\(\def\pd{\partial}

\def\Real{\mathbb{R}}

\)

- 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 \((I_+(t),I_-(t)),0\leq t\leq 1\). - 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}

\] - 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, \(xdy\wedge dz+…\). - Find the integral of

\[

\int\frac{1}{\sin{z}}

\]

over the contour \(\{|z|=2\}\) oriented counterclockwise. - 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}.

\]

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