Outline of the class 1 Vector calculus Vector

Outline of the class: 1. Vector calculus • Vector algebra • Scalar and vector fields • Differential calculus: Gradient, divergence, curl • Integral calculus: Line integrals, surface integrals (flux), volume integrals • Basic theorems: Green’s, divergence, Stokes’ 2. Fourier analysis: Series, transforms, generalization to ‘orthogonal functions’ (Bessel functions, Legendre/associate-Legendre polynomials, spherical harmonics…): The Sturm-Liouville theory 3. Partial differential equations (PDEs) • The wave equation • The diffusion (heat) equation • The Laplace and Poisson equations • The continuity equation 4. Complex analysis

Partial Differential Equations (Chapter 12) 1. Basic ideas and examples 2. The wave equation (Sec. 12. 2) • One spatial dimension: A violin string • Separation of variables, the solution as a Fourier series (Sec. 12. 3) • Example: A violin string plucked in the middle • One-dimensional waves in free space (an infinitely-long string) • Two spatial dimensions (Sec. 12. 8) • Vibration of a square membrane (Sec. 12. 9) • Laplacian in polar coordinates (Sec. 12. 10) • Vibrations of a circular membrane (a drum) • Bessel equation and Bessel functions • The Fourier-Bessel series • Three spatial dimensions (Sec. 12. 11) • Laplacian in spherical and cylindrical coordinates

Examples of partial differential equations (PDEs)

Partial differential equations • Equation in an unknown u(r) with r in a domain D • u(r) must satisfy boundary conditions (Laplace, Poisson) and/or initial (wave, continuity, diffusion) on a ‘surface’ in the domain For example: For a circular membrane (that is, a drum), described by its vertical displacement u from equilibrium at position (r, φ) at time t, u(r, φ, t), one must specify boundary conditions u(R, φ, t) = 0 (the membrane does not move on the rim of the drum at r=R) u(r, φ, t=0) = f(r, φ) (initial position of the membrane) initial condition(s) ∂ u(r, φ, t=0)/∂t = g(r, φ) (initial velocity of the membrane)

Partial differential equations • For a homogeneous linear problem L[u(r)] = 0 if u 1(r) and u 1(r) are two solutions, that is: L[u 1(r)] = 0 and L[u 2(r)] = 0, then: au 1(r)+b u 2(r) is also a solution, the same situation we have for ordinary differential equations