Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 37D / b

Table of contents
- i b
  - Solution i
- ii b
  - Solution ii

i

Solution

For plane polar coordinates,

x = r cos θ

and

y = r sin θ

. The chain rule gives

\partial_{r} = cos θ \partial_{x} + sin θ \partial_{y}, \partial_{θ} = - r sin θ \partial_{x} + r cos θ \partial_{y} .

Their Euclidean norms are

∥ \partial_{r} ∥ = 1

and

∥ \partial_{θ} ∥ = r

for

r > 0

.

ii

Solution

The unit fields are

r = \partial_{r}

and

θ = r^{- 1} \partial_{θ}

. Relation (3) and

[\partial_{r}, \partial_{θ}] = 0

give

[r, θ] = [\partial_{r}, \frac{1}{r} \partial_{θ}] = - \frac{1}{r ^{2}} \partial_{θ} = - \frac{1}{r} θ \neq = 0.

A coordinate basis must commute, so

(r, θ) is not a coordinate basis .

This is the noncommuting orthonormal polar frame.

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