# midterm 2

$$\def\pd{\partial} \def\Real{\mathbb{R}}$$

1. Consider the closed cycle $$\gamma:[0,1]\to\Real^2$$ shown on the left (we assume that $$\gamma(0)=\gamma(1)$$ is the dot on the positive $$x$$-axis). Define the functions
$I_-(t)=\int_0^t \left(\frac{(x+1)dy -ydx}{(x+1)^2+y^2}.\dot{\gamma}(t)\right)dt,\quad I_+(t)=\int_0^t \left(\frac{(x-1)dy -ydx}{(x-1)^2+y^2}.\dot{\gamma}(t)\right)dt,$
given as partial integrals over the cycle $$\gamma$$. Sketch the parametric plot of $$(I_+(t),I_-(t)),0\leq t\leq 1$$.
2. Are the following forms $$\omega_i$$ closed? If so, find their primitives, that is forms $$\alpha_i: \omega_i=d\alpha_i$$:
$\begin{array}{rl} \omega_1=2xydx+x^2dy+z^2dz&;\\ \omega_2=z^2dx+x^2dy-xzdz&;\\ \omega_3=-3zdx\wedge dz+2xdx\wedge dy&.\\ \end{array}$
3. Consider the 2-chain on the unit sphere in 3D given by the spherical cap
$\gamma=\{x^2+y^2+z^2=1, z\geq 1/2\}.$
Find
$\int_\gamma x^2 \sigma,$
where $$\sigma$$ is the standard area form on the sphere, $$xdy\wedge dz+…$$.
4. Find the integral of
$\int\frac{1}{\sin{z}}$
over the contour $$\{|z|=2\}$$ oriented counterclockwise.
5. Find the Laurent series expansion for
$f(z)=\frac{1}{(1-z^3)(4+z^2)}$
converging for $$1<|z|<2$$.

Hint: use elementary fraction decomposition
$f=\frac{P_1(z)}{1-z^3}+\frac{P_2(z)}{4+z^2}.$