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Agreement with the existing polar sphere measure

Statement

Assume ${A C}_{ω}$ and $n \geq 2$ . The chart surface measure on $S^{n - 1}$ equals the polar measure $σ$ . Orthogonal transformations preserve it. The map $ω \mapsto a + R ω$ , $R > 0$ , multiplies surface measure by $R^{n - 1}$ , and $∣ S^{n - 1} ∣ = n ∣ B_{1} ∣$ .

Facts & Assumptions

Given: Assume ${A C}_{ω}$ , $n \geq 2$ . Compare the chart measure and existing polar measure on the unit sphere. For scaling take $R > 0$ and $a \in R^{n}$ .

[F1]

Chart integrals are independent of charts and partitions. (Chart and partition independence of surface measure).

[F2]

The polar formula holds for every nonnegative Borel function. (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma).

[F3]

Borel substitution holds for C1 diffeomorphisms. (Borel change of variables from the compact-support formula and Radon uniqueness).

[F4]

Increasing simple approximation justifies the separated product integration and passage from set measures to integrals. (Monotone convergence for the integral).

Proof

1.1givenF1algebra

Write $m$ for chart measure and take a finite sphere atlas $X_{j} : U_{j} \to S^{n - 1}$ with a subordinate partition $χ_{j}$ . Since $∣ X_{j} ∣^{2} = 1$ , differentiation gives $X_{j} \cdot \partial_{i} X_{j} = 0$ . For $T_{j} (r, y) = r X_{j} (y)$ , $1 < r < 2$ , the derivative has columns $X_{j}, r \partial_{1} X_{j}, \dots, r \partial_{n - 1} X_{j}$ . Its Gram determinant is $r^{2 n - 2} \det (D X_{j}^{T} D X_{j})$ , hence $∣ \det D T_{j} ∣ = r^{n - 1} J_{j} (y)$ with $J_{j} = \sqrt{\det (D X_{j}^{T} D X_{j})} > 0$ . The inverse is $z \mapsto (∣ z ∣, X_{j}^{- 1} (z / ∣ z ∣))$ , so $T_{j}$ is a $C^{1}$ diffeomorphism onto its open annular sector.

2.1step 1.1F3F1algebraF4

For a Borel set $A \subset S^{n - 1}$ , apply F3 on each sector to $h_{j} (z) = 1_{A} (z / ∣ z ∣) χ_{j} (z / ∣ z ∣)$ . Its pullback times the Jacobian is the separated nonnegative function $r^{n - 1} 1_{A} (X_{j} (y)) χ_{j} (X_{j} (y)) J_{j} (y)$ . Integrating this product gives $C \int_{U_{j}} 1_{A} (X_{j}) χ_{j} (X_{j}) J_{j} d y$ , where $C = \int_{1}^{2} r^{n - 1} d r = (2^{n} - 1) / n > 0$ . The product integration identity follows first for simple functions of y by the defining product measure of rectangles, then by increasing simple approximation. Summing j, the partition identity shows that the volume of ${z : 1 < ∣ z ∣ < 2, z / ∣ z ∣ \in A}$ equals $C m (A)$ .

3.1step 2.1F2F1algebraF4

Applying F2 to the indicator of that same annular cone gives its volume as $C σ (A)$ . Thus $C m (A) = C σ (A)$ for every Borel A, and division by the strictly positive C gives $m = σ$ . For an orthogonal O, $D (O X)^{T} D (O X) = D X^{T} X$ , while $D (a + R X)^{T} D (a + R X) = R^{2} D X^{T} D X$ . Applying F1 chartwise proves orthogonal invariance and the factor $R^{n - 1}$ , also for integrals by simple approximation.

4.1step 3.1F2algebra∎

Use F2 with $1_{B_{1}}$ : $∣ B_{1} ∣ = \int_{0}^{1} r^{n - 1} σ (S^{n - 1}) d r = σ (S^{n - 1}) / n$ . A possible contribution at the origin is absent because the polar formula already applies to this indicator. Step 3.1 now yields the asserted chart area $∣ S^{n - 1} ∣ = n ∣ B_{1} ∣$ .

Source notes

Hunter §1.11, Proposition 1.45 and its preceding sphere parametrization, printed pp. 16–17 (PDF pp. 22–23). The identification with the existing cone-defined measure is proved here.

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Sources

Hunter, Notes on Partial Differential Equations (standard reference, not scraped)