1. §3.3.1 Sturm-Liouville theorem: orthogonal eigenfunctions
Christopher Crawford
PHY 416

2. Outline
Review of eigenvalue problem Linear function spaces: Sturm-Liouville theorem Review of rectangular BVP in term of vectors / eigenstuff
Separation of Cartesian variables: Plane waves: exponentials

3. Vectors vs. Functions
Functions can be added or stretched (pointwise operation)
Continuous vs. discrete vector space
Components: function value at each point
Visualization: graphs, not arrows ` `

4. Vectors vs. Functions ` `

5. Sturm-Liouville Theorem
Laplacian (self-adjoint) has orthogonal eigenfunctions
This is true in any orthogonal coordinate system!
Sturm-Liouville operator – eigenvalue problem
Theorem: eigenfunctions with different eigenvalues are orthogonal

6. Rectangular box: eigenfunctions
Boundary value problem: Laplace equation

7. Rectangular box: components
Boundary value problem: Boundary conditions 7