HW#8
—Phys374—Spring 2008
Prof. Ted Jacobson
Due before class, Friday, April 11, 2007
Room 4115, (301)4056020
www.physics.umd.edu/grt/taj/374b/
[email protected]
1. (a) Evaluate
z dz
along the following two contours connecting

1 to 1: (i) from

1
to 1 along the real axis, and (ii) along the semicircle of radius 1 in the complex plane.
(Do both contour integrals explicitly. Use the polar angle as the variable of integration
for (ii), and use
x
=
Re
(
z
) as the variable of integration for (i).) Explain how you
could have known the two integrals would agree without even evaluating them. (b)
Repeat part (a) for the integral
z
*
dz,
and explain why the two contour integrals do
not
agree in this case. [7+3=10 pts.]
2.
Fluid flow and analytic functions
: Problem 16.3 c,d,e,f [2+3+3+2=10 pts.]
3.
Vortex
: The velocity potential for a point source of fluid flow is given by the real
part of
h
(
z
) =
k
ln
z
(where
k
is a constant), as shown in the previous problem. Show
that the
imaginary part
of
h
(
z
) is the velocity potential for a point vortex. Do this by
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 Fall '10
 Jacobson
 Physics, Cartesian Coordinate System, Polar coordinate system, boundary condition

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