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Spherical coordinates have absolute Jacobian determinant $r^{2} \sin ϕ$ away from the axis and angular seam

Example

For $S (r, ϕ, θ) = (r \sin ϕ \cos θ, r \sin ϕ \sin θ, r \cos ϕ),$ one has $∣ \det D S ∣ = r^{2} \sin ϕ$ on $[1, 2] \times [π / 6, π / 3] \times [π / 6, π / 3] .$ The map is injective there, away from both polar axes and the angular seam.

Facts & Assumptions

Given: The spherical map and the compact parameter box in the statement.

[L1]

Sine and cosine have their standard derivatives and satisfy the Pythagorean identity (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 $[0, π]$ , and sine has no zero strictly between $0$ and $π$ (Signs, monotonicity intervals, and ranges of sine and cosine, The zero sets of sine and cosine and the least positive common period 2 pi). With [L1], sine is therefore positive on the displayed polar-angle interval.

[L3]

Compact-Jordan change of variables applies to injective $C^{1}$ maps with invertible derivative (Change of variables for an injective $C^{1}$ map on a compact Jordan set).

Verification

technique · computation

1.1

Differentiating the three coordinates and expanding by columns, then using [L1], gives $\det D S = r^{2} \sin ϕ$ . It is positive on the parameter box by [L2].

L1L2

2.1

The image norm recovers $r$ ; the quotient of the third coordinate by $r$ recovers $ϕ$ through strict cosine monotonicity; the first two normalized coordinates then recover $θ$ . The same recovery works when both angles range in $(π / 12, 5 π / 12)$ and $r \in (1 / 2, 5 / 2)$ , so $S$ is injective with nonzero determinant on an open neighborhood of the compact box.

L1L2step 1.1

3.1

The positive determinant and step 2.1 verify every hypothesis of [L3], so spherical integration on this box carries the factor $r^{2} \sin ϕ$ .

L3step 2.1∎

Depends on

Used by

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Sources

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

Spherical coordinates have absolute Jacobian determinant r2sin⁡ϕr2sinϕ away from the axis and angular seam