- Section 1.6: Gibbs' phenomenon and the sawtooth function
- Section 1.6: Gibbs' phenomenon in 3D (a complex-valued, piecewise smooth, discontinuous function and partial sums of its Fourier series)
- Section 2.3: "Thermal imaging" of heat flow in a bar, with Neumann boundary conditions (cf. Example 2.3.1. The initial position function is f(x)=x. Red portions of the bar are warmest; blue portions are coolest. See equation (2.64) for the analytic solution)
- Section 2.3: "Thermal imaging" of heat flow in a bar, with mixed boundary conditions (cf. Example 2.3.2. Same set-up, color coding, and initial temperature function as in the previous video. See equation (2.74) for the analytic solution)
- Section 2.4: "Thermal imaging" of heat flow in a bar, with periodic boundary conditions (cf. Example 2.4.1. Same set-up, color coding, and initial temperature function as in the previous video. See equation (2.87) for the analytic solution)
- Section 2.7: A solution to the wave equation, illustrating the principle (as reflected by the differential equation (2.149)) that acceleration is proportional to concavity
- Section 2.9: D'Alembert's solution the wave equation. We assume the initial velocity g to be zero for simplicity, so that d'Alembert's formula (2.187) reads y(x,t)=(f_odd(x+ct)+f_odd(x-ct))/2. That is, we have the graph of f_odd/2 moving left with velocity c, plus the graph of f_odd/2 moving right with velocity c. To make the interaction of these two waves more apparent, we have dashed them in beyond the actual, physical boundaries x=0 and x=ell of the string itself (which appears as a solid curve)
- Section 2.11: The Tacoma Narrows Bridge Collapse - a rather dramatic illustration of the phenomenon of resonance (or not, depending on whom you ask)
- Section 3.1: The Cauchy sequence of Figure 3.2: this sequence converges in norm to zero, but does not converge pointwise on ANY set of real numbers (to illustrate, the label at x=pi turns red when g_N(pi)=0, blue when g_N(pi) is strictly between 0 and 1, and green when g_N(pi)=1)
- Section 3.2: The Cauchy sequence of Figure 3.3 (f_1 through f_22 in red; the function f to which the f_N's converge pointwise in black)
- Section 4.2 (see also Section 2.4): "Thermal imaging" of heat flow in a bar, with Newton's law of cooling boundary conditions (cf. Examples 2.4.2 and 4.2.1. Same set-up, color coding, and initial temperature function as in the above thermal imaging examples from Chapter 2)
- Section 5.7: Convolution with approximate identities I. A discontinuous function f (in BLACK), and f*g_epsilon (in RED), with epsilon decreasing as time evolves. Here f and g are as in the top portion of Figure 5.3
- Section 5.7: Convolution with approximate identities II. A continuous function f (in BLACK), and f*g_epsilon (in RED), with epsilon decreasing as time evolves. Here f and g are as in the bottom portion of Figure 5.3
- Section 5.7: Limits of approximate identities I. The approximate identity g_epsilon of Figure 5.4, with epsilon decreasing as time evolves.
- Section 5.7: Limits of approximate identities II. The approximate identity g_epsilon generated by g(x)=C (4+x^2-3 x^4+ 8 x^5-x^6 +5 cos(6 x)-5 x^3 sin(7 x))/ (1+x^8) (C being the constant necessary to give g mass one), with epsilon decreasing as time evolves.