Question

Prob. 12.8 Criticize the following 'proof that r = 0. (a) Apply Green's theorem in a plane to the functions P(x, y) = tan-(y/x) and Q(x, y) = tan-'(x/y), taking the region R to be the unit circle centered on the origin. (b) The RHS of the equality so produced is S SR dxdy, which, either from symmetry considerations or by changing to plane polar coordinates, can be shown to have zero value. (c) In the LHS of the equality, set x = cos 6 and y = sin 0, yielding P(O) = 0 and QO) = /2–6. The line integral becomes 13" [(1 - 0) cos 6 – 6 sin 6] do, which has the value 2. (d) Thus 27 = 0 and the stated result follows.

Answer 1

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