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# problems1 - Math Review 1 1 Problem set#1 Basic Math review...

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Math Review 1 1 Problem set #1 Basic Math review, matlab warmup and Conservation equations Do all problems. 1. Consider the two dimensional function ψ ( x,y ) = sin( mπx ) sin( nπy ) on the unit square 0 x 1 , 0 y 1 . (a) Find F = ψ (b) Find V = × ψ k (c) Find · F = 2 ψ (d) Find · V and explain why we consider V to be “incompressible” (e) Use Matlab to create a figure that superposes a vector plot of V on a contour plot of ψ for the unit square. Show results for m = 4 and n = 1 . In 2-D, ψ is said to be the “streamfunction” for the velocity field V . Why? 2. Eigen fun (a) Find all solutions to the 2-point boundary value problem d 2 f dx 2 + λf = 0 on the unit line 0 x 1 with boundary conditions f (0) = f (1) = 0 . What are the eigenvalues λ ? (b) Now consider the discrete eigenvalue problem A N x = λ x where A N is the N × N tridiagonal matrix A N = ( N + 1) 2 2 - 1 - 1 2 - 1 - 1 2 - 1 .

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problems1 - Math Review 1 1 Problem set#1 Basic Math review...

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