Question: Verify by direct substitution that the functions (r^{n} cos (n theta)) etc. shown in the text satisfy the Laplace equation in polar coordinates. Calculate the

Verify by direct substitution that the functions $r^{n} \cos (n \theta)$ etc. shown in the text satisfy the Laplace equation in polar coordinates.

Calculate the streamfunction for the above-mentioned potential function and construct a contour-plot of streamlines for $n=1 / 2,1$, and 3/2. MATLAB or MATHEMATICA may be used to make these contour-plots.

Also construct a contour-plot of equipotential lines for any one of the above cases and show that these lines are perpendicular to the streamlines.

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