Exercises
Problem 1
Verify that u(x,y)=Exp[x] Cos[y] satisfies Laplace's equation. Use the
information on the Cauchy-Riemann equations to find the "stream
function" and plot the equipotential curves and the stream lines.
Problem 2
Verify that u(x,y)=1/6 x^4 - x^2 y^2 + 1/6 y^4 satisfies Laplace's equation.
Find the stream function and plot the equipotential curves and the streamlines.
Hint: Where is this in the list of polynomial solutions?
Problem 3
Why can't we use the Cauchy-Riemann equations to find the stream function for
the potential u(x,y)=x^2 y - x y^2 ?
Problem 4
Find the stream function and plot the equipotential curves and the streamlines
for u(x,y)=log[sqrt(x^2 + y^2)].
You may need the PlotPoints option to avois jagged equipotentials and
streamlines.
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