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  \begin{document}


  
  \vskip 1cm

  \centerline{\Large {Exterior derivative and the Stokes theorem}}
\vskip 1cm

\subsection*{Exterior derivative}

\begin{enumerate}[1.]

\item \label{arg}
  Define the following $(n-1)$-form $\omega$ on $\R^n\smallsetminus\{0\}$:
  $$\omega=\sum_{k=1}^n (-1)^k\frac{x_k~ dx_1\wedge dx_2\wedge\cdots
  \wedge \widehat{dx_k}\wedge\cdots\wedge
  dx_n}{(x_1^2+x_2^2+\cdots+x_n^2)^{n/2}}$$
  Note that the denominator is $\|x\|^n$.
  \begin{enumerate}[(i)]
    \item Show that $\omega$ is invariant under scaling, i.e. if
      $f:\R^n\smallsetminus\{0\}\to \R^n\smallsetminus\{0\}$ is
      defined by $f(x)=\lambda x$ for $\lambda\neq 0$, then
      $f^*(\omega)=\omega$.
    \item Show that $\omega$ is a closed form, i.e. $d\omega=0$.
  \end{enumerate}


\item Let $\omega$ be a 1-form on a manifold $M$ and $X,Y$ two smooth vector
  fields on $M$. As a check that you understand all definitions, show
  that
  $$2~ d\omega(X,Y)=X\omega(Y)-Y\omega(X)-\omega([X,Y])$$
  Here, e.g. $\omega(Y)$ is the smooth function on $M$ obtained by
  plugging in the vector $Y(p)$ into $\omega$ for every $p\in M$ and
  $X\omega(Y)$ is the derivative of this function along $X$. The
  bracket $[X,Y]$ is the Lie bracket. This identity can be used to
  define the exterior derivative of a 1-form without local
  coordinates. There are analogous and more complicated formulas for
  higher order forms, see Spivak. You should write it out in coordinates and for
  simplicity you may assume that $\dim M=2$. Thus $\omega=f~dx+g~dy$
  and by linearity you can assume e.g. $\omega=f~dx$.
  
\end{enumerate}    


\subsection*{Stokes Theorem and calculus}



\begin{enumerate}[resume]

\item Show that the Stokes theorem for a closed interval $[a,b]$ is
  just the Fundamental Theorem of Calculus.

\item Derive Green's formula in the plane from the Stokes theorem: If
  $W\subset\R^2$ is a compact surface with boundary $\gamma=\partial
  W$ and $f,g:\R^2\to\R$ are smooth, then
  $$\int_\gamma f~ dx+ g~ dy=\int_W \bigg(\frac{\partial g}{\partial
    x}-\frac {\partial f}{\partial y}\bigg) dx~ dy$$

\item Derive the Divergence theorem from the Stokes theorem: Let
  $W\subset\R^3$ be a compact 3-manifold and let $\vec{n}$ be the
  outward normal to $\partial W$. If $\vec{F}=(f_1,f_2,f_3)$ is a
  vector field in $\R^3$ then
  $$\int_W (div \vec{F}) dx~ dy~ dz=\int_{\partial
    W}(\vec{n}\cdot\vec{F})~ dA$$
  where $dA$ is the area form on $\partial A$.

\item Derive the (classical) Stokes theorem from the Stokes
  theorem(!):
  Let $S\subset\R^3$ be a compact orented surface with boundary and $\vec{F}=(f_1,f_2,f_3)$ a
  vector field in $\R^3$. Then
  $$\int_S (curl \vec{F}\cdot n) dA=\int_{\partial S}
  f_1dx_1+f_2dx_2+f_3dx_3$$
  where $\vec{n}$ is a unit normal to $S$ compatible with orientations
  and $dA$ is the area form.

\end{enumerate}

\subsection*{More Stokes}

\begin{enumerate}[resume]

  \item Let $W$ be a compact oriented manifold with boundary $\partial
    W=X$, and let $n=\dim X$. If $f:W\to Y$ is a smooth map and
    $\omega\in\Omega^n(Y)$ a closed form then $\int_X f^*(\omega)=0$.

  \item Let $f,g:X\to Y$ be smooth homotopic maps with $X$ compact,
    oriented, and $\partial X=\empty$. Let $n=\dim X$, and
    $\omega\in\Omega^n(Y)$ a closed form. Show that $\int_X
    f^*(\omega)=\int_X g^*(\omega)$.

  \item \label{integral}
    Let $M$ be a compact oriented $n$-manifold without
    boundary. Let $\omega\in\Omega^n(M)$, so $\omega$ is closed. Show
    that if $\omega$ is exact then $\int_M \omega=0$. More generally,
    show that if $\omega$ is an exact $p$-form on a manifold $X$ and
    $M\subset X$ is an oriented compact $p$-manifold without boundary
    then $\int_M \omega=0$.
    Normally,
    one shows that a form is not exact by computing an integral like
    this and showing it is not 0.

  \item \label{arg2}
    Show that the form from Problem \ref{arg} is not exact. Hint:
    By Problem \ref{integral}, it suffices to show that $\int_{S^{n-1}}\omega\neq
    0$. This might be a painful calculation, but here is a trick. The
    form $\eta=\sum_k (-1)^k x_k
    dx_1\wedge\cdots\wedge\widehat{dx_k}\wedge\cdots \wedge dx_n$ is
    defined on all of $\R^n$,
    agrees with $\omega$ on $S^{n-1}$ and $d\eta$ is a multiple of the
    volume form on $\R^n$, so apply Stokes on the unit ball.

    \item Show that the form $\omega$ (or equivalently $\eta$) from
      Problem \ref{arg2} is the volume form on $S^{n-1}$. This is of
      course harder than Problem \ref{arg2}. Hint: One approach is
      this. First check that this is true at the north pole
      $(0,0,\cdots,0,1)$. Then show that $\omega$ (and $\eta$) are
      invariant under rotations in 2 coordinates, i.e. under the maps
      such as $$(x_1,x_2,\cdots,x_n)\mapsto
      (x_1\cos\alpha+x_2\sin\alpha,-x_1\sin\alpha+x_2\cos\alpha,\cdots,x_n)$$
      and finally show that any point on the sphere can be rotated to
      the north pole by composing such 2-coordinate rotations.
  

\end{enumerate}



\end{document}
