Physics I / Change of Variables and the Jacobian

Practice question · Put in order

Order the steps of a change of variables in a double integral.

Compute the Jacobian $\partial(x,y)/\partial(u,v)$
Solve for $x(u,v)$ and $y(u,v)$
Evaluate ∬ f( $x(u,v)$ , $y(u,v)$ ) |J| du dv
Transform the integrand and limits
Define new variables $u = u(x,y), v = v(x,y)$

Hints

The substitution must be made before the scale factor is inserted.
The limits are rewritten in the new variables last.

Show the answer

Define new variables $u = u(x,y), v = v(x,y)$
Solve for $x(u,v)$ and $y(u,v)$
Compute the Jacobian $\partial(x,y)/\partial(u,v)$
Transform the integrand and limits
Evaluate ∬ f( $x(u,v)$ , $y(u,v)$ ) |J| du dv

Why

Change of variables: substitute, scale by |J|, adjust limits.

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