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If z is in the first quadrant then everything is in

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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 That is, Im[∞(F(z)) + i§(F(z))] = Const Since P= ∞ + i§ or §[F(z)] = const So, 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 -1 -1 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 -1 equipotential) on H. Then, by the proposition, F [z(t)] is a parametric representation of the associated streamline (or equipotential) on D. Examples (A) Plotting Isotherms Use technology to plo...
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