Math 2.0

Measure and integration Definition

Surface area

Builds on Parametrised surfaceTangent plane and normal vector of a surfaceDouble integralJordan measure

Definition

Let D⊆ℝ2D \subseteq \mathbb{R}^2 be a bounded Jordan measurable closed set, and let r ⁣:D→ℝ3r \colon D \to \mathbb{R}^3 be a parametrised surface that is injective on the interior of DD. The surface area of S=r(D)S = r(D) is area⁡(S)=∬D∥ru(u,v)×rv(u,v)∥ du dv.\operatorname{area}(S) = \iint_D \| r_u(u, v) \times r_v(u, v) \| \, du \, dv.

Remarks

The integrand is the length of the normal vector. It is the area of the parallelogram spanned by rur_u and rvr_v, which approximates the image of a small parameter rectangle. For the graph z=f(x,y)z = f(x, y) over DD, the area is ∬D1+fx2+fy2 dx dy\iint_D \sqrt{1 + f_x^2 + f_y^2} \, dx \, dy. For the unit sphere, ∥rφ×rθ∥=sin⁡φ\| r_\varphi \times r_\theta \| = \sin \varphi, and the area is ∫02π∫0πsin⁡φ dφ dθ=4π\int_0^{2\pi} \int_0^\pi \sin \varphi \, d\varphi \, d\theta = 4\pi. The value does not depend on the parametrisation. A piecewise smooth surface gets the sum of the areas of its pieces. Inscribed polyhedra do not give a good definition: the Schwarz lantern is a counterexample.

Reading path

To understand this page from the ground up, read these 23 pages in order. The foundations come first.

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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 areadefinition
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.
  • no evidence level
  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

Where this leads

29 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Surface_area, Definition
  2. Web https://en.wikipedia.org/wiki/Parametric_surface, Surface area
About this page
Tag0WQE3G7B88
URIm2://0WQE3G7B88
Kinddefinition
WritingW1 stated
MSC28A75
DefinesSurface area
UsesParametrised surface, Tangent plane and normal vector of a surface, Double integral, Jordan measure

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