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Cylindrical coordinates have absolute Jacobian determinant $r$ on an injective compact box

Example

The cylindrical-coordinate map $C (r, θ, z) = (r \cos θ, r \sin θ, z)$ is injective on $[1, 2] \times [π / 6, π / 3] \times [- 1, 1]$ and has absolute Jacobian determinant $r$ there.

Facts & Assumptions

Given: The cylindrical map on the displayed parameter box.

[L1]

Sine and cosine have their standard derivatives and satisfy $\sin^{2} + \cos^{2} = 1$ (The derivatives of sine and cosine are cosine and minus sine, Parity and the Pythagorean identity for sine and cosine).

[L2]

Cosine is strictly decreasing on $[0, π]$ (Signs, monotonicity intervals, and ranges of sine and cosine).

[L3]

Compact-Jordan change of variables applies on injective boxes with nonzero Jacobian (Change of variables for an injective $C^{1}$ map on a compact Jordan set).

Verification

technique · computation

1.1

The derivative matrix is block triangular over the polar block, and [L1] gives $\det D C = \det (\begin{array}{ccc} \cos θ & - r \sin θ & 0 \\ \sin θ & r \cos θ & 0 \\ 0 & 0 & 1 \end{array}) = r .$

2.1

The third image coordinate recovers $z$ . The first two recover $r$ from their Euclidean norm and then $θ$ from strict cosine monotonicity [L2]. The same recovery works on $(1 / 2, 5 / 2) \times (π / 12, 5 π / 12) \times (- 2, 2)$ , where $r > 0$ , so the map is injective with invertible derivative on an open neighborhood of the compact box.

L1L2step 1.1

3.1

Therefore [L3] applies on this seam-free compact box, and every transformed volume integral carries precisely the factor $r$ .

L3step 2.1∎

Depends on

Used by

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Sources

A. Leibman, Multidimensional Real Analysis, §5.5 (standard reference, not scraped)

Cylindrical coordinates have absolute Jacobian determinant rr on an injective compact box