Math 2.0

Differential geometry Definition

Orientable surface

Builds on Tangent plane and normal vector of a surfaceParametrised surfaceSurface integral

Definition

Let S⊆ℝ3S \subseteq \mathbb{R}^3 be a surface that has a tangent plane at each point. The surface SS is orientable if there is a continuous map n ⁣:S→ℝ3n \colon S \to \mathbb{R}^3 with the following property. For each p∈Sp \in S, n(p)n(p) is a unit normal vector of SS at pp. Such a map nn is an orientation of SS, and SS together with nn is an oriented surface.

Remarks

If SS is connected and orientable, then it has exactly two orientations, nn and −n-n. A regular injective parametrisation rr gives the orientation n=(ru×rv)/∥ru×rv∥n = (r_u \times r_v) / \| r_u \times r_v \|. Swapping uu and vv gives the opposite orientation. Spheres, planes and tori are orientable. The boundary of a bounded region in ℝ3\mathbb{R}^3 has the outward normal as its standard orientation. The Möbius strip is not orientable. Integrals of vector fields across a surface, such as flux, need an orientation. Reversing the orientation changes their sign. Integrals of scalar functions do not need one.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptInfimum, concept, no evidence levelInfimumconceptSupremum, concept, no evidence levelSupremumconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptJordan measure, definition, no evidence levelJordan measuredefinitionLimit 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 integraldefinitionGradient, definition, no evidence levelGradientdefinitionDifferentiable 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 surfacedefinitionTangent plane and normal vector of a surface, definition, no evidence levelTangent plane and nor…definitionSurface area, definition, no evidence levelSurface areadefinitionSurface integral, definition, no evidence levelSurface integraldefinitionOrientable surface, definition, no evidence levelOrientable 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. Infimum concept
  4. Supremum concept
  5. Fréchet derivative definition
  6. Function of several variables concept
  7. Vector-valued function concept
  8. Jordan measure definition
  9. Limit of a function of several variables concept
  10. Partial derivative concept
  11. Continuity in R^n definition
  12. Differentiable function of several variables definition
  13. Partial derivative definition
  14. Double integral definition
  15. Gradient definition
  16. Differentiable on an open set definition
  17. Higher partial derivatives definition
  18. Iterated derivative definition
  19. C^k function characterised by continuous derivatives statement
  20. Parametrised curve definition
  21. Parametrised surface definition
  22. Tangent plane and normal vector of a surface definition
  23. Surface area definition
  24. Surface integral definition
  25. Orientable surface definition

Where this leads

19 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Orientability, Orientable surfaces
  2. Web https://en.wikipedia.org/wiki/Orientability, Examples
  3. Web https://en.wikipedia.org/wiki/Normal_(geometry), Orientation
About this page
Tag0EKZW08FQ5
URIm2://0EKZW08FQ5
Kinddefinition
WritingW1 stated
MSC53A05
DefinesOrientable surface
UsesTangent plane and normal vector of a surface, Parametrised surface, Surface integral

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