Math 2.0

Geometry Definition

Polar coordinates

Builds on Euclidean spaceDiffeomorphism

Definition

The polar coordinate map is P ⁣:[0,∞)×ℝ→ℝ2P \colon [0, \infty) \times \mathbb{R} \to \mathbb{R}^2, P(r,θ)=(rcos⁡θ,rsin⁡θ)P(r, \theta) = (r \cos \theta, r \sin \theta). A pair (r,θ)(r, \theta) with P(r,θ)=(x,y)P(r, \theta) = (x, y) is a pair of polar coordinates of the point (x,y)(x, y). Here r=x2+y2r = \sqrt{x^2 + y^2} is the radial coordinate and θ\theta is the angular coordinate.

Remarks

Each point other than the origin has exactly one pair of polar coordinates with r>0r > 0 and θ∈[0,2π)\theta \in [0, 2\pi). The origin has r=0r = 0 and any θ\theta. For x>0x > 0, θ=arctan⁡(y/x)\theta = \arctan(y / x) is one valid angle. In general one uses the two-argument function atan2⁡(y,x)\operatorname{atan2}(y, x), with values in (−π,π](-\pi, \pi]. The restriction of PP to (0,∞)×(0,2π)(0, \infty) \times (0, 2\pi) is a C∞C^\infty diffeomorphism onto the plane without the half-line {(x,0):x≥0}\{(x, 0) : x \ge 0\}. Its Jacobian determinant is rr. The circle of radius aa about the origin is r=ar = a, and the ray at angle α\alpha is θ=α\theta = \alpha.

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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 derivativeconceptDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionJacobian matrix, definition, no evidence levelJacobian matrixdefinitionChain rule in several variables, statement, no evidence levelChain rule in several…statementHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionJacobian determinant, definition, no evidence levelJacobian determinantdefinitionDerivative of the inverse function, statement, no evidence levelDerivative of the inv…statementC^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 coordinatesdefinition
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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. Differentiable function of several variables definition
  9. Partial derivative definition
  10. Differentiable on an open set definition
  11. Jacobian matrix definition
  12. Chain rule in several variables statement
  13. Higher partial derivatives definition
  14. Iterated derivative definition
  15. Jacobian determinant definition
  16. Derivative of the inverse function statement
  17. C^k function characterised by continuous derivatives statement
  18. Inverse function theorem statement
  19. Diffeomorphism definition
  20. Polar coordinates definition

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Polar_coordinate_system, Converting between polar and Cartesian coordinates
  2. Web https://en.wikipedia.org/wiki/Polar_coordinate_system, Uniqueness of polar coordinates
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MSC51N20
DefinesPolar coordinates
UsesEuclidean space, Diffeomorphism

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