Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics III / 2 / b

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- Solution b

Solution

The multivariable chain rule first gives

g_{u} = u f_{x} + v f_{y}, g_{v} = - v f_{x} + u f_{y} .

Taking a second partial derivative in each variable, with the partial derivatives of

f

evaluated at

(x (u, v), y (u, v))

, gives

g_{uu} g_{vv} = f_{x} + u^{2} f_{xx} + uv (f_{x y} + f_{y x}) + v^{2} f_{yy}, = - f_{x} + v^{2} f_{xx} - uv (f_{x y} + f_{y x}) + u^{2} f_{yy} .

The mixed partial derivatives cancel even without using their equality. Hence the Laplacians satisfy

g_{uu} + g_{vv} = (u^{2} + v^{2}) (f_{xx} + f_{yy}) .

Away from the origin the factor

u^{2} + v^{2}

is positive, so $g$ is a harmonic function if and only if $f$ is a harmonic function on the corresponding local coordinate patch. This is the Laplacian in planar parabolic coordinates.

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