Math 2.0

Real functions Proposition

Jacobian of spherical coordinates

Builds on Spherical coordinatesJacobian determinantChange of variables in multiple integrals

Statement

Let S(ρ,φ,θ)=(ρsin⁡φcos⁡θ, ρsin⁡φsin⁡θ, ρcos⁡φ)S(\rho, \varphi, \theta) = (\rho \sin \varphi \cos \theta, \ \rho \sin \varphi \sin \theta, \ \rho \cos \varphi) be the spherical coordinate map. Its Jacobian determinant is det⁡DS(ρ,φ,θ)=ρ2sin⁡φ.\det DS(\rho, \varphi, \theta) = \rho^2 \sin \varphi. Hence, for a continuous function ff on a region E⊆ℝ3E \subseteq \mathbb{R}^3 described by a region E′E' of (ρ,φ,θ)(\rho, \varphi, \theta) with ρ≥0\rho \ge 0, 0≤φ≤π0 \le \varphi \le \pi, 0≤θ≤2π0 \le \theta \le 2\pi, ∭Ef dV=∭E′f(S(ρ,φ,θ)) ρ2sin⁡φ dρ dφ dθ.\iiint_E f \, dV = \iiint_{E'} f(S(\rho, \varphi, \theta)) \, \rho^2 \sin \varphi \, d\rho \, d\varphi \, d\theta.

Remarks

The Jacobian matrix has rows (sin⁡φcos⁡θ, ρcos⁡φcos⁡θ, −ρsin⁡φsin⁡θ)(\sin \varphi \cos \theta, \ \rho \cos \varphi \cos \theta, \ -\rho \sin \varphi \sin \theta), (sin⁡φsin⁡θ, ρcos⁡φsin⁡θ, ρsin⁡φcos⁡θ)(\sin \varphi \sin \theta, \ \rho \cos \varphi \sin \theta, \ \rho \sin \varphi \cos \theta) and (cos⁡φ, −ρsin⁡φ, 0)(\cos \varphi, \ -\rho \sin \varphi, \ 0). Expansion along the last row gives ρ2sin⁡φ\rho^2 \sin \varphi. The integral formula follows from the change of variables theorem. The set where SS is not injective, or where the determinant vanishes, has volume zero. Since 0≤φ≤π0 \le \varphi \le \pi, the factor ρ2sin⁡φ\rho^2 \sin \varphi is non-negative, so no absolute value is needed. Example: the volume of the ball of radius aa is ∫02π∫0π∫0aρ2sin⁡φ dρ dφ dθ=43πa3\int_0^{2\pi} \int_0^\pi \int_0^a \rho^2 \sin \varphi \, d\rho \, d\varphi \, d\theta = \tfrac{4}{3} \pi a^3.

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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 levelDiffeomorphismdefinitionPolar coordinates, definition, no evidence levelPolar coordinatesdefinitionChange of variables in multiple integrals, statement, no evidence levelChange of variables i…statementSpherical coordinates, definition, no evidence levelSpherical coordinatesdefinitionJacobian of spherical coordinates, statement, no evidence levelJacobian of spherical…statement
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  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. Polar coordinates definition
  23. Change of variables in multiple integrals statement
  24. Spherical coordinates definition
  25. Jacobian of spherical coordinates statement

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Further reading

  1. Web https://en.wikipedia.org/wiki/Multiple_integral, Spherical coordinates
  2. Web https://en.wikipedia.org/wiki/Spherical_coordinate_system, Integration and differentiation in spherical coordinates
About this page
Tag05WKRC7HWT
URIm2://05WKRC7HWT
Aliasesspherical-volume-element
Kindstatement, proposition
WritingW0 stub
MSC26B15
StatusThis copy of the site has no build data. Run m2 status 05WKRC7HWT.
UsesSpherical coordinates, Jacobian determinant, Change of variables in multiple integrals

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