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Unformatted text preview: ı 2πr ds,
taken around the circle, giving
ı 2πr (r dø) = c
0 So, the strength of the point source specified by F(z) = c
lnz is just c.
2π (B) Combining Sources and Sinks
Since the flux integral is zero away from any singularity, it follows that we can just add
fields like the above to get an arbitrary configurations of sources and sinks with specified
strengths ci by taking a sum of terms :
F(z) = £i i ln(z - zi)
lnz. Its imaginary part is given by the real part of the log, of
the magnitude, which tells us that the flow is circular. To see it exactly (and in which
direction it goes) compute the velocity field:
v = F'(z) =
2π(x + y )
which circulates counterclockwise if K is positive. To get the moment, we note that, since
the circulation is in the xy-plane, it only has one coordinate: the z-coordinate. Therefore
we need compute only one path integral (in the xy-plane). Now, actually we don't even
need to evaluate the path integral, because of the following facts:
(1) The complex path integral of 1/z around such a circle is equal to 2πi.
(2) The imaginary part of the integral of 1/z (namely, 2π) equals the negative of t...
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