Math 2.0

Real functions Functions of several real variables Theorem

Change of variables in multiple integrals

Builds on Jacobian determinantDiffeomorphismC^k function characterised by continuous derivativesDouble integral

Statement

Let U⊆ℝnU \subseteq \mathbb{R}^n be open, and let φ ⁣:U→ℝn\varphi \colon U \to \mathbb{R}^n be injective and of class C^1 with det⁡Dφ(u)≠0\det D\varphi(u) \ne 0 for each u∈Uu \in U. Let f ⁣:φ(U)→ℝf \colon \varphi(U) \to \mathbb{R} be continuous with compact support in φ(U)\varphi(U). Then ∫φ(U)f(v) dv=∫Uf(φ(u)) ∣det⁡Dφ(u)∣ du.\int_{\varphi(U)} f(v) \, dv = \int_U f(\varphi(u)) \, \lvert \det D\varphi(u) \rvert \, du.

Remarks

Here det⁡Dφ(u)\det D\varphi(u) is the Jacobian determinant of φ\varphi at uu. Under these hypotheses, φ(U)\varphi(U) is open and φ\varphi is a diffeomorphism from UU onto φ(U)\varphi(U). The factor ∣det⁡Dφ(u)∣\lvert \det D\varphi(u) \rvert is the ratio by which φ\varphi scales small volumes near uu. A linear map AA maps the unit cube to a parallelepiped of volume ∣det⁡A∣\lvert \det A \rvert. The formula extends to Lebesgue integrable ff on φ(U)\varphi(U), and to maps that fail the hypotheses only on a set of measure zero. Important cases are integration in polar coordinates, cylindrical coordinates and spherical coordinates.

Reading path

To understand this page from the ground up, read these 22 pages in order. The foundations come first.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFunction composition, concept, no evidence levelFunction compositionconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptContinuity in R^n, definition, no evidence levelContinuity in R^ndefinitionDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionDouble integral, definition, no evidence levelDouble integraldefinitionChain rule in several variables, statement, no evidence levelChain rule in several…statementDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionJacobian matrix, definition, no evidence levelJacobian matrixdefinitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionDerivative of the inverse function, statement, no evidence levelDerivative of the inv…statementIterated derivative, definition, no evidence levelIterated derivativedefinitionJacobian determinant, definition, no evidence levelJacobian determinantdefinitionC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementInverse function theorem, statement, no evidence levelInverse function theo…statementDiffeomorphism, definition, no evidence levelDiffeomorphismdefinitionChange of variables in multiple integrals, statement, no evidence levelChange of variables i…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Euclidean space object
  2. Function concept
  3. Function composition concept
  4. Fréchet derivative definition
  5. Function of several variables concept
  6. Limit of a function of several variables concept
  7. Partial derivative concept
  8. Continuity in R^n definition
  9. Differentiable function of several variables definition
  10. Partial derivative definition
  11. Double integral definition
  12. Chain rule in several variables statement
  13. Differentiable on an open set definition
  14. Jacobian matrix definition
  15. Higher partial derivatives definition
  16. Derivative of the inverse function statement
  17. Iterated derivative definition
  18. Jacobian determinant definition
  19. C^k function characterised by continuous derivatives statement
  20. Inverse function theorem statement
  21. Diffeomorphism definition
  22. Change of variables in multiple integrals statement

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Integration_by_substitution, Substitution for multiple variables
About this page
Tag00N5FHP1CV
URIm2://00N5FHP1CV
Aliasessubstitution-rule-several-variables, jacobian-change-of-variables
Kindstatement, theorem
WritingW0 stub
MSC26B15
StatusThis copy of the site has no build data. Run m2 status 00N5FHP1CV.
UsesJacobian determinant, Diffeomorphism, C^k function characterised by continuous derivatives, Double integral

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