Math 2.0

Geometry Definition

Spherical coordinates

Builds on Euclidean spacePolar coordinates

Definition

The spherical coordinate map is S ⁣:[0,∞)×[0,π]×ℝ→ℝ3S \colon [0, \infty) \times [0, \pi] \times \mathbb{R} \to \mathbb{R}^3, 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). A triple (ρ,φ,θ)(\rho, \varphi, \theta) with S(ρ,φ,θ)=(x,y,z)S(\rho, \varphi, \theta) = (x, y, z) is a triple of spherical coordinates of (x,y,z)(x, y, z). Here ρ=x2+y2+z2\rho = \sqrt{x^2 + y^2 + z^2} is the distance to the origin, φ\varphi is the polar angle from the positive zz-axis, and θ\theta is the azimuth.

Remarks

The pair (ρsin⁡φ,θ)(\rho \sin \varphi, \theta) is a pair of polar coordinates of (x,y)(x, y), and cos⁡φ=z/ρ\cos \varphi = z / \rho for ρ>0\rho > 0. The coordinates are unique for points off the zz-axis if ρ>0\rho > 0, 0<φ<π0 < \varphi < \pi and θ∈[0,2π)\theta \in [0, 2\pi). The sphere of radius aa is ρ=a\rho = a, and φ=π/4\varphi = \pi / 4 is a cone around the positive zz-axis. This is a common convention in mathematics. Physics texts often write (r,θ,φ)(r, \theta, \varphi), with θ\theta the polar angle and φ\varphi the azimuth.

Reading path

To understand this page from the ground up, read these 21 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 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 coordinatesdefinitionSpherical coordinates, definition, no evidence levelSpherical 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
  21. Spherical coordinates definition

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Spherical_coordinate_system, Definition
  2. Web https://en.wikipedia.org/wiki/Spherical_coordinate_system, Cartesian coordinates
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MSC51N20
DefinesSpherical coordinates
UsesEuclidean space, Polar coordinates

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