Points of the plane are given new labels and .

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

Points of the plane are given new labels

u = x + y

and

v = x - y

. ((a)) Write

x

and

y

in terms of

u

and

v

, and use the chain rule to express

(\partial f/\partial u)_v

and

(\partial f/\partial v)_u

through

f_x

and

f_y

for any

f(x,y)

. ((b)) Check both results on

f = x^2 - y^2

, which in the new labels is

uv

, at the point

x = 3

y = 1

. ((c)) Find

(\partial x/\partial u)_v

and

(\partial u/\partial x)_y

, and explain why their product is not

1

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