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§3.3. 1 Sturm-Liouville theorem: orthogonal eigenfunctions

§3.3. 1 Sturm-Liouville theorem: orthogonal eigenfunctions. Christopher Crawford PHY 416 2014-10-27. Outline. Review of eigenvalue problem Linear function spaces: Sturm-Liouville theorem Review of rectangular BVP in term of vectors / eigenstuff

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§3.3. 1 Sturm-Liouville theorem: orthogonal eigenfunctions

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  1. §3.3.1 Sturm-Liouville theorem:orthogonal eigenfunctions Christopher Crawford PHY 416 2014-10-27

  2. Outline • Review of eigenvalue problemLinear function spaces: Sturm-Liouville theoremReview 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

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