, if $𝐰 \neq 0$ , then $R_{90} (𝐰)$ spans the line perpendicular to $𝐰$ .

Figure 5.1: The operator

R_{90}

rotates vectors counterclockwise by

9 0^{\circ}

5.2 Signed curvature

Let

𝜸 : I \to R^{2}

be a regular plane curve, and write

𝐯 = 𝜸^{'}, 𝐚 = 𝜸^{″} .

The vector

R_{90} (\frac{𝐯}{| 𝐯 |})

is a unit vector spanning ${(span {𝐯})}^{⊥}$ . Orthogonal projection therefore gives

𝐚^{⊥} = ⟨ 𝐚, R_{90} (\frac{𝐯}{| 𝐯 |}) ⟩ R_{90} (\frac{𝐯}{| 𝐯 |}) .

Dividing by ${| 𝐯 |}^{2}$ , the curvature vector from Lecture 4 becomes

\frac{𝐚^{⊥}}{{| 𝐯 |}^{2}} = \underset{κ_{s}}{\underset{⏟}{⟨ \frac{𝐚}{{| 𝐯 |}^{2}}, R_{90} (\frac{𝐯}{| 𝐯 |}) ⟩}} R_{90} (\frac{𝐯}{| 𝐯 |}) .

The scalar coefficient records not only the amount of turning, but also its direction.

Definition 5.1: Signed curvature

Let $𝜸 : I \to R^{2}$ be a regular plane curve. Its signed curvature function is

κ_{s} (t) = ⟨ \frac{𝐚 (t)}{{| 𝐯 (t) |}^{2}}, R_{90} (\frac{𝐯 (t)}{| 𝐯 (t) |}) ⟩ = \frac{⟨ 𝐚 (t), R_{90} (𝐯 (t)) ⟩}{{| 𝐯 (t) |}^{3}} .

The curvature vector and the signed curvature are related by

\frac{𝐚^{⊥}}{{| 𝐯 |}^{2}} = κ_{s} R_{90} (\frac{𝐯}{| 𝐯 |}), κ = | κ_{s} | .

Thus $κ_{s} > 0$ means that the curve is turning counterclockwise (to the left), while $κ_{s} < 0$ means that it is turning clockwise (to the right).

5.2.1 The unit-speed case

If $𝜸$ has unit speed, then $| 𝐯 | = 1$ . Since differentiating $⟨ 𝐯, 𝐯 ⟩ = 1$ gives $⟨ 𝐚, 𝐯 ⟩ = 0$ , all of the acceleration is perpendicular to the velocity. Hence

𝐚 = κ_{s} R_{90} (𝐯), κ_{s} = ⟨ 𝐚, R_{90} (𝐯) ⟩, κ = | κ_{s} | .

Example 5.2 (A circle parametrized by arc length). Let $r > 0$ and

𝜸 (t) = (r \cos \frac{t}{r}, r \sin \frac{t}{r}) .

Then

𝐯 (t) = (- \sin \frac{t}{r}, \cos \frac{t}{r}), 𝐚 (t) = \frac{1}{r} (- \cos \frac{t}{r}, - \sin \frac{t}{r}) = \frac{1}{r} R_{90} (𝐯 (t)) .

Therefore $κ_{s} = 1 / r$ . Reversing the orientation of the circle changes the signed curvature to $- 1 / r$ , while its ordinary curvature remains $1 / r$ .

5.3 Angle functions

For a unit-speed plane curve, the velocity takes values in the unit circle

S^{1} = {𝐩 \in R^{2} : | 𝐩 | = 1} .

Thus we have a smooth map $𝐯 : I \to S^{1}$ . An angle function is a smooth lift of this map through the standard parametrization of the circle:

Definition 5.3: Angle function

Let $𝜸 : I \to R^{2}$ be a unit-speed plane curve with velocity $𝐯$ . An angle function for $𝜸$ is a smooth function $𝜃 : I \to R$ such that

𝐯 (t) = (\cos 𝜃 (t), \sin 𝜃 (t))

for every $t \in I$ .

A single inverse-trigonometric formula does not work globally: for example, $\arccos (v_{x})$ loses the sign of $v_{y}$ , and forcing $𝜃 \in [0, 2 π)$ creates jumps whenever the tangent direction passes through angle $2 π$ . Locally one may choose a suitable branch of $\arcsin$ or $\arccos$ ; the next result shows that these local choices can be assembled into a smooth function on the whole interval.

Proposition 5.4: Existence of a global angle function

Every unit-speed plane curve has a smooth angle function. It is unique up to addition of a constant integer multiple of $2 π$ . Moreover,

𝜃^{'} (t) = κ_{s} (t) .

Proof. Suppose first that $𝜃$ is a local angle function. Differentiating $𝐯 = (\cos 𝜃, \sin 𝜃)$ gives

𝐚 = 𝐯^{'} = 𝜃^{'} (- \sin 𝜃, \cos 𝜃) = 𝜃^{'} R_{90} (𝐯) .

Comparing this with $𝐚 = κ_{s} R_{90} (𝐯)$ shows that $𝜃^{'} = κ_{s}$ .

Choose $t_{0} \in I$ and $𝜃_{0} \in R$ with $𝐯 (t_{0}) = (\cos 𝜃_{0}, \sin 𝜃_{0})$ , and define

𝜃 (t) = 𝜃_{0} + \int_{t_{0}}^{t} κ_{s} (u) d 𝑢.

Let

A = {t \in I : 𝐯 (t) = (\cos 𝜃 (t), \sin 𝜃 (t))} .

The set $A$ contains $t_{0}$ and is closed by continuity. If $t \in A$ , choose a local inverse-trigonometric branch giving an angle function near $t$ . After adding a multiple of $2 π$ , the local angle agrees with $𝜃$ at $t$ ; since both have derivative $κ_{s}$ , they agree on a neighborhood of $t$ . Thus $A$ is also open. Because the interval $I$ is connected, $A = I$ . The same derivative argument proves uniqueness up to a constant multiple of $2 π$ . □

The formula

𝜃 (t) = 𝜃 (t_{0}) + \int_{t_{0}}^{t} κ_{s} (u) d u

says that signed curvature is the instantaneous counterclockwise turning rate of the unit tangent.

Remark 5.5 (A general parametrization). If the curve is regular but not necessarily unit speed, an angle function for its unit tangent

𝐭 (t) = \frac{𝐯 (t)}{| 𝐯 (t) |} = (\cos 𝜃 (t), \sin 𝜃 (t))

satisfies

𝜃^{'} (t) = κ_{s} (t) | 𝐯 (t) | .

This follows by passing to an orientation-preserving unit-speed reparametrization.

5.4 Rotation index

For a closed plane curve, the unit tangent returns to its initial direction. The angle function itself may nevertheless change by one or more full turns.

Definition 5.6: Rotation index

Let $𝜸 : [a, b] \to R^{2}$ be a unit-speed closed plane curve, and let $𝜃$ be an angle function. The rotation index of $𝜸$ is

rot (𝜸) = \frac{𝜃 (b) - 𝜃 (a)}{2 π} .

For a regular closed plane curve of variable speed, the rotation index means the rotation index of an orientation-preserving unit-speed reparametrization.

Because $𝐯 (a) = 𝐯 (b)$ , the difference $𝜃 (b) - 𝜃 (a)$ is an integer multiple of $2 π$ . Thus the rotation index is an integer. It counts the net number of times the unit tangent winds counterclockwise around the origin. In particular, a counterclockwise circle has rotation index $1$ , while the same circle with the opposite orientation has rotation index $- 1$ .

Corollary 5.7 (Total signed curvature). For a unit-speed closed plane curve,

rot (𝜸) = \frac{1}{2 π} \int_{a}^{b} κ_{s} (t) d 𝑡.

For a regular closed plane curve with arbitrary parameter,

rot (𝜸) = \frac{1}{2 π} \int_{a}^{b} κ_{s} (t) | 𝐯 (t) | d 𝑡.

Proof. Integrate $𝜃^{'} = κ_{s}$ in the unit-speed case. For a general parameter, integrate $𝜃^{'} = κ_{s} | 𝐯 |$ . □

5.5 Summary

$R_{90} (x, y) = (- y, x)$ rotates vectors counterclockwise by $9 0^{\circ}$ and produces the preferred perpendicular direction in the oriented plane.
Signed curvature is
$κ_{s} = \frac{⟨ 𝐚, R_{90} (𝐯) ⟩}{{| 𝐯 |}^{3}}, κ = | κ_{s} | .$
For a unit-speed curve, $𝐚 = κ_{s} R_{90} (𝐯)$ .
An angle function satisfies $𝐯 = (\cos 𝜃, \sin 𝜃)$ and $𝜃^{'} = κ_{s}$ .
The rotation index is the net number of counterclockwise turns made by the unit tangent along a closed plane curve.

Lecture 6Space Curves: Frenet Frame, Torsion, and Frenet Equations

Wednesday, September 2, 2026

Reading. [Tap16, § 1.7, pp. 41–47].

After class. Tapp, Exercises 1.62–1.66 and 1.68.

Goals. Review the cross product in $R^{3}$ , build the Frenet frame of a space curve, define torsion using Tapp’s sign convention, derive the Frenet equations, and explain why vanishing torsion characterizes planar curves.

6.1 The cross product

The special geometry of curves in $R^{3}$ depends on the cross product. Throughout this lecture, vectors and vector-valued functions are written in boldface, following Tapp.

Definition 6.1: Cross product

Let

𝐚 = (a_{1}, a_{2}, a_{3}), 𝐛 = (b_{1}, b_{2}, b_{3}) \in R^{3} .

Their cross product is

𝐚 \times 𝐛 = | \begin{matrix} 𝐢 & 𝐣 & 𝐤 \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \end{matrix} | = (a_{2} b_{3} - a_{3} b_{2}, a_{3} b_{1} - a_{1} b_{3}, a_{1} b_{2} - a_{2} b_{1}) .

The determinant in the middle is the standard mnemonic for the coordinate formula on the right.

For $𝐚, 𝐛, 𝐜 \in R^{3}$ , you can check that

⟨ 𝐜, 𝐚 \times 𝐛 ⟩ = \det (\begin{matrix} c_{1} & c_{2} & c_{3} \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \end{matrix}) .

(6.1)

Lemma 6.2 (Basic properties of the cross product). For $𝐚, 𝐛, 𝐜 \in R^{3}$ and $λ, μ \in R$ , the following hold.

$𝐚 \times 𝐛$ is orthogonal to both $𝐚$ and $𝐛$ . Thus
$𝐚 \times 𝐛 \in span {𝐚, 𝐛}^{⊥} .$
$𝐚 \times 𝐛 = - (𝐛 \times 𝐚)$ .
The cross product is linear in each input; for example,
$(λ 𝐚 + μ 𝐛) \times 𝐜 = λ (𝐚 \times 𝐜) + μ (𝐛 \times 𝐜) .$
${| 𝐚 \times 𝐛 |}^{2} = {| 𝐚 |}^{2} {| 𝐛 |}^{2} - {⟨ 𝐚, 𝐛 ⟩}^{2} .$

If $𝐚$ and $𝐛$ are nonzero and $𝜃 = ∠ (𝐚, 𝐛)$ , then
$| 𝐚 \times 𝐛 | = | 𝐚 | | 𝐛 | \sin 𝜃.$
The direction of $𝐚 \times 𝐛$ is determined by the right-hand rule.

Proof. The first statement follows from (6.1), because a repeated row makes the determinant zero. The algebraic identities follow by expanding the coordinate formula. Finally,

{| 𝐚 |}^{2} {| 𝐛 |}^{2} - {⟨ 𝐚, 𝐛 ⟩}^{2} = {| 𝐚 |}^{2} {| 𝐛 |}^{2} (1 - \cos^{2} 𝜃),

which gives the length formula. □

Figure 6.1: The cross product is perpendicular to the spanned plane, and its length is the area of the parallelogram.

6.1.1 Rotations and the cross product

A matrix $R \in M_{3 \times 3} (R)$ is a rotation matrix if

R^{T} R = I and \det R = 1 .

Proposition 6.3 (Rotations preserve the cross product). If $R$ is a rotation matrix, then

R (𝐚 \times 𝐛) = (R 𝐚) \times (R 𝐛)

for all $𝐚, 𝐛 \in R^{3}$ .

Proof. For arbitrary $𝐜 \in R^{3}$ , use preservation of the inner product and the scalar triple product:

\begin{array}{l} ⟨ 𝐜, (R 𝐚) \times (R 𝐛) ⟩ & = \det (𝐜, R 𝐚, R 𝐛) \\ = \det (R (R^{T} 𝐜), R 𝐚, R 𝐛) \\ = \det (R) \det (R^{T} 𝐜, 𝐚, 𝐛) \\ = ⟨ R^{T} 𝐜, 𝐚 \times 𝐛 ⟩ \\ = ⟨ 𝐜, R (𝐚 \times 𝐛) ⟩ . \end{array}

Since this holds for every $𝐜$ , the two vectors are equal. □

Remark 6.4. The condition $\det R = 1$ matters. More generally, for an orthogonal matrix $Q \in O (3)$ , i.e., $Q Q^{T} = Q^{T} Q = I$ ,

(Q 𝐚) \times (Q 𝐛) = \det (Q) Q (𝐚 \times 𝐛) .

Thus a reflection reverses the cross-product direction.

6.2 The Frenet frame

Let $𝜸 : I \to R^{3}$ be a regular space curve. At a time $t$ for which $κ (t) \neq 0$ , the unit tangent and unit normal are already defined by

𝐭 (t) = \frac{𝐯 (t)}{| 𝐯 (t) |}, 𝐧 (t) = \frac{𝐚^{⊥} (t)}{| 𝐚^{⊥} (t) |} = \frac{𝐭^{'} (t)}{| 𝐭^{'} (t) |} .

The cross product supplies a third vector.

Definition 6.5: Frenet frame

Let $𝜸 : I \to R^{3}$ be a regular space curve and suppose $κ (t) \neq 0$ . The unit binormal vector is

𝐛 (t) = 𝐭 (t) \times 𝐧 (t) .

The ordered orthonormal basis

{𝐭 (t), 𝐧 (t), 𝐛 (t)}

is called the Frenet frame at time $t$ .

Figure 6.2: The Frenet frame:

𝐭

and

𝐧

span a plane, with

𝐛

orthogonal to this plane.

6.3 The Frenet equations

Proposition 6.6: The Frenet equations

Let $𝜸 : I \to R^{3}$ be a regular space curve. At every time for which $κ \neq 0$ ,

\begin{aligned} 𝐭^{'} & = | 𝐯 | κ 𝐧, \\ 𝐧^{'} & = - | 𝐯 | κ 𝐭 + | 𝐯 | τ 𝐛, \\ 𝐛^{'} & = - | 𝐯 | τ 𝐧 . \end{aligned}

where $τ$ is defined to make the second equation be true. Equivalently,

{(\begin{matrix} 𝐭 \\ 𝐧 \\ 𝐛 \end{matrix})}^{'} = | 𝐯 | (\begin{matrix} 0 & κ & 0 \\ - κ & 0 & τ \\ 0 & - τ & 0 \end{matrix}) (\begin{matrix} 𝐭 \\ 𝐧 \\ 𝐛 \end{matrix}) .

Proof. The first Frenet equation was proved in Lecture 4, but let us recall quickly. Since $𝐯 = | 𝐯 | 𝐭$ , differentiation gives

𝐚 = \underset{𝐚^{∥}}{\underset{⏟}{{| 𝐯 |}^{'} 𝐭}} + \underset{𝐚^{⊥}}{\underset{⏟}{| 𝐯 | 𝐭^{'}}}

The identity $| 𝐭 | = 1$ implies $𝐭^{'} ⊥ 𝐭$ . The first term above is therefore parallel to $𝐯$ , while the second is perpendicular to $𝐯$ . Hence

𝐚^{⊥} = | 𝐯 | 𝐭^{'} .

Thus we have

\vec{κ} = \frac{𝐚^{⊥}}{| 𝐯 |^{2}} = \frac{| 𝐯 | 𝐭^{'}}{| 𝐯 |^{2}} = \frac{𝐭^{'}}{| 𝐯 |} = 𝜅𝔫.

Clearing denominators in the last equation gives $𝔱^{'} = | 𝐯 | 𝜅𝔫$ .

For the second Frenet equation, compute the three components of $𝐧^{'}$ in the orthonormal basis ${𝐭, 𝐧, 𝐛}$ :

\begin{array}{l} ⟨ 𝐧^{'}, 𝐭 ⟩ & = - ⟨ 𝐧, 𝐭^{'} ⟩ = - | 𝐯 | κ, \\ ⟨ 𝐧^{'}, 𝐧 ⟩ & = 0, \\ ⟨ 𝐧^{'}, 𝐛 ⟩ & = : | 𝐯 | 𝜏. \end{array}

In other words, we define $τ : = \frac{⟨ 𝐧^{'}, 𝐛 ⟩}{| 𝐯 |}$ .

This gives

𝐧^{'} = - | 𝐯 | κ 𝐭 + | 𝐯 | τ 𝐛 .

Finally, $𝐛^{'}$ is orthogonal to both $𝐛$ and $𝐭$ , so it is parallel to $𝐧$ . By the definition of $τ$ , we have

⟨ 𝐛^{'}, 𝐧 ⟩ = - ⟨ 𝔟, 𝔫^{'} ⟩ = - | 𝐯 | τ,

which proves the third equation. □

The definition of $τ$ above agrees with the definition in Tapp:

Definition 6.7: Torsion (Tapp)

Let $𝜸 : I \to R^{3}$ be a regular space curve, and let $t \in I$ satisfy $κ (t) \neq 0$ . The torsion of $𝜸$ at $t$ is

τ (t) = - \frac{⟨ 𝐛^{'} (t), 𝐧 (t) ⟩}{| 𝐯 (t) |} .

Remark 6.8 (Sign convention). This is Tapp’s convention. Some authors define torsion with the opposite sign. The factor $| 𝐯 |$ makes torsion independent of the speed of parametrization.

Remark 6.9 (Behavior under reparametrization). Torsion is independent of parametrization. Under an orientation-preserving reparametrization, all three Frenet vectors simply compose with the change of parameter. Under an orientation-reversing reparametrization, $𝐭$ and $𝐛$ change sign while $𝐧$ does not; the signs in the definition of $τ$ therefore cancel.

6.4 Some further topics

6.4.1 Zero torsion

Proposition 6.10. Let $γ : I \to R^{3}$ be a regular curve with $κ (t) \neq 0$ for all $t \in I$ . The trace of $γ$ is constrained to a plane if and only if $τ (t) = 0$ for all $t \in I$ .

Proof. If $γ$ is constrained to a plane, then it is easy to see that $𝔟^{'} = 0$ , so that using the Frenet equations, we have that $τ (t) = 0$ for all $t \in I$ .

Conversely, if $τ (t) = 0$ for all $t \in I$ , then the Frenet equations tell us that $𝔟^{'} = 0$ , so that $𝔟$ is constant. The claim is then that $⟨ γ (t), 𝔟 ⟩ = C$ is constant for all $t$ (which, since $𝔟$ is constant, implies that $γ$ is in a translate of the plane orthogonal to $𝔟$ ).

To see that $⟨ γ (t), 𝔟 ⟩$ is constant, we have

{⟨ γ (t), 𝔟 ⟩}^{'} = ⟨ 𝐯, 𝔟 ⟩ + ⟨ γ (t), 𝔟^{'} ⟩ = ⟨ 𝐯, 𝔟 ⟩ = 0

since $𝐯$ is a scalar multiple of $𝔱$ , and $𝔱 ⊥ 𝔟$ . □

6.4.2 Product rule

Lemma 6.11 (Product rule). If $𝜶, 𝜷 : I \to R^{3}$ are smooth, then

\frac{d}{d t} (𝜶 (t) \times 𝜷 (t)) = 𝜶^{'} (t) \times 𝜷 (t) + 𝜶 (t) \times 𝜷^{'} (t) .

Proof. Differentiate the coordinate formula component by component and apply the ordinary product rule. □

6.4.3 A curvature formula in $R^{3}$

Proposition 6.12: Curvature from the cross product

Let $𝜸 : I \to R^{3}$ be a regular space curve, with

𝐯 = 𝜸^{'}, 𝐚 = 𝜸^{″} .

Then

κ (t) = \frac{| 𝐯 (t) \times 𝐚 (t) |}{{| 𝐯 (t) |}^{3}} .

Proof. If $𝐚 = 0$ , both sides vanish. Otherwise, let $𝜃$ be the angle between $𝐯$ and $𝐚$ . Then $| 𝐚^{⊥} | = | 𝐚 | \sin 𝜃$ . Therefore

κ = \frac{| 𝐚^{⊥} |}{{| 𝐯 |}^{2}} = \frac{| 𝐯 | | 𝐚 | \sin 𝜃}{{| 𝐯 |}^{3}} = \frac{| 𝐯 \times 𝐚 |}{{| 𝐯 |}^{3}} .

□

6.5 Summary

The cross product satisfies
$𝐚 \times 𝐛 \in span {𝐚, 𝐛}^{⊥}, | 𝐚 \times 𝐛 | = area of the spanned parallelogram .$
For a regular space curve,
$κ = \frac{| 𝐯 \times 𝐚 |}{{| 𝐯 |}^{3}} .$
The Frenet frame is ${𝐭, 𝐧, 𝐛}$ , where $𝐛 = 𝐭 \times 𝐧$ .
Tapp’s torsion convention is
$τ = - \frac{⟨ 𝐛^{'}, 𝐧 ⟩}{| 𝐯 |} .$
The Frenet equations describe how the moving orthonormal frame changes along the curve.
When curvature is everywhere nonzero, a space curve has zero torsion exactly when its trace is contained in a plane.

Lecture 7Rigid Motions and Fundamental Theorems

Friday, September 4, 2026

Reading. [Tap16, §§ 1.8–1.9, pp. 48–60].

After class. HW 2 is due. Begin HW 3(a): Tapp, Exercises 1.70, 1.71, 1.75, 1.76, 1.78, and 1.83.

Goals. Define and classify rigid motions of Euclidean space, determine how curvature, signed curvature, and torsion behave under rigid motions, and state the fundamental theorems of plane and space curves. We prove the plane-curve theorem by reconstructing a curve from its signed curvature.

A rigid motion changes the position of a curve, and may also rotate or reflect it, without changing any distances. Thus rigid motions formalize the idea that two curves have the same shape but are viewed in different coordinate systems. We have been studying curves embedded in Euclidean space $R^{n}$ , and we will be interested in this lecture in understanding how our notions such as curvature and torsion change (or rather do not change) under rigid motions of the curve. In other words, we want to see that curvature and torsion are intrinsic to the curve in space, and not to its parameterization, or our coordinates (so long as we change coordinates in a way that preserves distances).

7.1 Rigid motions of Euclidean space

Definition 7.1: Rigid motion

A rigid motion of $R^{n}$ is a function

f : R^{n} \to R^{n}

that preserves distances. Equivalently,

| f (𝐩) - f (𝐪) | = | 𝐩 - 𝐪 |

for all $𝐩, 𝐪 \in R^{n}$ .

The inner product can be recovered from distances by the polarization identity

⟨ 𝐩, 𝐪 ⟩ = \frac{1}{2} ({| 𝐩 |}^{2} + {| 𝐪 |}^{2} - {| 𝐩 - 𝐪 |}^{2}) .

(7.1)

In particular, if a rigid motion fixes the origin, then it preserves norms and inner products.

For a matrix $A \in M_{n \times n} (R)$ , write

L_{A} (𝐱) = A 𝐱 .

Recall that $A$ is orthogonal if

A^{T} A = A A^{T} = 𝐼.

The set of orthogonal $n \times n$ matrices is denoted by $O (n)$ ; this is called the orthogonal group. The following conditions are equivalent (if they hold for all $𝐩, 𝐪 \in R^{n}$ ):

A \in O (n), ⟨ A 𝐩, A 𝐪 ⟩ = ⟨ 𝐩, 𝐪 ⟩, | A 𝐩 | = | 𝐩 | .

Thus the linear rigid motions are exactly the maps $L_{A}$ with $A \in O (n)$ .

For $𝐪 \in R^{n}$ , let

T_{𝐪} (𝐱) = 𝐱 + 𝐪

denote translation by $𝐪$ .

Theorem 7.2: Classification of rigid motions

Every rigid motion $f : R^{n} \to R^{n}$ has a unique expression

f = T_{𝐪} \circ L_{A},

where $𝐪 \in R^{n}$ and $A \in O (n)$ . Equivalently,

f (𝐱) = A 𝐱 + 𝐪 .

Proof. Set $𝐪 = f (0)$ and define

g = T_{- 𝐪} \circ 𝑓.

Then $g$ is a rigid motion with $g (0) = 0$ . By (7.1),

⟨ g (𝐩), g (𝐫) ⟩ = ⟨ 𝐩, 𝐫 ⟩

for all $𝐩, 𝐫 \in R^{n}$ .

Let $𝐞_{1}, \dots, 𝐞_{n}$ be the standard basis, and let $A$ be the matrix whose $i$ th column is $g (𝐞_{i})$ . These columns form an orthonormal basis, so $A \in O (n)$ . Define

h = L_{A}^{- 1} \circ 𝑔.

Then $h$ preserves inner products and fixes every $𝐞_{i}$ . If $𝐩 = \sum_{i} p_{i} 𝐞_{i}$ and $h (𝐩) = \sum_{i} b_{i} 𝐞_{i}$ , then

b_{i} = ⟨ h (𝐩), 𝐞_{i} ⟩ = ⟨ h (𝐩), h (𝐞_{i}) ⟩ = ⟨ 𝐩, 𝐞_{i} ⟩ = p_{i} .

Thus $h$ is the identity, so $g = L_{A}$ and $f = T_{𝐪} \circ L_{A}$ .

For uniqueness, evaluating at $0$ determines $𝐪 = f (0)$ , after which the linear part $A$ is determined as well. □

Since $A^{T} A = I$ ,

1 = \det (A^{T} A) = \det {(A)}^{2},

so $\det A = 1$ or $\det A = - 1$ .

Recall that ${GL}_{n} (R)$ is the set of $n \times n$ matrices with non-zero determinant, and is called the general linear group. A matrix $A \in {GL}_{n} (R)$ is said to be orientation preserving if $\det (A) > 0$ (and is orientation reversing if $\det (A) < 0$ ). A rigid motion is called orientation preserving if the matrix $A$ in the theorem above has determinant $1$ .

Definition 7.3: Proper and improper rigid motions

Suppose $f = T_{𝐪} \circ L_{A}$ is a rigid motion. Following Tapp, $f$ is called proper if $\det A = 1$ , and improper if $\det A = - 1$ . Proper rigid motions preserve orientation; improper rigid motions reverse orientation.

Figure 7.1: A rigid motion changes the position of a curve but preserves the distance between every pair of corresponding points.

7.2 Geometric quantities under rigid motion

Let $𝜸 : I \to R^{n}$ be regular, let $f = T_{𝐪} \circ L_{A}$ be a rigid motion, and define

\hat{𝜸} = f \circ 𝜸 = 𝐴𝛾 + 𝐪 .

Writing $𝐯 = 𝜸^{'}$ and $𝐚 = 𝜸^{″} = 𝐯^{'}$ , and similarly for $\hat{γ}$ , differentiation gives

\begin{align} \hat{𝐯} & : = {\hat{γ}}^{'} = {(𝐴𝛾 + 𝐪)}^{'} = A^{'} γ + A γ^{'} + 𝐪^{'} = A γ^{'} = A 𝐯, & (7.2) \\ \hat{𝐚} & : = {(\hat{𝐯})}^{'} = {(A 𝐯)}^{'} = A^{'} 𝐯 + A 𝐯^{'} = A 𝐚, & (7.3) \end{align}

since $A$ and $𝐪$ are constant.

Because $A$ is orthogonal, it preserves inner products, lengths, and orthogonality.

Proposition 7.4: Invariance under rigid motions

For a regular curve, the following statements hold.

Curvature is invariant under every rigid motion:
$\hat{κ} = 𝜅.$
For a space curve, torsion is preserved by proper rigid motions and changes sign under improper rigid motions:
$\hat{τ} = \det (A) 𝜏.$
For a plane curve, signed curvature is preserved by proper rigid motions and changes sign under improper rigid motions:
${\hat{κ}}_{s} = \det (A) κ_{s} .$

Proof. Equation (7.3) implies

{\hat{𝐚}}^{⊥} = A 𝐚^{⊥}, | \hat{𝐯} | = | 𝐯 | .

Therefore, since $A$ preserves lengths,

\hat{κ} = \frac{| {\hat{𝐚}}^{⊥} |}{{| \hat{𝐯} |}^{2}} = \frac{| A 𝐚^{⊥} |}{{| A 𝐯 |}^{2}} = \frac{| 𝐚^{⊥} |}{{| 𝐯 |}^{2}} = 𝜅.

For a space curve at a point where $κ \neq 0$ ,

\hat{𝐭} = A 𝐭, \hat{𝐧} = A 𝐧 .

The transformation rule for the cross product is

(A 𝐱) \times (A 𝐲) = \det (A) A (𝐱 \times 𝐲),

\hat{𝐛} = \det (A) A 𝐛 .

Using Tapp’s torsion convention,

\hat{τ} = - \frac{⟨ {\hat{𝐛}}^{'}, \hat{𝐧} ⟩}{| \hat{𝐯} |} = - \det (A) \frac{⟨ A 𝐛^{'}, A 𝐧 ⟩}{| 𝐯 |} = \det (A) 𝜏.

In the plane,

R_{90} (A 𝐰) = \det (A) A R_{90} (𝐰) .

Substituting this identity into the formula for signed curvature gives

{\hat{κ}}_{s} = \frac{⟨ A 𝐚, R_{90} (A 𝐯) ⟩}{{| A 𝐯 |}^{3}} = \det (A) κ_{s} .

□

Thus ordinary curvature records bending without orientation, while signed curvature and torsion also detect handedness. A reflection preserves the amount of bending but reverses the relevant sign.

7.3 The fundamental theorems of curves

The preceding proposition shows that curvature data are unchanged by proper rigid motions. The converse is the fundamental result: the appropriate curvature data determine the curve up to a proper rigid motion.

Theorem 7.5: The fundamental theorems of plane and space curves

Let $I \subseteq R$ be an interval.

If $κ_{s} : I \to R$ is smooth, then there exists a unit-speed plane curve $𝜸 : I \to R^{2}$ whose signed curvature is $κ_{s}$ . If $𝜸$ and $\hat{𝜸}$ are two such curves, then
$\hat{𝜸} = f \circ 𝜸$

for some proper rigid motion $f$ of $R^{2}$ .
If $κ, τ : I \to R$ are smooth and $κ > 0$ , then there exists a unit-speed space curve $𝜸 : I \to R^{3}$ whose curvature and torsion are $κ$ and $τ$ . If $𝜸$ and $\hat{𝜸}$ are two such curves, then
$\hat{𝜸} = f \circ 𝜸$

for some proper rigid motion $f$ of $R^{3}$ .

The theorem has two parts. Existence says that allowable curvature data come from a curve. Uniqueness says that these data determine the curve’s shape: the only remaining freedom is its initial position and initial orientation.

7.3.1 The plane-curve construction

Proof of the plane-curve statement. Fix $t_{0} \in I$ . We first construct the curve with initial conditions

𝜸 (t_{0}) = (0, 0), 𝜸^{'} (t_{0}) = (1, 0) .

Define

𝜃 (t) : = \int_{t_{0}}^{t} κ_{s} (u) d u,

(7.4)

then define the velocity and the curve by

𝐯 (t) : = (\cos 𝜃 (t), \sin 𝜃 (t)), 𝜸 (t) : = \int_{t_{0}}^{t} 𝐯 (u) d 𝑢.

(7.5)

The vector integral is taken componentwise. The construction can be remembered as

κ_{s} \to_{}^{integrate} 𝜃 \to_{}^{(\cos 𝜃, \sin 𝜃)} 𝐯 \to_{}^{integrate} 𝜸 .

By the Fundamental Theorem of Calculus,

𝜸^{'} = 𝐯, | 𝐯 | = 1,

so $𝜸$ has unit speed. Furthermore,

𝐚 = 𝐯^{'} = 𝜃^{'} (- \sin 𝜃, \cos 𝜃) = κ_{s} R_{90} (𝐯) .

For a unit-speed plane curve, this equation says exactly that the signed curvature is $κ_{s}$ . This proves existence. Adding a constant to $𝜃$ and a constant vector to $𝜸$ gives arbitrary initial direction and initial position.

For uniqueness, let $𝜸$ and $\hat{𝜸}$ be unit-speed plane curves with the same signed curvature $κ_{s}$ , and choose angle functions $𝜃$ and $\hat{𝜃}$ . After applying a rigid motion to each curve, we can assume that $γ (t_{0}) = \hat{γ} (t_{0}) = 0$ , and $γ^{'} (t_{0}) = {\hat{γ}}^{'} (t_{0}) = [\begin{matrix} 1 \\ 0 \end{matrix}]$ . Since

𝜃^{'} = κ_{s} = {\hat{𝜃}}^{'},

there is a constant $𝜃_{0}$ such that

\hat{𝜃} = 𝜃 + 𝜃_{0} .

But then, plugging into $𝐯 (t) : = (\cos 𝜃 (t), \sin 𝜃 (t))$ and $\hat{𝐯} (t) : = (\cos \hat{𝜃} (t), \sin \hat{𝜃} (t))$ , and setting $t = t_{0}$ , we see that $𝜃_{0}$ is a multiple of $2 π$ , so we can take $𝜃_{0} = 0$ . Therefore $𝐯 = \hat{𝐯}$ .

Finally, since

γ^{'} (t) = : 𝐯 (t) = \hat{v} (t) : = {\hat{γ}}^{'} (t)

we see that $\hat{γ} (t) = γ (t) + γ_{0}$ . Setting $t = t_{0}$ , and using that $γ (t) = \hat{γ} (t) = 0$ , we see that $\hat{γ} = γ$ . □

Remark 7.6 (Sketch of proof of the space-curve statement and ODE). Tapp states the space-curve theorem without proof because its proof uses existence and uniqueness for ordinary differential equations. For unit speed, the Frenet equations are

𝐭^{'} = κ 𝐧, 𝐧^{'} = - κ 𝐭 + τ 𝐛, 𝐛^{'} = - τ 𝐧 .

In matrix form, in the form you would use the result from ODE, this is (in block form)

\underset{𝐅^{'} (t)}{\underset{⏟}{{(\begin{matrix} 𝐭 \\ 𝐧 \\ 𝐛 \end{matrix})}^{'}}} = \underset{Ω}{\underset{⏟}{(\begin{matrix} 0 & κ & 0 \\ - κ & 0 & τ \\ 0 & - τ & 0 \end{matrix})}} \underset{𝐅 (t)}{\underset{⏟}{(\begin{matrix} 𝐭 \\ 𝐧 \\ 𝐛 \end{matrix})}},

where the $9 \times 9$ (skew symmetric) matrix $Ω$ and the $9 \times 1$ column matrix $𝐅 (t)$ are defined as indicated.

Given $κ > 0$ , $τ$ , and an initial positively oriented orthonormal frame $𝐅_{0} = (𝔱_{0}, 𝔫_{0}, 𝔟_{0})$ , this linear first-order system has a unique solution $𝐅 (t) = (𝔱 (t), 𝔫 (t), 𝔟 (t))$ with $𝐅 (t_{0}) = (𝔱 (t_{0}), 𝔫 (t_{0}), 𝔟 (t_{0})) = 𝐅_{0} = (𝔱_{0}, 𝔫_{0}, 𝔟_{0})$ .

Note that since $Ω = - Ω^{T}$ , we have that

{(𝐅 {(t)}^{T} 𝐅 (t))}^{'} = {(𝐅^{T})}^{'} 𝐅 + 𝐅^{T} 𝐅^{'} = {(𝐅^{'})}^{T} 𝐅 + 𝐅^{T} 𝐅^{'} = {(Ω 𝐅)}^{T} 𝐅 + 𝐅^{T} (Ω 𝐅) = 𝐅^{T} Ω^{T} 𝐅 + 𝐅^{T} Ω 𝐅 = 0,

so that $𝐅^{T} (t) 𝐅 (t)$ is constant. Since $𝐅 {(t_{0})}^{T} 𝐅 (t_{0}) = I$ (since $𝔱_{0}$ , $𝔫_{0}$ , and $𝔟_{0}$ form an orthonormal basis), we see that $𝐅^{T} (t) 𝐅 (t) = I$ for all $t$ , so that for all $t$ , we have that $𝔱 (t)$ , $𝔫 (t)$ , and $𝔟 (t)$ form an orthonormal basis. For the orientation, make a new $3 \times 3$ matrix $𝐆 (t)$ with columns given by $𝔱$ , $𝔫$ , and $𝔟$ . We know that $𝐆^{T} 𝐆 = I$ for all $t$ , so that $𝐆 (t) \in O (3)$ for all $t$ ; therefore $\det (𝐆 (t)) = \pm 1$ . Since the determinant is is a polynomial in the entries, it is continuous, and therefore since $\det (𝐆 (t_{0})) = 1$ , we know that $𝔱 (t)$ , $𝔫 (t)$ , and $𝔟 (t)$ are positively oriented for all $t$ .

Defining

𝜸 (t) = 𝐩_{0} + \int_{t_{0}}^{t} 𝐭 (u) d u

then produces a curve with $γ (t_{0}) = 𝐩_{0}$ , and, from the Fundamental Theorem of Calculus, we have that $γ^{'} (t) = 𝐭$ , so that the curve has unit speed. Then the differential equations above imply that $𝔱^{'} = 𝜅𝔫$ , so that the unit normal for $γ$ is equal to $𝔫$ , and the curvature is $κ$ . The differential equations above also give us that $𝔫^{'} = - 𝜅𝔱 + 𝜏𝔟$ , so that the torsion is $τ$ .

Uniqueness for the ODE, after using a proper rigid motion to align the initial points and Frenet frames, gives uniqueness of the curve up to a proper rigid motion.

7.4 A short overview of curvature formulas

Tapp’s § 1.9 collects the formulas developed in the preceding lectures. For a regular curve in $R^{n}$ , (Theorem 4.6)

κ = \frac{| 𝐚^{⊥} |}{{| 𝐯 |}^{2}} = \frac{| 𝐭^{'} |}{| 𝐯 |}, κ = | 𝐚 | when the curve has unit speed .

For a plane curve $𝜸 (t) = (x (t), y (t))$ ,

κ_{s} = \frac{⟨ 𝐚, R_{90} (𝐯) ⟩}{{| 𝐯 |}^{3}} = \frac{x^{'} y^{″} - x^{″} y^{'}}{{({(x^{'})}^{2} + {(y^{'})}^{2})}^{3 / 2}} .

In particular, for the graph parametrization $𝜸 (t) = (t, f (t))$ ,

κ_{s} (t) = \frac{f^{″} (t)}{{(1 + {(f^{'} (t))}^{2})}^{3 / 2}} .

For a space curve (see Theorem 6.12),

κ = \frac{| 𝐯 \times 𝐚 |}{{| 𝐯 |}^{3}} .

7.5 Summary

Every rigid motion of $R^{n}$ has the form $f (𝐱) = A 𝐱 + 𝐪$ with $A \in O (n)$ .
Curvature is preserved by all rigid motions. Signed curvature and torsion are preserved by proper rigid motions and change sign under improper rigid motions.
A smooth signed-curvature function determines a unit-speed plane curve up to a proper rigid motion.
Smooth functions $κ > 0$ and $τ$ determine a unit-speed space curve up to a proper rigid motion.

Lecture 8Hopf’s Umlaufsatz and the Jordan Curve Theorem

Wednesday, September 9, 2026

Reading. [Tap16, § 2.1, pp. 61–67].

After class. HW 3(b): Tapp, Exercises 1.52, 1.55, 1.58, 2.1, 2.2, and 2.3.

Goals. Interpret the rotation index as the degree of the unit tangent map, prove Hopf’s Umlaufsatz using a family of secant directions, and use winding number together with a tubular neighborhood to explain the Jordan curve theorem. We conclude with the piecewise-regular formula recorded in class.

The results in the preceding lectures were principally local: they described what a curve is doing near one point. The two theorems in this lecture are global. They constrain the behavior of an entire simple closed plane curve. The German word Umlaufsatz means, roughly, a “going-around theorem.”

Theorem 8.1: Theorems of Hopf and Jordan

Let $𝜸 : [a, b] \to R^{2}$ be a simple closed plane curve, and let

C = 𝜸 ([a, b])

denote its trace.

Hopf’s Umlaufsatz. The rotation index of $𝜸$ is either $1$ or $- 1$ .
Jordan curve theorem. The complement $R^{2} - C$ has exactly two path-connected components. Their common boundary is $C$ . One component is bounded and is called the interior; the other is unbounded and is called the exterior.

8.1 Degree and the unit tangent map

Write

S^{1} = {(x, y) \in R^{2} : x^{2} + y^{2} = 1} .

The rotation index introduced in Lecture 5 is a special case of a topological integer attached to any closed path in the circle.

Definition 8.2: Degree of a circle-valued loop

Let $f : [a, b] \to S^{1}$ be continuous and suppose that $f (a) = f (b)$ . A continuous angle function for $f$ is a continuous function $𝜃 : [a, b] \to R$ such that

f (t) = (\cos 𝜃 (t), \sin 𝜃 (t)) .

Such an angle function exists and is unique up to adding a constant integer multiple of $2 π$ . The degree of $f$ is

\deg (f) = \frac{𝜃 (b) - 𝜃 (a)}{2 π} \in 𝑍.

The integer condition follows from $f (a) = f (b)$ : the two endpoint angles differ by an integer multiple of $2 π$ . Existence of a global angle function can be proved by covering $S^{1}$ with semicircles, choosing local inverse branches of $𝜃 \mapsto (\cos 𝜃, \sin 𝜃)$ , and matching the branches on finitely many subintervals.

If $𝜸$ is a regular closed plane curve, its unit tangent map is

𝐭 (t) = \frac{𝜸^{'} (t)}{| 𝜸^{'} (t) |} \in S^{1} .

The rotation index is exactly

rot (𝜸) = \deg (𝐭) .

For a unit-speed curve, the formula $𝜃^{'} = κ_{s}$ gives

rot (𝜸) = \frac{1}{2 π} \int_{a}^{b} κ_{s} (t) d 𝑡.

For an arbitrary regular parametrization, the corresponding formula is

rot (𝜸) = \frac{1}{2 π} \int_{a}^{b} κ_{s} (t) | 𝜸^{'} (t) | d 𝑡.

Figure 8.1: The rotation index is the degree of the unit tangent map.

The next lemma is the basic stability principle used in both global proofs.

Lemma 8.3 (Antipodal-value lemma). Let $f_{0}, f_{1} : [a, b] \to S^{1}$ be continuous loops. If $\deg (f_{0}) \neq \deg (f_{1})$ , then for some $t_{0} \in [a, b]$ ,

f_{0} (t_{0}) = - f_{1} (t_{0}) .

Equivalently, if the two vectors are never opposite, then the two maps have the same degree.

Proof. Choose angle functions $𝜃_{0}$ and $𝜃_{1}$ , and set $δ = 𝜃_{1} - 𝜃_{0}$ . If the degrees differ, then

| δ (b) - δ (a) | \geq 2 𝜋.

Consequently the interval between $δ (a)$ and $δ (b)$ contains an odd multiple of $π$ . By the intermediate value theorem, $δ (t_{0})$ is an odd multiple of $π$ for some $t_{0}$ , which is exactly the assertion that $f_{0} (t_{0}) = - f_{1} (t_{0})$ . □

Corollary 8.4 (Homotopy invariance of degree). Suppose $H : [0, 1] \times [a, b] \to S^{1}$ is continuous and $H (s, a) = H (s, b)$ for every $s$ . Then

\deg (H (s, \cdot))

is independent of $s$ .

Proof. Uniform continuity implies that for nearby values of $s$ , the two loops are pointwise close and hence never antipodal. Lemma 8.3 therefore shows that their degrees agree. Thus the integer-valued degree is locally constant as a function of $s$ , and hence constant on $[0, 1]$ . □

8.2 Hopf’s Umlaufsatz

Proof of Hopf’s Umlaufsatz. Reparametrize $𝜸$ by arc length. Choose a point $𝐩 \in C$ such that the entire trace lies on one side of the tangent line at $𝐩$ . Such a point exists: place the curve inside a large circle and shrink the circle until it first touches the trace. After shifting the parameter, assume

𝜸 (a) = 𝜸 (b) = 𝐩 .

Consider the closed triangle

T = {(t_{1}, t_{2}) : a \leq t_{1} \leq t_{2} \leq b} .

Define the secant-direction map $𝝍 : T \to S^{1}$ by

𝝍 (t_{1}, t_{2}) = {\begin{matrix} \frac{𝜸 (t_{2}) - 𝜸 (t_{1})}{| 𝜸 (t_{2}) - 𝜸 (t_{1}) |}, & t_{1} < t_{2}, (t_{1}, t_{2}) \neq (a, b), \\ 𝐭 (t_{1}), & t_{1} = t_{2}, \\ - 𝐭 (a), & (t_{1}, t_{2}) = (a, b) . \end{matrix}

The derivative definition of $𝐭$ gives continuity along the diagonal, and the special value at $(a, b)$ makes the map continuous at the remaining corner.

Let $𝜶_{0}$ traverse the diagonal of $T$ from $(a, a)$ to $(b, b)$ . Then

𝝍 \circ 𝜶_{0} = 𝐭,

\deg (𝝍 \circ 𝜶_{0}) = rot (𝜸) .

Let $𝜶_{1}$ instead travel from $(a, a)$ to $(a, b)$ and then from $(a, b)$ to $(b, b)$ . Since $T$ is convex, the paths

𝜶_{s} = (1 - s) 𝜶_{0} + s 𝜶_{1}, 0 \leq s \leq 1,

remain in $T$ and give a homotopy through loops in $S^{1}$ after composing with $𝝍$ . Corollary 8.4 gives

\deg (𝝍 \circ 𝜶_{0}) = \deg (𝝍 \circ 𝜶_{1}) .

Along the first edge of $𝜶_{1}$ , the secant vectors point from $𝐩$ to the rest of the curve. Because the curve lies on one side of the supporting tangent line, these vectors remain in one semicircle and turn from $𝐭 (a)$ to $- 𝐭 (a)$ . Along the second edge, their negatives traverse the other semicircle. The total change in angle is therefore $2 π$ or $- 2 π$ , depending on the orientation. Hence

rot (𝜸) = 1 or rot (𝜸) = - 1 .

□

A supporting tangent line at $𝐩$ .

The diagonal path is deformed to the two outer edges.

Figure 8.2: The two geometric ingredients in the proof of Hopf’s Umlaufsatz.

8.3 The Jordan curve theorem

The proof uses two ideas from the handwritten notes: a narrow tubular neighborhood of the curve and the winding number of the curve about a point.

Proposition 8.5: Tubular neighborhood of a simple closed curve

After an arc-length reparametrization, let $𝐭$ be the unit tangent of the simple closed plane curve $𝜸$ . For sufficiently small $𝜀 > 0$ , the map

Φ : (- 𝜀, 𝜀) \times [a, b) \to R^{2}, Φ (s, t) = 𝜸 (t) + s R_{90} (𝐭 (t)),

is injective.

Its image is a tubular neighborhood of $C$ . Removing $C$ leaves two path-connected strips,

Φ ((0, 𝜀) \times [a, b)) and Φ ((- 𝜀, 0) \times [a, b)) .

Tapp postpones the proof of injectivity until the inverse function theorem is available; locally it follows from regularity, while compactness and simplicity prevent different normal segments from meeting globally.

For $𝐩 \in R^{2} - C$ , define

f_{𝐩} (t) = \frac{𝜸 (t) - 𝐩}{| 𝜸 (t) - 𝐩 |} \in S^{1},

and define the winding number of $C$ about $𝐩$ by

W (𝐩) = \deg (f_{𝐩}) .

Lemma 8.6. The function

W : R^{2} - C \to Z

is locally constant. Consequently it is constant on every path-connected component of $R^{2} - C$ .

Proof. Fix $𝐩 \notin C$ . Because $C$ is compact, $d = dist (𝐩, C) > 0$ . If $𝐪$ is sufficiently close to $𝐩$ , the segment from $𝐩$ to $𝐪$ does not meet $C$ . Moving the observation point along that segment gives a homotopy from $f_{𝐩}$ to $f_{𝐪}$ . Their degrees are therefore equal by Corollary 8.4. □

Proof of the Jordan curve theorem. First choose a point $𝐪$ very far from $C$ . All vectors $𝜸 (t) - 𝐪$ then point into a single open semicircle, so $f_{𝐪}$ has degree zero:

W (𝐪) = 0 .

Next choose two nearby points $𝐩_{+}$ and $𝐩_{-}$ on opposite sides of $C$ inside a tubular neighborhood. Zooming in near the intervening point of $C$ , the curve is nearly its tangent line. Over the short local arc, the two direction maps traverse opposite semicircles; away from that arc they are arbitrarily close. The antipodal-value lemma allows these maps to be perturbed without changing degree, and the local semicircle computation gives

| W (𝐩_{+}) - W (𝐩_{-}) | = 1 .

Thus $𝐩_{+}$ and $𝐩_{-}$ lie in different path-connected components, so the complement has at least two components.

Now let $𝐩 \in R^{2} - C$ be arbitrary. Since $C$ is compact, there is a shortest line segment from $𝐩$ to $C$ . Before reaching its endpoint, that segment enters the tubular neighborhood and remains on one of its two sides. Within that side strip it can be joined to either $𝐩_{+}$ or $𝐩_{-}$ . Hence every point in the complement lies in one of these two components, and there are exactly two.

A sufficiently large ball contains $C$ . The component containing the complement of that ball is unbounded and is the exterior; the other is bounded and is the interior. Finally, every point of $C$ has tubular-neighborhood points from both components arbitrarily close to it, so $C$ is the common boundary of the two components. □

Figure 8.3: The winding number is constant on each complementary component, and it changes by one when one crosses the curve.

The two theorems give equivalent ways to choose an orientation.

Definition 8.7: Positive orientation

A simple closed plane curve is positively oriented if its rotation index is $1$ . Equivalently, the interior stays on one’s left while the curve is traversed: for every $t$ , the vector $R_{90} (𝐭 (t))$ points into the interior. A curve with the opposite orientation has rotation index $- 1$ .

The equivalence follows from the same local picture used above. The vector $R_{90} (𝐭)$ points consistently into one of the two tubular strips; at the supporting point used in the proof of Hopf’s theorem, the sign of the single full turn identifies that strip as the interior or exterior.

8.4 A first look at the piecewise-regular formula

The final pages of the handwritten notes record the version needed for curves with corners. We include that formula here before turning to differential forms and line integrals in Lecture 9.

Definition 8.8: Piecewise-regular curve and signed corner angle

A continuous curve $𝜸 : [a, b] \to R^{2}$ is piecewise regular if there is a partition

a = t_{0} < t_{1} < \dots < t_{m} = b

such that each restriction to $[t_{i}, t_{i + 1}]$ is regular. At a corner $t_{i}$ , let $𝐯_{i}^{-}$ and $𝐯_{i}^{+}$ be the one-sided velocity vectors. The signed angle $α_{i} \in [- π, π]$ is the smallest signed rotation carrying $𝐯_{i}^{-}$ to $𝐯_{i}^{+}$ , positive for counterclockwise rotation and negative for clockwise rotation.

Theorem 8.9: Generalized Hopf Umlaufsatz

Let $𝜸 : [a, b] \to R^{2}$ be a positively oriented, unit-speed, piecewise-regular simple closed plane curve. If $κ_{s}$ is its signed curvature on the smooth pieces and $α_{i}$ are its signed corner angles, then

\int_{a}^{b} κ_{s} (t) d t + ∑_{i} α_{i} = 2 𝜋.

The integral means the sum of the integrals over the regular pieces.

The idea is to round each corner in a very small neighborhood. The signed curvature accumulated along the rounded portion approaches the signed corner angle. The resulting smooth simple closed curve has total signed curvature $2 π$ by Hopf’s Umlaufsatz.

Example 8.10 (A rectangle). Orient the boundary of a rectangle counterclockwise. The signed curvature is zero on every open side. At each of the four corners the tangent turns counterclockwise through $π / 2$ . Therefore

\int κ_{s} d t + ∑_{i = 1}^{4} α_{i} = 0 + 4 (\frac{π}{2}) = 2 𝜋.

Figure 8.4: For a positively oriented rectangle, all of the turning occurs at the four corners.

8.5 Summary

The degree of a loop in $S^{1}$ records its net number of counterclockwise turns, and the rotation index is the degree of the unit tangent map.
Degree is unchanged under a homotopy through loops.
Hopf’s Umlaufsatz says that the tangent of a simple closed plane curve turns exactly once: its rotation index is $1$ or $- 1$ .
The winding number $W (𝐩)$ is locally constant on the complement of the trace. Together with a tubular neighborhood, this yields the interior and exterior in the Jordan curve theorem.
For a positively oriented piecewise-regular simple closed curve, the smooth turning and the corner turning add to $2 π$ .

Lecture 9Differential Forms, Vector Fields, and Line Integrals

Friday, September 11, 2026

Reading. Related reading: [Tap16, § 2.4, pp. 81–89].

After class. HW 3 due; review the wedge-product and line-integral computations from class.

Goals. Introduce differential forms in Euclidean coordinates, compute with the wedge product and exterior derivative, identify vector fields with one-forms using the Euclidean inner product, and define line integrals in a parametrization-independent way.

The handwritten lecture begins the language needed for Green’s theorem. Rather than starting immediately with circulation and flux, we first record a coordinate model for differential forms. In this lecture, the symbols $d x_{1}, \dots, d x_{n}$ are algebraic objects that encode oriented infinitesimal directions. The wedge product keeps track of the order of those directions.

9.1 Coordinate forms and the wedge product

Let $x_{1}, \dots, x_{n}$ denote the standard coordinate functions on $R^{n}$ . Associated to them are the coordinate one-forms (just consider these as symbols) $d x_{1}, \dots, d x_{n}$ . Let $V$ be the real vector space with basis $d x_{1}, \dots, d x_{n}$ . Then for any non-negative integer $k$ , consider the real vector space, denoted by $\otimes^{k} V$ , which is defined to be the real vector space with basis the symbols

d x_{i_{1}} \otimes d x_{i_{2}} \otimes \dots \otimes d x_{i_{k}},

where $i_{1}, i_{2}, \dots, i_{k} \in {1, \dots, n}$ . This vector space has dimension $n^{k}$ . By convention, $\otimes^{0} V = R$ .

Now, inside of $\otimes^{k} V$ , for any $i_{1}, \dots, i_{k}$ , with $i_{1}, \dots, i_{k} \in (1, \dots, n)$ , then if $σ$ is a permutation of ${1, \dots, k}$ that interchanges exactly two of the numbers (e.g., $(1, 2, 3, 4) \mapsto (2, 1, 3, 4)$ ), consider the vector

d x_{i_{σ (1)}} \otimes \dots \otimes d x_{i_{σ (k)}} + d x_{i_{1}} \otimes \dots \otimes d x_{i_{k}} .

The span of these vectors is a vector subspace $S_{k}$ of $\otimes^{k} V$ . This subspace defines an equivalence relation on $\otimes^{k} V$ ; two vectors in $\otimes^{k} V$ are equivalent if their difference lies in $S_{k}$ . We define $\land^{k} V$ to be the vector space that has elements that are equivalence classes of elements in $\otimes^{k} V$ . We denote the equivalence class of $d x_{i_{1}} \otimes d x_{i_{2}} \otimes \dots \otimes d x_{i_{k}}$ by

d x_{i_{1}} \land d x_{i_{2}} \land \dots \land d x_{i_{k}} .

The basic consequence of this construction is that we have the following relation between the symbols above: Given $i_{1}, \dots, i_{k}$ , with $i_{1}, \dots, i_{k} \in (1, \dots, n)$ , then if $σ$ is a permutation of ${1, \dots, k}$ that interchanges exactly two of the numbers (e.g., $(1, 2, 3, 4) \mapsto (2, 1, 3, 4)$ ), we have

d x_{i_{σ (1)}} \land \dots \land d x_{i_{σ (k)}} = - d x_{i_{1}} \land \dots \land d x_{i_{k}} .

In other words, for example, we have

d x_{i} \land d x_{j} = - d x_{j} \land d x_{i} .

This implies that

d x_{i} \land d x_{i} = - d x_{i} \land d x_{i},

d x_{i} \land d x_{i} = 0 .

Thus any wedge product containing a repeated coordinate differential is zero.

For example, in $R^{3}$ ,

\begin{array}{l} d x_{2} \land d x_{1} \land d x_{3} & = - d x_{1} \land d x_{2} \land d x_{3}, \\ d x_{3} \land d x_{1} \land d x_{2} & = d x_{1} \land d x_{2} \land d x_{3}, \\ d x_{3} \land d x_{2} \land d x_{1} & = - d x_{1} \land d x_{2} \land d x_{3} . \end{array}

I = (i_{1}, \dots, i_{k}),

we use the abbreviation

d x_{I} = d x_{i_{1}} \land \dots \land d x_{i_{k}} .

Examples

The coordinate basis forms in low dimensional examples are:

\begin{matrix} one-forms & two-forms & three-forms \\ R^{1} & d x_{1} & 0 & 0 \\ R^{2} & d x_{1}, d x_{2} & d x_{1} \land d x_{2} & 0 \\ R^{3} & d x_{1}, d x_{2}, d x_{3} & d x_{1} \land d x_{2}, d x_{1} \land d x_{3}, d x_{2} \land d x_{3} & d x_{1} \land d x_{2} \land d x_{3} . \end{matrix}

There are no nonzero $k$ -forms on $R^{n}$ when $k > n$ , because every wedge of more than $n$ coordinate differentials must repeat an index.

Exercise 9.1. Show that a basis for $\land^{k} V$ is given by the collection of $d x_{I}$ for the set of $(i_{1}, \dots, i_{k})$ with $1 \leq i_{1} < i_{2} < \dots < i_{k} \leq n$ . Therefore the dimension of $\land^{k} V$ is $(\binom{n}{k})$ .

9.1.1 Wedge products of forms

One can take wedges of differential forms. If $ω$ is a $k$ -form and $η$ is an $ℓ$ -form, then one gets a $(k + ℓ)$ -form

ω \land 𝜂.

For instance, if

ω = f_{1} (x_{1}, x_{2}, x_{3}) d x_{1} + f_{2} (x_{1}, x_{2}, x_{3}) d x_{2}

and

η = g_{13} (x_{1}, x_{2}, x_{3}) d x_{1} \land d x_{3},

then

ω \land η = (f_{1} (x_{1}, x_{2}, x_{3}) d x_{1} + f_{2} (x_{1}, x_{2}, x_{3}) d x_{2}) \land (g_{13} (x_{1}, x_{2}, x_{3}) d x_{1} \land d x_{3})

= f_{1} (x_{1}, x_{2}, x_{3}) d x_{1} \land g_{13} (x_{1}, x_{2}, x_{3}) d x_{1} \land d x_{3} + f_{2} (x_{1}, x_{2}, x_{3}) d x_{2} \land g_{13} (x_{1}, x_{2}, x_{3}) d x_{1} \land d x_{3}

= 0 + f_{2} (x_{1}, x_{2}, x_{3}) g_{13} (x_{1}, x_{2}, x_{3}) d x_{2} \land d x_{1} \land d x_{3}

= - f_{2} (x_{1}, x_{2}, x_{3}) g_{13} (x_{1}, x_{2}, x_{3}) d x_{1} \land d x_{2} \land d x_{3} .

You can check, in this example, and in general, that if $ω$ is a $k$ -form and $η$ is an $ℓ$ -form, then

ω \land η = {(- 1)}^{𝑘ℓ} η \land 𝜔.

This is called graded commutativity.

9.2 Smooth differential forms

Definition 9.2: Differential form in Euclidean coordinates

Let $U \subseteq R^{n}$ be open. A smooth $k$ -form on $U$ is an expression

ω = ∑_{| I | = k} f_{I} (𝐱) d x_{I},

where the sum runs over ordered index sets $I = (i_{1} < \dots < i_{k})$ , and every coefficient $f_{I} : U \to R$ is smooth. The vector space of smooth $k$ -forms on $U$ is denoted by $Ω^{k} (U)$ .

A smooth zero-form is simply a smooth function:

Ω^{0} (U) = C^{\infty} (U) .

The low-dimensional forms in the notes can therefore be written as follows.

Example 9.3 (Forms on $R$ ). Every one-form on an interval has the form

ω = f (x) d 𝑥.

Example 9.4 (Forms on $R^{2}$ ). Writing the coordinates as $(x, y)$ , a one-form and a two-form have the respective forms

ω = P (x, y) d x + Q (x, y) d y

and

η = h (x, y) d x \land d 𝑦.

Example 9.5 (Forms on $R^{3}$ ). Writing the coordinates as $(x, y, z)$ , the possible degrees are

\begin{array}{l} Ω^{0} (U) : & f, \\ Ω^{1} (U) : & P d x + Q d y + R d z, \\ Ω^{2} (U) : & A d y \land d z + B d z \land d x + C d x \land d y, \\ Ω^{3} (U) : & h d x \land d y \land d 𝑧. \end{array}

9.3 The exterior derivative

The exterior derivative turns a $k$ -form into a $(k + 1)$ -form.

Definition 9.6: Exterior derivative

For a smooth function $f : U \to R$ , define

d f = ∑_{i = 1}^{n} \frac{∂𝑓}{\partial x_{i}} d x_{i} .

For a general differential form

ω = ∑_{I} f_{I} d x_{I},

define

d ω = ∑_{I} d f_{I} \land d x_{I} .

For example, if

ω = P (x, y) d x + Q (x, y) d y

is a one-form on $R^{2}$ , then

\begin{array}{l} d ω & = d P \land d x + d Q \land d y \\ = (P_{x} d x + P_{y} d y) \land d x + (Q_{x} d x + Q_{y} d y) \land d y \\ = (Q_{x} - P_{y}) d x \land d 𝑦. \end{array}

The coefficient $Q_{x} - P_{y}$ is the planar curl that will appear in Green’s theorem.

The exterior derivative obeys the graded product rule recorded in the handwritten notes.

Proposition 9.7 (Graded Leibniz rule). If $ω \in Ω^{k} (U)$ and $η \in Ω^{ℓ} (U)$ , then

d (ω \land η) = d ω \land η + {(- 1)}^{k} ω \land d 𝜂.

For $k = 0$ , this is the ordinary product rule

d (𝑓𝜂) = d f \land η + f d 𝜂.

The sign in the general rule occurs because $d$ must pass across a $k$ -form before differentiating the second factor.

9.4 Vector fields and one-forms

Tapp formulates line integrals using vector fields. The Euclidean inner product identifies vector fields with one-forms.

Definition 9.8: Vector field and associated one-form

A smooth vector field on an open set $U \subseteq R^{n}$ is a smooth map

𝐅 = (F_{1}, \dots, F_{n}) : U \to R^{n} .

Its associated one-form is

ω_{𝐅} = F_{1} d x_{1} + \dots + F_{n} d x_{n} .

At $𝐩 \in U$ , this one-form acts on $𝐮 \in R^{n}$ by

{(ω_{𝐅})}_{𝐩} (𝐮) = ⟨ 𝐅 (𝐩), 𝐮 ⟩ .

Thus a vector field and its associated one-form contain the same information once the Euclidean inner product has been fixed. The one-form notation is especially useful because it makes the wedge product and exterior derivative available.

9.5 Line integrals

Let $𝜸 : [a, b] \to U$ be a smooth curve. The line integral of a one-form measures the form on the oriented velocity vector and then integrates over the parameter interval.

Definition 9.9: Line integral of a one-form

If $ω \in Ω^{1} (U)$ , define

\int_{𝜸} ω = \int_{a}^{b} ω_{𝜸 (t)} (𝜸^{'} (t)) d 𝑡.

If $ω = ω_{𝐅}$ is associated to a vector field, then

\int_{𝜸} ω_{𝐅} = \int_{𝜸} 𝐅 \cdot d 𝜸 = \int_{a}^{b} ⟨ 𝐅 (𝜸 (t)), 𝜸^{'} (t) ⟩ d 𝑡.

The formal notation $d 𝜸 = 𝜸^{'} (t) d t$ encodes this formula.

Figure 9.1: A line integral accumulates the component of the vector field in the oriented tangent direction of the curve.

The definition depends on the orientation of the curve, but not on the particular orientation-preserving parametrization.

Proposition 9.10 (Change of parameter). Let $ϕ : [α, β] \to [a, b]$ be a smooth diffeomorphism and let $\tilde{𝜸} = 𝜸 \circ 𝜙.$ If $ϕ$ is increasing, then

\int_{\tilde{𝜸}} ω = \int_{𝜸} 𝜔.

If $ϕ$ is decreasing, then

\int_{\tilde{𝜸}} ω = - \int_{𝜸} 𝜔.

Proof. For an increasing reparametrization and $ω = ω_{𝐅}$ , the chain rule gives

\begin{array}{l} \int_{\tilde{𝜸}} ω & = \int_{α}^{β} ⟨ 𝐅 (𝜸 (ϕ (s))), 𝜸^{'} (ϕ (s)) ϕ^{'} (s) ⟩ d s \\ = \int_{a}^{b} ⟨ 𝐅 (𝜸 (t)), 𝜸^{'} (t) ⟩ d t, \end{array}

where the second equality is the substitution $t = ϕ (s)$ . If $ϕ$ is decreasing, the same substitution reverses the limits of integration and introduces a minus sign. □

Example 9.11 (The work–energy identity). Suppose a particle of mass $m$ travels along $𝜸 : [a, b] \to R^{n}$ . Let

𝐯 = 𝜸^{'}, 𝐚 = 𝜸^{″}, 𝐅 = m 𝐚 .

The work done by the force along the path is

\begin{array}{l} \int_{𝜸} 𝐅 \cdot d 𝜸 & = m \int_{a}^{b} ⟨ 𝐚 (t), 𝐯 (t) ⟩ d t \\ = m \int_{a}^{b} ⟨ 𝐯^{'} (t), 𝐯 (t) ⟩ d t \\ = \frac{m}{2} \int_{a}^{b} \frac{d}{d t} {| 𝐯 (t) |}^{2} d t \\ = \frac{m}{2} {| 𝐯 (b) |}^{2} - \frac{m}{2} {| 𝐯 (a) |}^{2} . \end{array}

Thus the work equals the change in kinetic energy.

9.6 Summary

The wedge product is alternating, and a smooth $k$ -form is a sum $\sum_{| I | = k} f_{I} d x_{I}$ with smooth coefficient functions.
The exterior derivative is determined by $d f = \sum_{i} f_{x_{i}} d x_{i}$ and the graded Leibniz rule; the Euclidean inner product identifies a vector field $𝐅$ with the one-form $ω_{𝐅} = \sum_{i} F_{i} d x_{i} .$
The line integral $\int ⟨ 𝐅 (𝜸), 𝜸^{'} ⟩ d t$ is unchanged by orientation-preserving reparametrization, changes sign under orientation reversal, and represents work when $𝐅$ is a force field.

Bibliography

[Bro96] Andrew Browder. Mathematical Analysis: An Introduction. Undergraduate Texts in Mathematics. Springer, New York, first edition, 1996.

[Tap16] Kristopher Tapp. Differential Geometry of Curves and Surfaces. Undergraduate Texts in Mathematics. Springer, Cham, first edition, 2016.

Index

The numbers in this index refer to pages in the complete typeset print edition.

$C^{\infty} (I)$ , 2
$O (n)$ , 41
$R_{90}$ , 29
$Ω^{k} (U)$ , 56

acceleration, 15
    orthogonal to velocity, 13
    perpendicular component, 16
    tangential component, 16
angle
    between vectors, 8
angle function, 31, 48
antisymmetry, 55
arc length, 5
    invariance under reparametrization, 20
    lower bound by endpoint distance, 13
arc-length parametrization, 20

binormal vector, 36
Borel’s lemma, 2

Cauchy–Schwarz inequality, 8
chain rule
    for reparametrization, 19
circle
    curvature of, 24
closed curve, 21
    periodic extension, 21
component
    of a vector, 10
constant-speed curve, 13
corner
    of a curve, 52
cross product, 34
curvature
    cross-product formula, 38
    formulas for, 46
    invariance under rigid motion, 42
    of a curve, 23
    signed, 30
curve
    acceleration, 15
    arc length, 5
    closed, 21
    curvature, 23
    nonregular, 4
    parametrized, 3
    regular, 4
    reparametrization, 18
    speed, 4
    trace, 18
    unit normal vector, 24
    unit tangent vector, 24
cusp
    semicubical, 4

degree
    of a map to the circle, 48
    winding number, 51
differential
    of a map, 2
differential form, 56
    coordinate form, 55
distance
    Euclidean, 1
    preserved by a rigid motion, 40
dot product, 7

Euclidean norm, 1
extension
of a smooth function, 2
exterior
of a Jordan curve, 47
exterior derivative, 57

flat function, 2
Frenet equations, 37
Frenet frame, 36
fundamental theorem
of plane curves, 44
of space curves, 44

graded commutativity, 56
Gram–Schmidt process, 12
graph
as a parametrized curve, 3

Hopf’s Umlaufsatz, 47
generalized, 52

inner product
Euclidean, 7
interior
of a Jordan curve, 47
interval, 1

Jordan curve theorem, 47

kinetic energy, 59

Leibniz rule
    graded, 57
line integral, 58
    reparametrization invariance, 59
line segment
    shortest path, 13

negative orientation, 51

one-form
    associated to a vector field, 58
orientation
    of a curve, 19
    of a simple closed plane curve, 51
    preserved by a rigid motion, 42
    reversal of a line integral, 59
orthogonal complement, 11
orthogonal decomposition, 10
    of acceleration, 16
orthogonal matrix, 41
orthogonal vectors, 8
orthonormal basis, 11
orthonormal set, 11

parallel vectors, 8
parametrized curve, 3
periodic curve, 21
piecewise-regular curve, 52
positive orientation, 51
product rule
    for the inner product, 12
projection
    in a direction, 10
    onto a subspace, 11

regular curve, 4
reparametrization, 18
    orientation-preserving, 19
    orientation-reversing, 19
rigid motion, 40
    classification, 41
    improper, 42
    proper, 42
rotation index, 32
    as a degree, 48
rotation matrix, 29
    cross product, 35

signed angle
    at a corner, 52
signed curvature, 30
    invariance under rigid motion, 42
simple closed curve, 21, 47
smooth function, 2
smooth map, 2
smoothness
    at a boundary point, 2
speed, 4
    derivative of, 17
sphere
    velocity tangent to, 13

Taylor series
    formal, 2
torsion, 37
    invariance under rigid motion, 42
total signed curvature, 33, 52
trace of a curve, 18
translation, 41
tubular neighborhood
    of a curve, 50
turning number, 32

unit binormal vector, 36
unit normal vector
    of a curve, 24
unit tangent
    angle function, 31
unit tangent map, 48
unit tangent vector, 24
unit-speed curve
    existence of reparametrization, 20

vector field, 58
velocity of a curve, 4

wedge product, 55
winding number, 51
of the unit tangent, 32
work integral, 58
work–energy identity, 59

Preface

Organization of the notes

Course Information

Scope

How to use these notes

Conventions and Notation

Contents

Lecture 1Parametrized Curves: Regularity, Speed, and Arc Length

1.1 Review

1.1.1 Number systems and Euclidean space

1.1.2 Intervals

1.1.3 Smooth functions and maps

1.2 Parametrized curves

1.3 Velocity, regularity, and speed

1.4 Arc length

1.5 Summary

Lecture 2Inner Products, Projections, and Orthonormal Bases

2.1 The Euclidean inner product

2.2 Orthogonal projection

2.2.1 Review: Components and projections (not covered in class)

2.2.2 Review: Orthonormal sets and bases

2.3 Differentiating inner products

2.4 A geometric application: straight lines are shortest (not covered in class)

2.5 Summary

Lecture 3Acceleration and Reparametrization

3.1 Acceleration

3.1.1 Tangential and perpendicular acceleration

3.2 Trace and reparametrization

3.3 More on reparameterization (not covered in class)

3.4 Reparametrization by arc length (Covered on Friday August 28)

3.5 Closed curves (not covered in class)

3.6 Summary

Lecture 4Curvature

4.1 The curvature function

4.2 Unit tangent and unit normal vectors

4.3 Curvature of a graph at a critical point (not covered in class)

4.3.1 Computational formula

4.4 A worked example (not covered in class)

4.5 Further topics

4.6 Summary

Lecture 5Plane Curves

5.1 Rotation matrices

5.2 Signed curvature

5.2.1 The unit-speed case

5.3 Angle functions

5.4 Rotation index

5.5 Summary

Lecture 6Space Curves: Frenet Frame, Torsion, and Frenet Equations

6.1 The cross product

6.1.1 Rotations and the cross product

6.2 The Frenet frame

6.3 The Frenet equations

6.4 Some further topics

6.4.1 Zero torsion

6.4.2 Product rule

6.4.3 A curvature formula in R3

6.5 Summary

Lecture 7Rigid Motions and Fundamental Theorems

7.1 Rigid motions of Euclidean space

7.2 Geometric quantities under rigid motion

7.3 The fundamental theorems of curves

7.3.1 The plane-curve construction

7.4 A short overview of curvature formulas

7.5 Summary

Lecture 8Hopf’s Umlaufsatz and the Jordan Curve Theorem

8.1 Degree and the unit tangent map

8.2 Hopf’s Umlaufsatz

8.3 The Jordan curve theorem

8.4 A first look at the piecewise-regular formula

8.5 Summary

Lecture 9Differential Forms, Vector Fields, and Line Integrals

9.1 Coordinate forms and the wedge product

Examples

9.1.1 Wedge products of forms

9.2 Smooth differential forms

9.3 The exterior derivative

9.4 Vector fields and one-forms

9.5 Line integrals

9.6 Summary

Bibliography

6.4.3 A curvature formula in $R^{3}$