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§3.3. 1 Sturm-Liouville theorem: orthogonal eigenfunctions

Explore the Sturm-Liouville Theorem and orthogonal eigenfunctions in linear function spaces. Review and compare rectangular boundary value problems in vectors and eigenstuff. Learn about Cartesian variables separation and the fundamental differences between vectors and functions. Discover how functions can be manipulated pointwise and the significance of continuous versus discrete vector spaces. Visualize components as function values at points and grasp the essence of the Sturm-Liouville Theorem.

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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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