Math 2.0

Measure and integration Theorem

Volume of a ball in R^n

Builds on Euclidean spaceChange of variables in multiple integralsFubini's theorem

Statement

Let n≥1n \ge 1, let R>0R > 0, and let Bn(R)={x∈ℝn:∥x∥≤R}B_n(R) = \{x \in \mathbb{R}^n : \|x\| \le R\}. Then vol⁡(Bn(R))=πn/2Γ ⁣(n2+1) Rn,\operatorname{vol}(B_n(R)) = \frac{\pi^{n/2}}{\Gamma\!\left(\frac{n}{2} + 1\right)} \, R^n, where Γ(s)=∫0∞ts−1e−t dt\Gamma(s) = \int_0^\infty t^{s - 1} e^{-t} \, dt is the gamma function.

Remarks

The change of variables x=Rux = R u shows that vol⁡(Bn(R))=Rnvol⁡(Bn(1))\operatorname{vol}(B_n(R)) = R^n \operatorname{vol}(B_n(1)). Since Γ(1)=1\Gamma(1) = 1, Γ(1/2)=π\Gamma(1/2) = \sqrt{\pi} and Γ(s+1)=s Γ(s)\Gamma(s + 1) = s \, \Gamma(s), the formula gives 2R2R, πR2\pi R^2, 43πR3\tfrac{4}{3} \pi R^3 and 12π2R4\tfrac{1}{2} \pi^2 R^4 for n=1,2,3,4n = 1, 2, 3, 4. One proof computes ∫ℝne−∥x∥2 dx=πn/2\int_{\mathbb{R}^n} e^{-\|x\|^2} \, dx = \pi^{n/2} in two ways: as a product of nn Gaussian integrals by Fubini's theorem, and by integration over spheres of radius rr. The volume of the unit ball tends to 00 as n→∞n \to \infty.

Reading path

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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…conceptIterated integral, definition, no evidence levelIterated integraldefinitionPartial 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…definitionFubini's theorem, statement, no evidence levelFubini's theoremstatementDerivative 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…statementVolume of a ball in R^n, statement, no evidence levelVolume of a ball in R…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. Iterated integral definition
  8. Partial derivative concept
  9. Continuity in R^n definition
  10. Differentiable function of several variables definition
  11. Partial derivative definition
  12. Double integral definition
  13. Chain rule in several variables statement
  14. Differentiable on an open set definition
  15. Jacobian matrix definition
  16. Higher partial derivatives definition
  17. Fubini's theorem statement
  18. Derivative of the inverse function statement
  19. Iterated derivative definition
  20. Jacobian determinant definition
  21. C^k function characterised by continuous derivatives statement
  22. Inverse function theorem statement
  23. Diffeomorphism definition
  24. Change of variables in multiple integrals statement
  25. Volume of a ball in R^n statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Volume_of_an_n-ball, Closed form
  2. Web https://en.wikipedia.org/wiki/Volume_of_an_n-ball, Direct integration in spherical coordinates
  3. Web https://en.wikipedia.org/wiki/Volume_of_an_n-ball, Gaussian integrals
About this page
Tag00YH28R457
URIm2://00YH28R457
Aliasesvolume-of-an-n-ball
Kindstatement, theorem
WritingW0 stub
MSC28A75
StatusThis copy of the site has no build data. Run m2 status 00YH28R457.
UsesEuclidean space, Change of variables in multiple integrals, Fubini's theorem
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