Toronto Math Forum
MAT3342018F => MAT334Tests => Thanksgiving bonus => Topic started by: Victor Ivrii on October 06, 2018, 05:08:29 AM

Read section 2.1.1* and solve the problem (10 karma points). As a sample see solutions to odd numbered problems
Problem 8. Decide whether or not the given function represents a locally sourceless and/or irrotational flow. For those that do, decide whether the flow is globally sourceless and/or irrotational. Sketch some of the streamlines.
$$
\cosh(x)\cos(y)+i\sinh(x)\sin(y).
$$

$cosh(x)cos(y)+isinh(x)sin(y)$
Then
$u=cosh(x)cos(y)$
$v=sinh(x)sin(y)$
$\frac{du}{dx} = sinh(x)cos(y)$
$\frac{du}{dy}= cosh(x)sin(y)$
$\frac{dv}{dx} = cosh(x)sin(y)$
$\frac{dv}{dy} = sinh(x)cos(y)$
since $\frac{du}{dx} \neq  \frac{dv}{dy}$
$\frac{du}{dy} \neq \frac{dv}{dx}$
Thus, the flow is not globally sourceless and is not irrotational.

sorry, there is a typo. The solution should be the given function doesn't represent a locally sourceless and an irrotational flow

First, correct your typesetting, replacing cos by \cos and so on
Also not \frac{du}{dx} but \frac{\partial u}{\partial x} and so on (on Quiz/Test/Exam in any math discipline starting from Calculus II it will cost you.
On Page 92 (not in problems) there are 2 errors in formulae. Find them and type a correct version.

Let $f(u,v) = u + iv$ where $u = \cosh{x}\cos{y}$ and $v=\sinh{x}\sin{y}$
Note that
\begin{equation*}
\begin{aligned}
\frac{\partial u}{\partial x} &= \sinh{x}\cos{y}\\
\frac{\partial v}{\partial y} &= \sinh{x}\cos{y}\\
\frac{\partial u}{\partial y} &= \cosh{x}\sin{y}\\
\frac{\partial v}{\partial x} &= \cosh{x}\sin{y}
\end{aligned}
\end{equation*}
Since
\begin{equation*}
\begin{aligned}
\frac{\partial u}{\partial x} &= \frac{\partial v}{\partial y} \\
\frac{\partial u}{\partial y} &\neq \frac{\partial v}{\partial x}
\end{aligned}
\end{equation*}
We know that $f$ is not analytic on $D$ by the contrapositve of CauchyRiemann Theorem.
So, $f$ is not sourceless or irrotational on $D$.
The error you are talking about is that the textbook used $dx$ instead of $\partial x$
Thus, the proper equations should be
\begin{equation*}
\begin{aligned}
\int_{\gamma} u dx+v dy &= \iint_{\Omega} \left[\frac{\partial v}{\partial x}\frac{\partial u}{\partial y}\right]dxdy=0 \\
\int_{\gamma} u dy  v dx &= \iint_{\Omega} \left[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right]dxdy=2 \text{ area}(\Omega)
\end{aligned}
\end{equation*}