Here is where I will be posting some suggested exercises.

Exercises:

In class, we proved Weyl’s law for one and two dimensional rectangles. Prove it for any n-dimensional rectangle.

Let U and V be domains in R^n with piecewise smooth boundary such that U is a subset of V and the boundaries of U and V are disjoint. Let f be a smooth function in U that vanishes on the boundary. Show that f extends by 0 to an element of H^1_0(V).

Verify that the map we constructed in class (see Figure 17 in the Gordon and Webb reference if needed) takes Dirichlet eigenvectors to Dirichlet eigenvectors.

Verify that the divergence of a vector field on a Riemannian manifold (M,g) agrees with the usual definition of divergence when (M,g) is R^n with its standard metric.

Spend some time thinking about why the expansion for the heat kernel that we proposed in class (k_t = \sum_k e^{-t \lambda_k} \psi_k(x) \psi_k(y) ) is reasonable.

Show that on any (M,g), div(f X) = f div(X) + g( grad(f), X ), for any function f and vector field X.

Derive the expression for the divergence in coordinates that was stated in class.

Show that the operator on \ell^2 that sends (x_1,x_2, \ldots) to (0,x_1/2, x_2/3, \ldots) is compact but has no eigenvalues

Show that the operator on L^2[0,1] mapping f(t) to tf(t) is self-adjoint but has no eigenvalues.

Roe 5.33

Roe 1.31

Roe 3.28

Roe 3.29

Roe 3.32

Roe 5.34

Roe 5.35

Roe 5.36

Roe 3.28

Roe 3.29

Roe 4.32

Roe 4.33

Roe 9.23

Roe 9.24