Math 2.0

Geometry Definition

Cylindrical coordinates

Builds on Polar coordinatesEuclidean space

Definition

The cylindrical coordinate map is C ⁣:[0,∞)×ℝ×ℝ→ℝ3C \colon [0, \infty) \times \mathbb{R} \times \mathbb{R} \to \mathbb{R}^3, C(r,θ,z)=(rcos⁡θ, rsin⁡θ, z).C(r, \theta, z) = (r \cos \theta, \ r \sin \theta, \ z). A triple (r,θ,z)(r, \theta, z) with C(r,θ,z)=(x,y,z)C(r, \theta, z) = (x, y, z) is a triple of cylindrical coordinates of (x,y,z)(x, y, z). Here r=x2+y2r = \sqrt{x^2 + y^2} is the distance to the zz-axis, θ\theta is the azimuth, and zz is the height.

Remarks

The pair (r,θ)(r, \theta) is a pair of polar coordinates of (x,y)(x, y). So, away from the zz-axis, the coordinates are unique if r>0r > 0 and θ∈[0,2π)\theta \in [0, 2\pi). The Jacobian determinant of CC is rr, so the volume element is dV=r dr dθ dzdV = r \, dr \, d\theta \, dz. Example: the solid cylinder x2+y2≤a2x^2 + y^2 \le a^2, 0≤z≤h0 \le z \le h has volume ∫0h∫02π∫0ar dr dθ dz=πa2h\int_0^h \int_0^{2\pi} \int_0^a r \, dr \, d\theta \, dz = \pi a^2 h. Some texts write ρ\rho or ss for rr, and φ\varphi for θ\theta.

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 coordinatesdefinitionCylindrical coordinates, definition, no evidence levelCylindrical coordinat…definition
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.
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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. Cylindrical coordinates definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Cylindrical_coordinate_system, Definition
  2. Web https://en.wikipedia.org/wiki/Cylindrical_coordinate_system, Line and volume elements
About this page
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WritingW1 stated
MSC51N20
DefinesCylindrical coordinates
UsesPolar coordinates, Euclidean space

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