The map , sends coordinates to (it is the real and imaginary parts of with…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch Differential forms and the exterior derivative

Problem

The map

x = u^2 - v^2

y = 2uv

sends coordinates

(u,v)

(x,y)

(it is the real and imaginary parts of

w \mapsto w^2

with

w = u+iv

). ((a)) Compute

\mathrm{d}x

and

\mathrm{d}y

in terms of

\mathrm{d}u, \mathrm{d}v

. ((b)) Form

\mathrm{d}x \wedge \mathrm{d}y

and express it as

J(u,v)\,\mathrm{d}u \wedge \mathrm{d}v

, giving

J(u,v)

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