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\def\d{{\mathrm d}}
\def\e{{\mathrm e}}
\def\R{{\mathbb R}}
\def\ba{\boldsymbol{a}}
\def\bb{\boldsymbol{b}}
\def\bc{\boldsymbol{c}}
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\def\bn{\boldsymbol{n}}
\def\bg{\boldsymbol{g}}
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\begin{document}

\title{Advanced Calculus and \\Methods of Mathematical Physics}
\author{Homework 9}
\date{Due via Gradescope, Friday, May 9, before 20:00}
\maketitle

\emph{Note:} Assignments marked (*) will not be graded.  Do not turn
them in.  However, they will be discussed in the tutorial and example
solutions can be found in the appendix of Kantorovitz' book.  

\begin{enumerate}

\item Prove the following identities for vectors $\ba, \bb, \bc, \bd
\in \R^3$. 
\begin{enumerate}
\item The
\emph{``BAC--CAB-identity''}
\[
  \ba \times (\bb \times \bc)
  = \bb \, (\ba \cdot \bc) - \bc \, (\ba \cdot \bb) \,.
\]
\item The \emph{Jacobi identity} in three dimensions
\[
  \ba \times (\bb \times \bc)
  + \bb \times (\bc \times \ba)
  + \bc \times (\ba \times \bb) = 0 \,.
\]
  
\item The \emph{Cauchy--Binet formula} in three dimensions
\[
  (\ba \times \bb) \cdot (\bc \times \bd)
  = (\ba \cdot \bc) \, (\bb \cdot \bd)
  - (\ba \cdot \bd) \, (\bb \cdot \bc) \,.
\]
\end{enumerate}
All three identities are important and worth memorizing.  Note that
(c) directly implies the identity
\[
  \lVert \ba \times \bb \rVert^2
  = \lVert \ba \rVert^2 \, \lVert \bb \rVert^2
  - (\ba \cdot \bb)^2
\]
we used in class.  

\item *(Kantorovitz, Exercises 4.5.7, Problem 6.)  A smooth curve is
given in the $yz$-plane by the parameterization
\[
  \gamma(t) = (0, y(t), z(t)) \,, \quad t \in [a,b] \,.
\]
The surface $M$ is obtained by revolving $\gamma$ about the $z$-axis.
\begin{enumerate}
\item Show that $M$ has surface area
\[
  \sigma(M)
  = 2 \pi \int_\gamma \lvert y \rvert \, \d s 
  = 2 \pi \int_a^b
    \lvert y(t) \rvert \, \lVert \gamma' (t) \rVert \, \d t \,.
\]
\item Take $\gamma$ to be the circle centered at $(0,R,0)$ with radius
$r \in (0,R)$ in the $yz$-plane, so that $M$ is a torus.  Find the
area of the torus $M$.
\end{enumerate}

\item Prove \emph{Pappus' theorem}: If $M$ is the surface obtained by
revolving a plane curve $\gamma$ of length $L = \Lambda(\gamma)$ about
an axis in the plane of the curve at distance $h$ from the centroid of
the curve, then
\[
  \sigma(M) = 2 \pi Lh \,.
\]

\emph{Hint:} Use the formula from Question~2(a).


\item Show, by explicit computation, that the surface area of a smooth
regular surface $M$ with parameterization $f \in C^1 (U \to \R^3)$,
\[
  \sigma(M) = \int_U \lVert n \rVert \, \d S
  = \int_U \bigg\lVert \frac{\partial f}{\partial u_1} \times
  \frac{\partial f}{\partial u_2}  \bigg\rVert \, \d u
\]
is independent of the parameterization.  I.e., if $g \in C^1(V, \R^3)$
is another smooth regular parameterization with $g = f \circ \phi$ for
some $phi \in C^1(V,U)$, then
\[
  \sigma(M) 
  = \int_V \bigg\lVert \frac{\partial g}{\partial v_1} \times
  \frac{\partial g}{\partial v_2}  \bigg\rVert \, \d v \,.
\]
\emph{Hint:} Use chain rule, change-of-variable formula, and the
properties of the cross-product.


\item *(Kantorovitz, Exercises 4.5.7, Problem 4.)  Let $F=(xy, 0,
-z^2)$, $D = [0,1]^3$, and $M=\partial D$ oriented such that the
normal vector points outwards.  Calculate the flux
\[
  \Phi = \int_M F \cdot n \, \d \sigma
\]
\begin{enumerate}
\item by applying the divergence theorem,
\item directly.
\end{enumerate}

\item *(Kantorovitz, Exercises 4.5.7, Problem 7.)  Let $\gamma$ be the
closed curve parameterized by
\[
  \gamma(t) = (\cos t, \sin t, \cos 2 t) \,, \quad t \in [0, 2 \pi] \,.
\]
Let $M$ be the portion of the hyperbolic paraboloid $S$ defined by the
equation
\[
  z = x^2 - y^2
\]
with boundary $\gamma$ (note that $\gamma$ lies on $S$!).  Calculate
the line integral
\[
  \int_\gamma F \cdot \d x
\]
for the vector field $F=(x^2+z^2,y,z)$
\begin{enumerate}
\item by applying Stokes' theorem,
\item directly.
\end{enumerate}


\item Use Stokes' theorem to compute the line integral
\[
  \int_\gamma x^2 \, y^3 \, \d x + \d y + z \, \d z
\]
where $\gamma$ is the circle $x^2+y^2=a^2$ in the $xy$-plane.

\item Let $D \subset \R^k$ be the simplex with $k$ of its faces faces
on the coordinate hyperplanes, namely
\[
  S_j \subset \{x \in \R^k \colon x_j = 0 \}\,, \quad j = 1, \dots, k \,,
\]
and the final face $S$ in the interior of the first orthant, i.e.,
\[
  S \subset \{ x \in \R^k \colon x_i \geq 0, i=1, \dots, k \} \,.
\]
Show that the outward normal vector $n$ on $S$ has coordinates
\[
  n_j = \frac{\sigma(S_j)}{\sigma(S)} \,.
\]

\emph{Hint}: Use the divergence theorem for smartly chosen vector fields.



\end{enumerate}
\end{document}
