Math 2.0

Real functions Definition

Flux through a surface

Builds on Orientable surfaceSurface integralTangent plane and normal vector of a surfaceParametrised surfaceVector field

Definition

Let S⊆ℝ3S \subseteq \mathbb{R}^3 be an oriented surface with unit normal field nn, and let FF be a continuous vector field on SS. The flux of FF through SS is the surface integral ∬SF⋅n dS.\iint_S F \cdot n \, dS.

Remarks

If r ⁣:D→ℝ3r \colon D \to \mathbb{R}^3 is a regular parametrisation of SS with ru×rvr_u \times r_v pointing in the direction of nn, then the flux is ∬DF(r(u,v))⋅(ru×rv) du dv\iint_D F\big( r(u, v) \big) \cdot \big( r_u \times r_v \big) \, du \, dv. Reversing the orientation of SS changes the sign of the flux. The notation ∬SF⋅d𝐒\iint_S F \cdot d\mathbf{S}, with d𝐒=n dSd\mathbf{S} = n \, dS, is common. Example: for F(x,y,z)=(x,y,z)F(x, y, z) = (x, y, z) and the unit sphere with the outward normal n(p)=pn(p) = p, we get F⋅n=1F \cdot n = 1, so the flux is the area 4π4\pi. If FF is the velocity field of a fluid, the flux is the volume of fluid that crosses SS per unit time. Flow in the direction of nn counts as positive.

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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 curvedefinitionVector field, definition, no evidence levelVector fielddefinitionParametrised 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 surfacedefinitionFlux through a surface, definition, no evidence levelFlux through a 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. Vector field definition
  22. Parametrised surface definition
  23. Tangent plane and normal vector of a surface definition
  24. Surface area definition
  25. Surface integral definition
  26. Orientable surface definition
  27. Flux through a surface definition

Where this leads

17 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Flux, General mathematical definition (surface integral)
  2. Web https://en.wikipedia.org/wiki/Surface_integral, Surface integrals of vector fields
  3. Web https://en.wikipedia.org/wiki/Surface_integral, Dependence on parametrization
About this page
Tag0YPRY7T4PK
URIm2://0YPRY7T4PK
Kinddefinition
WritingW0 stub
MSC26B15
DefinesFlux through a surface
UsesOrientable surface, Surface integral, Tangent plane and normal vector of a surface, Parametrised surface, Vector field

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