PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Calculus of change and accumulation · Calculus of change and accumulation
Mathematical Methods for Physics, Ch Partial differentiation, chain rules and Jacobians
Problem
The map x=u2−v2, y=2uv sends each point (u,v) to a point (x,y).
((a)) Find its Jacobian J=∂(x,y)/∂(u,v), and say where it vanishes.
((b)) Show that (u,v) and (−u,−v) always land on the same point. J is not zero at either of them unless both are the origin; explain why this does not contradict the statement that where J=0 the labels are one to one near that point.
((c)) Using the value of J at the centre of the small square 1≤u≤1.02, 0≤v≤0.02, estimate the area of its image, and compare it with the area of the square.