Day 15

Thursday, March 30, 2017

Section 5.5, cont'd

3 proofs of Sturm-Liouville facts

orthogonality of eigenfunctions

reality of eigenvalues

uniqueness of eigenfunctions

2017_03_30/b1.png 2017_03_30/b2.png 2017_03_30/b3.png 2017_03_30/b4.png 2017_03_30/b5.png

Section 5.6: Rayleigh quotient for SL problems

Derivation

Conditions guaranteeing positivity of eigenvalues

Minimization principle: statement, utility, proof

2017_03_30/b6.png 2017_03_30/b7.png 2017_03_30/b8.png

Correction: the class of functions we're minimizing over can be extended to functions that are continuous on [a,b], have a piecewise continuous derivative on (a,b), and satisfy the BCs.

Next: Examples and proof of minimization principle