Existing in Mathlib · Analysis
Change of Variables Formula
An injective map that is differentiable within a measurable set transforms integration over that set into integration over its image, with the absolute determinant of the derivative correcting for local volume change.
- change of variables
- Bochner integral
- Fréchet derivative
- Jacobian determinant
- additive Haar measure
- measurable embedding
Exact theorem
Exact Mathlib statement
theorem MeasureTheory.integral_image_eq_integral_abs_det_fderiv_smul {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {s : Set E} {f : E → E} {f' : E → E →L[ℝ] E} [MeasurableSpace E] [BorelSpace E] (μ : Measure E) [IsAddHaarMeasure μ] (hs : MeasurableSet s) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) (g : E → F) : ∫ x in f '' s, g x ∂μ = ∫ x in s, |(f' x).det| • g (f x) ∂μThe theorem at a glance
Change of variables at a glance
A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.
Change of variables at a glance
Detailed visual description
The poster foregrounds the measurable-set, within-set differentiability, and injectivity hypotheses with the editorial integral formula. Its central plate pairs a bounded source region with a non-self-overlapping image region and compares one local source cell with its stretched image cell. The retained transcript then names the source route through restriction, weighted-measure transport, measurable-embedding integration, and scalar-density rewriting, while the scope footer rules out a global diffeomorphism or derivative-invertibility assumption.
Statement structure
Statement and scope
Read the exact Mathlib declaration
This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.
Six determinant-weighted local transports
Detailed visual description
Two bounded warm-paper mesh regions contain exactly six blue source point-cell pairs and six corresponding image point-cell pairs. Six left-to-right noncrossing arrows preserve their vertical order, the image cells vary in area, and six separate antique-gold contour disks at the far left vary in intensity.
Why it matters
A mathematical landmark
The change-of-variables formula is one of the central bridges between differentiation, geometry, measure, and integration. Mathlib's selected theorem is notable for its precise set-level form: within-set differentiability and injectivity support a vector-valued integral identity under weaker hypotheses than the global diffeomorphism version often taught first.
ProofAtlas record
What has been checked
Mathlib is the source of the theorem; the local Lean replay and page review are separate.
Claim boundary
No new theorem is claimed
This page indexes Mathlib's finite-dimensional Bochner-integral change-of-variables declaration for a measurable set s and a map f that is injective and differentiable within s with recorded derivative f'. The endpoint is exactly the equality between the integral of an F-valued function g on f '' s and the source integral weighted by the absolute determinant of f'. It does not assume or conclude a global diffeomorphism, global injectivity or surjectivity, an open domain, a continuously differentiable map, or an everywhere-invertible derivative. The formal declaration quantifies over g : E → F without a separate measurability or integrability hypothesis; its exact Mathlib set-integral semantics remain authoritative.
- The selected declaration concerns the image of one measurable set and does not state that f is a global diffeomorphism or even globally injective.
- The hypotheses specify a derivative within s but do not separately assume that f is continuously differentiable or that its derivative is invertible at every point.
- The determinant appears through its absolute value as a real scalar acting on an F-valued function; this is not restricted to scalar-valued integrands.
- The selected formal type has no separate measurability or integrability assumption on g; readers should use the exact Mathlib set-integral semantics rather than silently importing a textbook side condition.
- ProofAtlas is indexing an existing Mathlib theorem, and generated explanations and visuals are not proof evidence.
Source and local evidence
Where the theorem comes from
- Existing declaration
MeasureTheory.integral_image_eq_integral_abs_det_fderiv_smulin mathlib- Relationship
- The checked artifact is the upstream declaration itself
- Local Lean evidence
artifact.library.mathlib.change-of-variables-formula.v001