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Polar coordinates are a local diffeomorphism away from zero radius

Example

For

$P (r, θ) = (r \cos θ, r \sin θ),$

the restriction to either half-plane $r > 0$ or $r < 0$ is a local diffeomorphism. It is locally orientation-preserving when $r > 0$ and locally orientation-reversing when $r < 0$ . On $(0, \infty) \times (- π, π)$ it is a diffeomorphism onto its open image, but on $(0, \infty) \times R$ it is not injective.

Facts & Assumptions

Given: The polar map above, the inverse-function consequence An injective regular $C^{1}$ map is a diffeomorphism onto its image, the Pythagorean identity Parity and the Pythagorean identity for sine and cosine, the fundamental period of sine and cosine The zero sets of sine and cosine and the least positive common period 2 pi, and their bijective parametrization of the unit circle on a half-open interval $t \mapsto (\cos t, \sin t)$ is a bijection from $[0, 2 π)$ onto the real unit circle.

[L1]

The functions $\sin$ and $\cos$ are differentiable on $R$ , with $(\sin x)^{'} = \cos x$ and $(\cos x)^{'} = - \sin x$ (The derivatives of sine and cosine are cosine and minus sine).

[L2]

A regular $C^{1}$ map is locally orientation-preserving where $\det D f > 0$ and locally orientation-reversing where $\det D f < 0$ (Local orientation of a regular $C^{1}$ Euclidean map).

[L3]

Products of differentiable real functions are differentiable and satisfy the product rule (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

[L4]

A $C^{1}$ Euclidean map with invertible derivative at a point restricts to a $C^{1}$ diffeomorphism between neighbourhoods of that point and its image (The Euclidean inverse function theorem).

Verification

technique · direct

1.1

By [L1] and [L3], $D P (r, θ) = (\begin{array}{cc} \cos θ & - r \sin θ \\ \sin θ & r \cos θ \end{array}), \det D P (r, θ) = r .$ Thus [L4] and [L2] give the asserted local diffeomorphism and orientation wherever $r \neq 0$ .

L1L2L3L4algebra

2.1

On the principal strip, equality of two images first gives equality of the positive radii by the Pythagorean identity. Translating each negative angle by $2 π$ puts both angles into the half-open interval of the unit-circle parametrization without changing sine or cosine; its injectivity then gives equality of the original angles. The injective regular-map theorem makes this restriction a diffeomorphism onto its open image. On the full positive-radius domain, $(r, θ)$ and $(r, θ + 2 π)$ are distinct with the same image, so global injectivity fails there.

step 1.1givenalgebra∎

Depends on

Used by

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Sources

J. Lebl, Basic Analysis II, Exercise 8.5.8 (standard reference, not scraped)
University of Toronto MAT237, §3.3 (standard reference, not scraped)