 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2015/prelims/mathematics-iii/2/a/solution

In a partial derivative, the variable held fixed matters. Holding

u

fixed gives

x_{v} = - v

. To find

v_{y}

, hold

x

fixed and invert the Jacobian matrix:

(d x d y) = (u v - v u) (d u d v), (d u d v) = \frac{1}{u ^{2} + v ^{2}} (u - v v u) (d x d y) .

Thus

\frac{\partial x}{\partial v}_{u} = - v, \frac{\partial v}{\partial y}_{x} = \frac{u}{u ^{2} + v ^{2}} .

The second formula holds on each local inverse branch away from

(u, v) = (0, 0)

(u, v)

and

(- u, - v)

represent the same

(x, y)

, so no global choice of inverse is implicit.

 New to topics? Read the docs here!