Math 2.0

Real functions Definition

Surface integral

Builds on Parametrised surfaceSurface areaTangent plane and normal vector of a surfaceDouble integral

Definition

Let r ⁣:D→ℝ3r \colon D \to \mathbb{R}^3 be a parametrised surface as in the definition of surface area, let S=r(D)S = r(D), and let f ⁣:S→ℝf \colon S \to \mathbb{R} be continuous. The surface integral of ff over SS is ∬Sf dS=∬Df(r(u,v)) ∥ru(u,v)×rv(u,v)∥ du dv.\iint_S f \, dS = \iint_D f\big( r(u, v) \big) \, \| r_u(u, v) \times r_v(u, v) \| \, du \, dv.

Remarks

The expression dS=∥ru×rv∥ du dvdS = \| r_u \times r_v \| \, du \, dv is the surface element. With f=1f = 1, the integral is the area of SS. The value does not depend on the parametrisation of SS. For a surface made of several pieces, add the integrals over the pieces. For the graph z=g(x,y)z = g(x, y) over DD, ∬Sf dS=∬Df(x,y,g(x,y))1+gx2+gy2 dx dy\iint_S f \, dS = \iint_D f(x, y, g(x, y)) \sqrt{1 + g_x^2 + g_y^2} \, dx \, dy. Example: on the unit sphere SS, symmetry gives ∬Sx2 dS=∬Sy2 dS=∬Sz2 dS\iint_S x^2 \, dS = \iint_S y^2 \, dS = \iint_S z^2 \, dS. These add up to the area 4π4\pi, so ∬Sz2 dS=4π/3\iint_S z^2 \, dS = 4\pi / 3.

Reading path

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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 integraldefinition
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

Where this leads

20 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Surface_integral, Surface integrals of scalar fields
  2. Web https://en.wikipedia.org/wiki/Surface_integral, Dependence on parametrization
About this page
Tag0DQ8ERJZNS
URIm2://0DQ8ERJZNS
Kinddefinition
WritingW1 stated
MSC26B15
DefinesSurface integral
UsesParametrised surface, Surface area, Tangent plane and normal vector of a surface, Double integral

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