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
    \[
    f=\frac{P_1(z)}{1-z^3}+\frac{P_2(z)}{4+z^2}.
    \]

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