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Polar change of variables on a compact annular sector gives the Jacobian factor and its area
Example
On , the polar map is injective with Jacobian factor . Its annular-sector image has area .
Facts & Assumptions
Given: The polar map and compact parameter rectangle .
Sine and cosine have their standard derivatives and satisfy (The derivatives of sine and cosine are cosine and minus sine, Parity and the Pythagorean identity for sine and cosine).
Cosine is strictly decreasing on (Signs, monotonicity intervals, and ranges of sine and cosine).
Compact-Jordan change of variables uses the absolute Jacobian determinant (Change of variables for an injective map on a compact Jordan set).
Verification
Differentiation and [L1] give
Equality of two images first gives equality of radii by [L1], then equality of cosines; [L2] gives equality of angles. The same recovery works on the open neighborhood , where never vanishes. Thus the compact theorem's neighborhood hypotheses hold.
Applying [L3] to the constant-one function and integrating gives
Depends on
- Change of variables for an injective $C^1$ map on a compact Jordan set
- The Jacobian determinant of a square-dimensional $C^1$ map is the determinant of its Jacobian matrix
- The derivatives of sine and cosine are cosine and minus sine
- Parity and the Pythagorean identity for sine and cosine
- The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G' = f$ and $f$ is integrable, then $\int_a^b f = G(b)-G(a)$
- For a natural $n \ge 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $\iota(n)\,x^{\,n-1}$; for $n = 0$ it is the constant $1$, with derivative $0$; for a natural $n \ge 1$ the function $x \mapsto x^{-n}$ is differentiable at every $x \ne 0$ with derivative $-\iota(n)\,x^{-n-1}$; consequently every polynomial function is differentiable at every real, with the derivative computed term by term
- Signs, monotonicity intervals, and ranges of sine and cosine
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
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Sources
- A. Leibman, Multidimensional Real Analysis, §5.5 (standard reference, not scraped)