Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
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 · next 3 levels
Direct dependencies and their dependencies through the next three levels: 174 results over 30 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- A. Leibman, Multidimensional Real Analysis, §5.5 (standard reference, not scraped)