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\centerline{Mathematics 173 -- Applied Mathematics 1}
\centerline{Problem Set 3 -- September 28, 1999}
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\noindent
I.  Recall the definitions of the hyperbolic cosine and sine functions
from class on 9/28:
$$\cosh x = {e^x + e^{-x}\over 2} \qquad \sinh x = {e^x - e^{-x}\over 2}$$
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\item{A)}  Show that for all $x \in {\bf R}$, $\cosh^2 x - \sinh^2 x = 1$.
Consider the parametric curve $\alpha(t) = (\cosh t, \sinh t)$ in the 
plane.
How does this curve help explain the names {\it hyperbolic cosine\/} and 
{\it sine\/}?
\item{B)}  Express the derivatives of the hyperbolic functions
$\cosh x, \sinh x$, and the functions $\tanh x = {\sinh x\over \cosh x}, 
\coth x = {\cosh x\over \sinh x}, {\rm sech} x = {1\over \cosh x}, 
{\rm csch} x = {1 \over \sinh x}$ analogous to the other trig functions
 in terms of the same hyperbolic functions.
\item{C)} Show that, in strong analogy with standard trig identities,  
we have
$$\cosh 2x = \cosh^2 x + \sinh^2 x \qquad \sinh 2x = 2 \sinh x \cosh x$$
and
$$\cosh^2 x = {1 + \cosh 2x\over 2} \qquad \sinh^2 x = {-1 + \cosh 2x \over 2}$$
\item{D)}  Show that $\sinh x$ has an inverse function defined for 
all $x$: 
$$\sinh^{-1} x = \ln(x + \sqrt{x^2 + 1}).$$
\item{E)}  Here is another way the hyperbolic functions are often 
used.  Consider the indefinite integral
$$\int \sqrt{1 + x^2}\ dx$$
Make the substitution $x = \sinh t$ and use the identities in parts C and D
to express this integral as a function of $x$.
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\noindent
II.  Suppose we have a thin metal ring with constant
$c, \rho, K_0$ in the shape of 
$D = \{(r, \theta) : a \le r \le b, 0 \le \theta \le 2\pi\}$
in the polar coordinate plane.  The ring has a radially symmetric 
equilibrium temperature function $u = u(r)$ that satisfies
$${\partial u\over \partial r}(a) = \beta, \quad {\partial u\over \partial r}(b) = 
1.$$
Show that $\beta = b/a$ and interpret this result physically.  
(Note: As for the case of rectangular coordinates, 
${\partial u\over \partial r}(a)$ is proportional to the heat flux
across the inner boundary of the ring.  Similarly at $r = b$.)
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\noindent
From the text: Problems 2.3.1, 2.3.2, 2.3.3 a,d.
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