MATH 396 PROBLEMS 6

IGOR KRIZ

Regular problems:

1. Calculate the line integral (=curve integral of second kind) of the function

f ( \begin{matrix}x \\ y \end{matrix} ) = ( \begin{matrix}x^{2} − 2xy \\ y^{2} − 2xy \end{matrix} ) from ( \begin{matrix}−1 \\ 1 \end{matrix} ) to ( \begin{matrix}1 \\ 1 \end{matrix} ) along the parabola y = x^{2}.

2. A force field (i.e function assigning to each point a force vector) in ℝ^{3} is given by the formula

f ( \begin{matrix}x \\ y \\ z \end{matrix} ) = ( \begin{matrix}yz \\ xz \\ x(y + 1) \end{matrix} ) .

Calculate the work done by f in moving a particle once around the triangle with

vertices ( \begin{matrix}0 \\ 0 \\ 0 \end{matrix} ), ( \begin{matrix}1 \\ 1 \\ 1 \end{matrix} ), ( \begin{matrix}−1 \\ 1 \\ −1 \end{matrix} ) in that order.

3. Calculate the complex curve integral of the function f(z) = 1/z counterclockwise over

(a) the circle in ℂ with center in the origin and radius r

(b) the square with vertices in the points 1 + i, −1 + i, −1 − i, 1 − i

(c) the circle with radius 1 and center 2.

4. Using Greene’s formula, evaluate the line integral (=curve integral of sec-

ond kind) of the function f ( \begin{matrix}x \\ y \end{matrix} ) = ( \begin{matrix}xe^{−y^{2}} \\ −x^{2}ye^{−y^{2}} + 1/(x^{2} + y^{2}) \end{matrix} ) around the

boundary of the square consisting of all points ( \begin{matrix}x \\ y \end{matrix} ) such that |x| ≤ 1, |y| ≤ 1.

Challenge problem:

5. Calculate the complex curve integral of the function z^{n}, n ∈ ℤ over the unit circle in ℂ.