Fundamental nuclear symmetries meet classical electrodynamic ...

Fundamental nuclear symmetries meet classical electrodynamic ...

3.3.1 Separation of Cartesian variables: Orthogonal functions Christopher Crawford PHY 311 2014-02-24 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 Laplaces 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 Boundary value problem: Boundary conditions 6 Example: rectangular box Boundary value problem:

Laplace equation 7 Example: rectangular box Boundary value problem: Boundary conditions

8 Vectors vs. Functions 9 Vectors vs. Functions

10 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

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