§ Separation of Cartesian variables: Orthogonal functions Christopher Crawford PHY

Outline Analogies from Chapter 1 Inner, outer products Projections, eigenstuff Example boundary value problem – square box Separation of variables Solution of boundary conditions Discrete vs. continuous linear spaces Basis, components; index notation Inner product; orthogonal projection; rotations Eigenvalues – Sturm-Liouville theorem 2

Analogies from Chapter 1 Inner vs. outer product – projections & components Linear operators – stretches & rotations 3

Separation of variables technique Goal: solve Laplace’s equation (PDE) by converting it into one ODE per variable Method: separate the equation into separate terms in x, y, z start with factored solution V(x,y,z) = X(x) Y(y) Z(z) Trick: if f (x) = g (y) then they both must be constant Endgame: form the most general solution possible as a linear combination of single-variable solutions. Solve for coefficients using the boundary conditions. Trick 2: use orthogonal functions to project out coefficients Analogy: the set of all solutions forms a vector space the basis vectors are independent individual solutions 4

Example: rectangular box Boundary value problem: Laplace equation 5

Example: rectangular box 6 Boundary value problem: Boundary conditions 6

Example: rectangular box Boundary value problem: Laplace equation 7

Example: rectangular box 8 Boundary value problem: Boundary conditions 8

Vectors vs. Functions 9

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Vectors vs. Functions 11

Sturm-Liouville Theorem Laplacian (self-adjoint) has orthogonal eigenfunctions – This is true in any orthogonal coordinate system! Sturm-Liouville operator – eigenvalue problem 12