Math 2.0

Differential geometry Definition

First fundamental form

Builds on Regular surfaceTangent plane and normal vector of a surfaceDot productQuadratic form

Definition

Let S⊆ℝ3S \subseteq \R^3 be a regular surface and let p∈Sp \in S. Let TpS⊆ℝ3T_p S \subseteq \R^3 be the tangent plane of SS at pp, viewed as a 22-dimensional linear subspace. The first fundamental form of SS at pp is the quadratic form Ip ⁣:TpS→ℝI_p \colon T_p S \to \R, Ip(w)=⟨w,w⟩I_p(w) = \ip{w}{w}, where ⟨⋅,⋅⟩\ip{\cdot}{\cdot} is the dot product of ℝ3\R^3. In a local parametrisation x(u,v)x(u, v) of SS, its coefficients are E=⟨xu,xu⟩,F=⟨xu,xv⟩,G=⟨xv,xv⟩.E = \ip{x_u}{x_u}, \qquad F = \ip{x_u}{x_v}, \qquad G = \ip{x_v}{x_v}. For w=axu+bxvw = a x_u + b x_v, this gives Ip(w)=Ea2+2Fab+Gb2I_p(w) = E a^2 + 2F ab + G b^2.

Remarks

One writes I=E du2+2F du dv+G dv2I = E \, du^2 + 2F \, du \, dv + G \, dv^2. The length of a curve t↦x(u(t),v(t))t \mapsto x(u(t), v(t)), t∈[a,b]t \in [a, b], is ∫abEu′2+2Fu′v′+Gv′2 dt\int_a^b \sqrt{E u'^2 + 2F u' v' + G v'^2} \, dt, and the area of x(R)x(R) is ∬REG−F2 du dv\iint_R \sqrt{EG - F^2} \, du \, dv. Example: for the unit sphere with x(θ,φ)=(sin⁡θcos⁡φ,sin⁡θsin⁡φ,cos⁡θ)x(\theta, \varphi) = (\sin \theta \cos \varphi, \sin \theta \sin \varphi, \cos \theta), one gets E=1E = 1, F=0F = 0 and G=sin⁡2θG = \sin^2 \theta. For 0<θ<π0 < \theta < \pi the form is positive definite, as it must be, since EG−F2=sin⁡2θ>0EG - F^2 = \sin^2 \theta > 0. Non-example: at θ=0\theta = 0 the same formulas give G=0G = 0, a degenerate form; the map is not a local parametrisation there, because xφ=0x_\varphi = 0.

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Coordinate space, object, no evidence levelCoordinate spaceobjectEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptRegular surface, concept, no evidence levelRegular surfaceconceptVector space, definition, no evidence levelVector spacedefinitionDot product, definition, no evidence levelDot productdefinitionFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptBilinear form, definition, no evidence levelBilinear formdefinitionLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptQuadratic form, definition, no evidence levelQuadratic formdefinitionContinuity 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 …definitionGradient, definition, no evidence levelGradientdefinitionHigher 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…definitionFirst fundamental form, definition, no evidence levelFirst fundamental formdefinition
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  1. Coordinate space object
  2. Euclidean space object
  3. Function concept
  4. Regular surface concept
  5. Vector space definition
  6. Dot product definition
  7. Fréchet derivative definition
  8. Function of several variables concept
  9. Vector-valued function concept
  10. Bilinear form definition
  11. Limit of a function of several variables concept
  12. Partial derivative concept
  13. Quadratic form definition
  14. Continuity in R^n definition
  15. Differentiable function of several variables definition
  16. Partial derivative definition
  17. Differentiable on an open set definition
  18. Gradient definition
  19. Higher partial derivatives definition
  20. Iterated derivative definition
  21. C^k function characterised by continuous derivatives statement
  22. Parametrised curve definition
  23. Parametrised surface definition
  24. Tangent plane and normal vector of a surface definition
  25. First fundamental form definition

Further reading

  1. Web https://en.wikipedia.org/wiki/First_fundamental_form
About this page
TagBTX4G1FDF6
URIm2://BTX4G1FDF6
Kinddefinition
WritingW0 stub
MSC53A05
DefinesFirst fundamental form
UsesRegular surface, Tangent plane and normal vector of a surface, Dot product, Quadratic form
Changed2 November 2026

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