MATH 396 PROBLEMS 8

IGOR KRIZ

Regular problems:

1. Compute the area of a sphere of radius a using surface integrals.

2. If S is the surface of the sphere x^{2} = y^{2} + z^{2} = a^{2}, compute the surface integral of second kind

∫_{S}(xz, yz, x^{2}).

3. The cylinder x^{2} + y^{2} = 2x cuts out a portion of a surface S from the upper nappe of the cone x^{2} + y^{2} = z^{2}. Compute the value of the surface integral of the first kind

∫_{S}(x^{4} − y^{4} + y^{2}z^{2} − z^{2}x^{2} + 1).

4. Let S be the hemisphere x^{2} + y^{2} + z^{2} = 1, z ≥ 0. Using Stokes’ theorem, calculate

∫_{S}(y^{2}, xy, xz).

Challenge problems:

5. Let A be a k × n matrix, k ≤ n. Prove that the sum of the squares of the determinants of all k × k submatrices of A is equal to det(AA^{T}).

6. Can you find a function f : ℝ → ℝ such that f(f(x)) = e^{x} for all x?