MATH 396 PROBLEMS 7

IGOR KRIZ

Regular problems:

1. Prove that ℝ^{3} with the cross-product

( \begin{matrix}x \\ y \\ z \end{matrix} ) × ( \begin{matrix}t \\ u \\ v \end{matrix} ) = det ( \begin{matrix}e_{1} & x & t \\ e_{2} & y & u \\ e_{3} & z & v \end{matrix} )

is a Lie algebra, i.e. satisfies

\overrightarrow{x} × \overrightarrow{y} = −\overrightarrow{y} × \overrightarrow{x},
(\overrightarrow{x} × \overrightarrow{y}) × \overrightarrow{z} + (\overrightarrow{y} × \overrightarrow{z}) × \overrightarrow{x} + (\overrightarrow{z} × \overrightarrow{x}) × \overrightarrow{y} = 0.

2. Using Green’s formula, explain how you would find the potential of a function f : U → ℝ^{2} (U ⊆ ℝ^{2} open), i.e. a function F : U → ℝ such that f ( \begin{matrix}x \\ y \end{matrix} ) = ( \begin{matrix}∂F/∂x \\ ∂F/∂y \end{matrix} ), if one exists. Apply to the function f which assigns to a point ( \begin{matrix}x \\ y \end{matrix} ) the vector which is a positive multiple of ( \begin{matrix}x \\ y \end{matrix} ) and has length 1/(x^{2} + y^{2}).

3. Let f : ℝ^{2} → ℝ^{2} be given by f ( \begin{matrix}x \\ y \end{matrix} ) = ( \begin{matrix}xe^{−y^{2}} \\ −x^{2}ye^{−y^{2}} + 1/(x^{2} + y^{2}) \end{matrix} ). Cal-

culate ∫_{L} f where L is the boundary of the square with vertices ( \begin{matrix}−a \\ −a \end{matrix} ), ( \begin{matrix}a \\ −a \end{matrix} ),

( \begin{matrix}a \\ a \end{matrix} ), ( \begin{matrix}−a \\ a \end{matrix} ).

4. Calculate the area of the parametrized surface

ϕ ( \begin{matrix}u \\ v \end{matrix} ) = ( \begin{matrix}u cos v \\ u sin v \\ u^{2} \end{matrix} ) ,

0 ≤ u ≤ 4, 0 ≤ v ≤ 2π.

Challenge problem:

5. Prove (find an example) that the Green formula is false for a continuous function f : ℝ^{2} − {0} → ℝ^{2} with continuous partial derivatives, if f is not required to be bounded. [Hint: recall some complex curve integrals.]