Math 2.0

Real functions Theorem

Integration in polar coordinates

Builds on Change of variables in multiple integralsPolar coordinatesDouble integral

Statement

Let 0≤r1<r20 \le r_1 < r_2 and α<β≤α+2π\alpha < \beta \le \alpha + 2\pi, and let D={(rcos⁡θ,rsin⁡θ):r1≤r≤r2, α≤θ≤β}D = \{(r \cos \theta, r \sin \theta) : r_1 \le r \le r_2, \ \alpha \le \theta \le \beta\}. Let f ⁣:D→ℝf \colon D \to \mathbb{R} be continuous. Then ∬Df(x,y) dx dy=∫αβ∫r1r2f(rcos⁡θ,rsin⁡θ) r dr dθ.\iint_D f(x, y) \, dx \, dy = \int_\alpha^\beta \int_{r_1}^{r_2} f(r \cos \theta, r \sin \theta) \, r \, dr \, d\theta.

Remarks

This is the change of variables formula for the polar coordinate map, whose Jacobian determinant is rr. The boundary rays and the origin have area zero and do not change the integral. Example: the area of the disc of radius aa is ∫02π∫0ar dr dθ=πa2\int_0^{2\pi} \int_0^a r \, dr \, d\theta = \pi a^2. For a region α≤θ≤β\alpha \le \theta \le \beta, 0≤r≤g(θ)0 \le r \le g(\theta) with gg continuous, the formula gives the area 12∫αβg(θ)2 dθ\tfrac{1}{2} \int_\alpha^\beta g(\theta)^2 \, d\theta. The Gaussian integral follows from ∬ℝ2e−(x2+y2) dx dy=∫02π∫0∞e−r2r dr dθ=π\iint_{\mathbb{R}^2} e^{-(x^2 + y^2)} \, dx \, dy = \int_0^{2\pi} \int_0^\infty e^{-r^2} r \, dr \, d\theta = \pi. Hence ∫−∞∞e−x2 dx=π\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}.

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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…statementIntegration in polar coordinates, statement, no evidence levelIntegration in polar …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
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  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. Integration in polar coordinates statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Polar_coordinate_system, Area
  2. Web https://en.wikipedia.org/wiki/Multiple_integral, Polar coordinates
About this page
Tag0T89W1QZ9K
URIm2://0T89W1QZ9K
Aliasespolar-change-of-variables
Kindstatement, theorem
WritingW0 stub
MSC26B15
StatusThis copy of the site has no build data. Run m2 status 0T89W1QZ9K.
UsesChange of variables in multiple integrals, Polar coordinates, Double integral

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