MATH 396 PROBLEMS 9

IGOR KRIZ

Regular problems:

1. Let S be the image of [0, 1]×[0, 1] under the map ℝ^{2} → ℝ^{3} given by x = r+s^{2}, y = r − s. Using Stokes’ theorem, calculate the integral of f(x, y, z) = (x^{2} + y, z − y, y^{3}) over the boundary of S.

2. Prove that if f : ℂ → ℂ satisfies f(z) ∈ ℝ for all z ∈ ℂ, and f has complex derivative everywhere, then f is constant. [Consider the total differential.]

3. Using Gauss’ formula, calculate the surface integral of second kind of the function f(x, y, z) = (xy, xz, z^{2}) over the surface of the unit ball {(x, y, z)|x^{2} + y^{2} + z^{2} ≤ 1} (oriented so that the normal vector points outward).

4. (a) Using the definition, calculate the complex derivative of the function f(z) = \frac{1}{z−i}.

(b) A function f : ℂ → ℂ is given by f(x+iy) = e^{x^{2}+y^{2}+1}(cos(2xy)+i sin(2xy)). Is it holomorphic? [Use Cauchy-Riemann conditions.] Challenge problem:

5. Consider the 2-form in ℝ^{4}

ω = (x + t + z)^{2}dx ∧ dy + e^{x}dy ∧ dt + (x − y)dx ∧ dt + 3xydx ∧ dz.

(a) Let S be the image of the square [0, 1] × [0, 1] under the map ℝ^{2} → ℝ^{4} given by x = rs, y = r^{2} + s^{2}, z = r − s, t = s. Calculate ∫_{S} ω.

(b) Using Stokes’ theorem, calculate the integral of ω over the boundary of the cube {(x, y, z, t)|0 ≤ x, y, z ≤ 1, t = 2}.