 Solution

ID: past-exam-of-the-geometry-course-of-the-university-of-oxford/2011/3/a/solution

For a regular parametrization

r (u, v)

I = d r \cdot d r = E d u^{2} + 2 F d u d v + G d v^{2}, E = r_{u} \cdot r_{u}, F = r_{u} \cdot r_{v}, G = r_{v} \cdot r_{v} .

It measures squared lengths of tangent vectors and therefore lengths of curves. In the plane put

x = ρ cos θ

y = ρ sin θ

. The derivatives are orthogonal, of lengths one and

ρ

, so

d x^{2} + d y^{2} = d ρ^{2} + ρ^{2} d θ^{2} .

These polar coordinates are regular for

ρ > 0

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