Math 2.0

Differential geometry Definition

Parametrised surface

Builds on Partial derivativeC^k function characterised by continuous derivativesParametrised curve

Definition

Let D⊆ℝ2D \subseteq \mathbb{R}^2 be a set that is the closure of an open set. A parametrised surface is a map r ⁣:D→ℝ3r \colon D \to \mathbb{R}^3, r(u,v)=(x(u,v),y(u,v),z(u,v))r(u, v) = \big( x(u, v), y(u, v), z(u, v) \big), of class C^1 on an open set that contains DD. Its image S=r(D)S = r(D) is the surface that rr parametrises. The parametrisation is regular at (u,v)(u, v) if the partial derivatives ru=∂r/∂ur_u = \partial r / \partial u and rv=∂r/∂vr_v = \partial r / \partial v are linearly independent, that is, ru×rv≠0r_u \times r_v \ne 0.

Remarks

The graph of a function f ⁣:D→ℝf \colon D \to \mathbb{R} is parametrised by r(x,y)=(x,y,f(x,y))r(x, y) = (x, y, f(x, y)). Here rx×ry=(−fx,−fy,1)≠0r_x \times r_y = (-f_x, -f_y, 1) \ne 0, so this parametrisation is regular. The unit sphere is parametrised by r(φ,θ)=(sin⁡φcos⁡θ,sin⁡φsin⁡θ,cos⁡φ)r(\varphi, \theta) = (\sin \varphi \cos \theta, \sin \varphi \sin \theta, \cos \varphi) on [0,π]×[0,2π][0, \pi] \times [0, 2\pi]. It is not regular at the poles φ=0\varphi = 0 and φ=π\varphi = \pi. The cylinder of radius R>0R > 0 about the zz-axis is parametrised by r(θ,z)=(Rcos⁡θ,Rsin⁡θ,z)r(\theta, z) = (R \cos \theta, R \sin \theta, z). One surface has many parametrisations. Areas and surface integrals do not depend on the choice.

Reading path

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptContinuity 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 derivativedefinitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementParametrised curve, definition, no evidence levelParametrised curvedefinitionParametrised surface, definition, no evidence levelParametrised surfacedefinition
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. Fréchet derivative definition
  4. Function of several variables concept
  5. Vector-valued function concept
  6. Limit of a function of several variables concept
  7. Continuity in R^n definition
  8. Differentiable function of several variables definition
  9. Partial derivative definition
  10. Differentiable on an open set definition
  11. Higher partial derivatives definition
  12. Iterated derivative definition
  13. C^k function characterised by continuous derivatives statement
  14. Parametrised curve definition
  15. Parametrised surface definition

Where this leads

36 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Parametric_surface, lead section
  2. Web https://en.wikipedia.org/wiki/Parametric_surface, Examples
  3. Web https://en.wikipedia.org/wiki/Parametric_surface, Tangent plane and normal vector
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Kinddefinition
WritingW1 stated
MSC53A05
DefinesParametrised surface
UsesPartial derivative, C^k function characterised by continuous derivatives, Parametrised curve

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