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Unformatted text preview: the streamlines and equipotentials on D.
Proof The associated complex potential on D is, as we have seen, given by composition:
Q = PõF
Therefore its streamlines are specified by setting imaginary part equal to a constant:
Im(P(F(z)) = Const
Im[∞(F(z)) + i§(F(z))] = Const
Since P= ∞ + i§
§[F(z)] = const
z is in a specific streamline on D
¤ §[F(z)] = K (K a specific constant)
¤ w = F(z) is in the associated streamline §(w) = K on H.
In other words, streamlines go under F to streamlines. Put another way, streamlines in H
map to streamlines of D under F (since F is the inverse of F, under which streamlines
correspond to streamlines.) The argument for equipotentials is similar.
Consequence: How to graph streamlines & equipotentials
Suppose that z(t) = x(t) + iy(t) is a parametric representation of a streamline (or
equipotential) on H. Then, by the proposition, F [z(t)] is a parametric representation of
the associated streamline (or equipotential) on D.
(A) Plotting Isotherms
Use technology to plo...
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