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\title{Advanced Calculus and \\Methods of Mathematical Physics}
\author{Homework 8}
\date{Due via Gradescope, Friday, April 24 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 Recall the definition for a line integral of a vector field $F
\in C(D,\R^k)$ on a domain $D \subset \R^k$ along a smooth curve
$\gamma \in C^1([a,b],D)$,
\[
  \int_\gamma F \cdot \d x
  = \int_a^b F(\gamma(t)) \cdot \gamma'(t) \, \d t \,.
\]
Prove, by explicit calculation, that this definition is independent of
the choice of parametrization of the curve.


\item *(Kantorovitz, Exercises 4.3.15, Problem 1.) Let $\gamma$ be the
helix parameterized by
\[
  \gamma(t) = (a \cos t, a \sin t, b t)
\]
with $a,b>0$.
\begin{enumerate}
\item Find the arc length $s(t)$ of the arc $\{ \gamma(\tau) \colon 0
\leq \tau \leq t \}$.
\item Find the length of one turn of the helix.
\item Let $F=(-y,x,z)$ be a vector field in $\R^3$.  Calculate the
line integral
\[
  \int_\gamma F \cdot \d x
\]
over the turn of the helix $t \in [0, 2\pi]$.

\item Calculate the line integral
\[
  \int_\gamma \frac1{\lVert x \rVert} \, \d s
\]
over the same turn of the helix.
\end{enumerate}


\item *(Kantorovitz, Exercises 4.3.15, Problem 3.)  The curve $\gamma
\subset \R^2$ is given in polar coordinates by the $C^1$ function
\[
  r = g(\theta) \,,
  \quad \theta \in [a,b] \,.
\]
\begin{enumerate}
\item Show that the arc length function on $\gamma$ is given by
\[
  s(\theta)
  = \int_a^\theta \sqrt{g'(\tau)^2 + g(\tau)^2} \, \d \tau \,.
\]
\item For $g (\theta) = 1-\cos \theta$, $[a,b]=[0,2\pi]$, find the
length of the curve $\gamma$.
\end{enumerate}


\item (Kantorovitz, Exercises 4.3.15, Problem 4.)  Let $\gamma$ be the
circle centered at the origin with radius $r$.  Calculate
\[
  \int_\gamma F \cdot \d x
\]
for
\begin{enumerate}
\item $F(x) = (x_1-x_2,x_1+x_2)$,
\item $F(x) = \nabla \phi$ with $\phi(x) = \arctan(x_2/x_1)$.
\end{enumerate}
Explain the difference between case (a) and case (b) in light of the
theory.

\item (Kantorovitz, Exercises 4.3.15, Problem 6.)  Let $\gamma$ be the
ellipse with equation
\[
  \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \,,
\]
with counter-clockwise orientation.  Let
\[
  F(x,y)
  = (y^2 / (1+x^2), 2 y \arctan x) \,.
\]
Calculate
\[
  \int_\gamma F \cdot \d x \,.
\]

\item *(Kantorovitz, Exercises 4.4.5, Problem 1.) Find the area $S(D)$
of the domain $D\subset \R^2$ with given boundary $\partial D =
\gamma$, where $\gamma$ is the \emph{cardioid} parameterized by
\[
  \gamma(t) = (1-\cos t) (\cos t, \sin t) \,,
  \quad t \in [0,2\pi].
\]

\item *(Kantorovitz, Exercises 4.4.5, Problem 2.) As a above, where
$\gamma$ is the \emph{hypocycloid} satisfying the equation
\[
  \Bigl( \frac{x}a \Bigr)^{\tfrac23}
  + \Bigl( \frac{y}b \Bigr)^{\tfrac23} = 1
\]
with $a,b>0$.

\emph{Hint:} find a trigonometric parametrization.

\item (Kantorovitz, Exercises 4.4.5, Problem 3.) As a above, where
$\gamma$ is the curve with parametrization
\[
  \gamma(t) = (t^2-1) (1,t) \,,
  \quad t \in [-1,1] \,.
\]

\item (From Moskowitz/Paliogiannis, p.~460.) Evaluate the following
line integrals along a curve $\gamma$, both directly and by using
Green's theorem:
\begin{enumerate}
\item $\displaystyle \int_\gamma (1-x^2) \, y \, \d x + (1+y^2) \, x
\, \d y$, where $\gamma$ is the unit circle in anti-clockwise orientation.
\item $\displaystyle \int_\gamma x \, y^2 \, \d y - x^2 \, y \, \d x$,
where $\gamma$ is the boundary of the annulus $1 \leq x^2 + y^2 \leq
4$, in standard orientation.
\item $\displaystyle \int_\gamma  (y - \sin x) \, \d x + \cos x \, \d
y$, where $\gamma$ is the perimeter of the triangle with vertices
$(0,0)$, $(\tfrac\pi2,0)$, and $(\tfrac\pi2,1)$ in counter-clockwise
orientation. 
\end{enumerate}
\end{enumerate}
\end{document}
