∇
VECTOR Calc 1–4
4d Streak
120 XP
Mode A: Guided Applet
x₀ = 1.00| f(x₀) = 0.750| f'(x₀) = 0.250
Dynamic Parameters
Target Point x₀ (x0x_0)1.0
Epsilon Corridor ε (ε\varepsilon)0.50
Delta Window δ (δ\delta)0.35
Visual Overlays & Layers
Chapter 5 Theory

Fourier Series & 3D Partial Differential Equations (Heat & Wave)

Eigenfunction superposition, harmonic modes, and 3D surface evolution

Scrub time t on the 3D surface u(x,y,t) to watch 2D standing wave modes cos(nx)cos(my)cos(cωt) and thermal diffusion e^(-αkt) evolve in 3D.

1. Heat Diffusion vs. Wave Propagation in 3D

In the Heat Equation u_t = α∇²u, the rate of temperature change is proportional to spatial curvature (Laplacian), smoothing sharp peaks exponentially via e^(-αλt). In the Wave Equation u_tt = c²∇²u, acceleration is proportional to curvature, creating standing waves.

∂u∂t=α∇2u(Heat),∂2u∂t2=c2∇2u(Wave)\frac{\partial u}{\partial t} = \alpha \nabla^2 u \quad (\text{Heat}), \qquad \frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u \quad (\text{Wave})

Quick Reference & Cheat Sheet

Fourier Sine & Cosine Coefficients
an=1L∫−LLf(x)cos⁡(nπxL)dx,bn=1L∫−LLf(x)sin⁡(nπxL)dxa_n = \frac{1}{L}\int_{-L}^L f(x)\cos\left(\frac{n\pi x}{L}\right)dx, \quad b_n = \frac{1}{L}\int_{-L}^L f(x)\sin\left(\frac{n\pi x}{L}\right)dx
Orthogonal projection of f(x) onto harmonic basis waves.