The map takes the quarter-plane onto the upper half-plane

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch Manifolds, charts and tangent spaces

Problem

The map F(x,y)=(x2−y2, 2xy)F(x,y) = (x^2-y^2,\,2xy) takes the quarter-plane x,y>0x, y > 0 onto the upper half-plane. At the point (x,y)=(2,1)(x,y) = (2,1) a tangent vector v=3 ∂x−∂yv = 3\,\partial_x - \partial_y is given. Find the components of the pushforward F∗vF_*v in the target coordinates (u,w)=(x2−y2,2xy)(u,w) = (x^2-y^2, 2xy).

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