- First fundamental form
In

differential geometry , the**first fundamental form**is theinner product on thetangent space of asurface in three-dimensionalEuclidean space which is inducedcanonical ly from thedot product of**R**^{"3"}. It permits the calculation ofcurvature and metric properties of a surface such as length and area in a manner consistent with theambient space . The first fundamental form is denoted by the Roman numeral I,:$!mathrm\{I\}(v,w)=\; langle\; v,w\; angle.$Let "X"("u", "v") be a

parametric surface . Then the inner product of twotangent vector s is:$egin\{align\}\; \{\}\; quad\; mathrm\{I\}(aX\_u+bX\_v,cX\_u+dX\_v)\; \backslash \; =\; ac\; langle\; X\_u,X\_u\; angle\; +\; (ad+bc)\; langle\; X\_u,X\_v\; angle\; +\; bd\; langle\; X\_v,X\_v\; angle\; \backslash \; =\; Eac\; +\; F(ad+bc)\; +\; Gbd,end\{align\}$

where "E", "F", and "G" are the

**coefficients of the first fundamental form**.The first fundamental form may be represented as a

symmetric matrix .:$!mathrm\{I\}(v,w)\; =\; v^Tegin\{pmatrix\}E\; F\; \backslash F\; Gend\{pmatrix\}w$

**Further notation**When the first fundamental form is written with only one argument, it denotes the inner product of that vector with itself.:$!mathrm\{I\}(v)=\; langle\; v,v\; angle\; =\; |v|^2$

The first fundamental form is often written in the modern notation of the

metric tensor . The coefficients may then be written as $g\_\{ij\}$: :$left(g\_\{ij\}\; ight)\; =\; egin\{pmatrix\}g\_\{11\}\; g\_\{12\}\; \backslash g\_\{21\}\; g\_\{22\}end\{pmatrix\}\; =egin\{pmatrix\}E\; F\; \backslash F\; Gend\{pmatrix\}$The components of this tensor are calculated as the scalar product of tangent vectors "X"

_{1}and "X"_{2}::$g\_\{ij\}\; =\; X\_i\; cdot\; X\_j$

for "i", "j" = 1, 2. See example below.

**Calculating lengths and areas**The first fundamental form completely describes the metric properties of a surface. Thus, it enables one to calculate the lengths of curves on the surface and the areas of regions on the surface. The

line element may be expressed in terms of the coefficients of the first fundamental form as:$ds^2\; =\; Edu^2+2Fdudv+Gdv^2\; ,$.The classical area element given by $dA\; =\; |X\_u\; imes\; X\_v|\; du,\; dv$ can be expressed in terms of the first fundamental form with the assistance of

Lagrange's identity ,:$dA\; =\; |X\_u\; imes\; X\_v|\; du,\; dv=\; sqrt\{\; langle\; X\_u,X\_u\; angle\; langle\; X\_v,X\_v\; angle\; -\; langle\; X\_u,X\_v\; angle^2\; \}\; du,\; dv\; =\; sqrt\{EG-F^2\}\; ,\; du,\; dv.$

**Example**The unit

sphere in**R**^{"3"}may be parametrized as:$X(u,v)\; =\; egin\{pmatrix\}\; cos\; u\; sin\; v\; \backslash \; sin\; u\; sin\; v\; \backslash \; cos\; v\; end\{pmatrix\},\; (u,v)\; in\; [0,2pi)\; imes\; [0,pi).$

Differentiating $X(u,v)$ with respect to u and v yields

:$X\_u\; =\; egin\{pmatrix\}\; -sin\; u\; sin\; v\; \backslash \; cos\; u\; sin\; v\; \backslash \; 0\; end\{pmatrix\},\; X\_v\; =\; egin\{pmatrix\}\; cos\; u\; cos\; v\; \backslash \; sin\; u\; cos\; v\; \backslash \; -sin\; v\; end\{pmatrix\}.$

The coefficients of the first fundamental form may be found by taking the dot product of the

partial derivatives .:$E\; =\; X\_u\; cdot\; X\_u\; =\; sin^2\; v$:$F\; =\; X\_u\; cdot\; X\_v\; =\; 0$:$G\; =\; X\_v\; cdot\; X\_v\; =\; 1$

**Length of a curve on the sphere**The

equator of the sphere is a parametrized curve given by $(u(t),v(t))=(t,frac\{pi\}\{2\})$ with t ranging from 0 to $2pi$. The line element may be used to calculate the length of this curve.:$int\_0^\{2pi\}\; sqrt\{\; Eleft(frac\{du\}\{dt\}\; ight)^2\; +\; 2Ffrac\{du\}\{dt\}frac\{dv\}\{dt\}\; +\; Gleft(frac\{dv\}\{dt\}\; ight)^2\; \}\; ,dt\; =\; int\_0^\{2pi\}\; sin\; v\; ,dt\; =\; 2pi\; sin\; v\; =\; 2pi$

**Area of a region on the sphere**The area element may be used to calculate the area of the sphere.

:$int\_0^\{pi\}\; int\_0^\{2pi\}\; sqrt\{\; EG-F^2\; \}\; du,\; dv\; =\; int\_0^\{pi\}\; int\_0^\{2pi\}\; sin\; v\; ,\; du,\; dv\; =\; 2pi\; left\; [-cos\; v\; ight]\; \_0^\{pi\}\; =\; 4pi$

**Gaussian curvature**The

Gaussian curvature of a surface is given by:$K\; =\; frac\{det\; II\}\{det\; I\}\; =\; frac\{\; LN-M^2\}\{EG-F^2\; \},$

where "L", "M", and "N" are the coefficients of the

second fundamental form .Theorema egregium of Gauss states that the Gaussian curvature of a surface can be expressed solely in terms of the first fundamental form and its derivatives, so that "K" is in fact an intrinsic invariant of the surface. An explicit expression for the Gaussian curvature in terms of the first fundamental form is provided by the Brioschi formula.**ee also***

Metric tensor

*Second fundamental form **External links*** [

*http://mathworld.wolfram.com/FirstFundamentalForm.html First Fundamental Form — from Wolfram MathWorld*]

* [*http://planetmath.org/encyclopedia/FirstFundamentalForm.html PlanetMath: first fundamental form*]

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