Inner product of a surface in 3D, induced by the dot product

In differential geometry, the **first fundamental form** is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of **R**^{3}. It permits the calculation of curvature and metric properties of a surface such as length and area in a manner consistent with the ambient space. The first fundamental form is denoted by the Roman numeral I,

$\mathrm {I} (x,y)=\langle x,y\rangle .$

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Definition

Let *X*(*u*, *v*) be a parametric surface. Then the inner product of two tangent vectors is

${\begin{aligned}&{}\quad \mathrm {I} (aX_{u}+bX_{v},cX_{u}+dX_{v})\\&=ac\langle X_{u},X_{u}\rangle +(ad+bc)\langle X_{u},X_{v}\rangle +bd\langle X_{v},X_{v}\rangle \\&=Eac+F(ad+bc)+Gbd,\end{aligned))$

where E, F, and G are the **coefficients of the first fundamental form**.
The first fundamental form may be represented as a symmetric matrix.

$\mathrm {I} (x,y)=x^{\mathsf {T)){\begin{bmatrix}E&F\\F&G\end{bmatrix))y$

##
Further notation

When the first fundamental form is written with only one argument, it denotes the inner product of that vector with itself.

$\mathrm {I} (v)=\langle v,v\rangle =|v|^{2))$

The first fundamental form is often written in the modern notation of the metric tensor. The coefficients may then be written as g_{ij}:

$\left(g_{ij}\right)={\begin{pmatrix}g_{11}&g_{12}\\g_{21}&g_{22}\end{pmatrix))={\begin{pmatrix}E&F\\F&G\end{pmatrix))$

The components of this tensor are calculated as the scalar product of tangent vectors *X*_{1} and *X*_{2}:

$g_{ij}=\langle X_{i},X_{j}\rangle$

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 *ds* may be expressed in terms of the coefficients of the first fundamental form as

$ds^{2}=E\,du^{2}+2F\,du\,dv+G\,dv^{2}\,.$

The classical area element given by *dA* = |*X*_{u} × *X*_{v}| *du* *dv* can be expressed in terms of the first fundamental form with the assistance of Lagrange's identity,

$dA=|X_{u}\times X_{v}|\ du\,dv={\sqrt {\langle X_{u},X_{u}\rangle \langle X_{v},X_{v}\rangle -\left\langle X_{u},X_{v}\right\rangle ^{2))}\,du\,dv={\sqrt {EG-F^{2))}\,du\,dv.$

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Example: curve on a sphere

A spherical curve on the unit sphere in **R**^{3} may be parametrized as

$X(u,v)={\begin{bmatrix}\cos u\sin v\\\sin u\sin v\\\cos v\end{bmatrix)),\ (u,v)\in [0,2\pi )\times [0,\pi ].$

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

${\begin{aligned}X_{u}&={\begin{bmatrix}-\sin u\sin v\\\cos u\sin v\\0\end{bmatrix)),\\X_{v}&={\begin{bmatrix}\cos u\cos v\\\sin u\cos v\\-\sin v\end{bmatrix)).\end{aligned))$

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

${\begin{aligned}E&=X_{u}\cdot X_{u}=\sin ^{2}v\\F&=X_{u}\cdot X_{v}=0\\G&=X_{v}\cdot X_{v}=1\end{aligned))$

so:

${\begin{bmatrix}E&F\\F&G\end{bmatrix))={\begin{bmatrix}\sin ^{2}v&0\\0&1\end{bmatrix)).$

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Length of a curve on the sphere

The equator of the unit sphere is a parametrized curve given by

$(u(t),v(t))=(t,{\tfrac {\pi }{2)))$

with t ranging from 0 to 2π. The line element may be used to calculate the length of this curve.
$\int _{0}^{2\pi }{\sqrt {E\left({\frac {du}{dt))\right)^{2}+2F{\frac {du}{dt)){\frac {dv}{dt))+G\left({\frac {dv}{dt))\right)^{2))}\,dt=\int _{0}^{2\pi }\left|\sin v\right|dt=2\pi \sin {\tfrac {\pi }{2))=2\pi$

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Area of a region on the sphere

The area element may be used to calculate the area of the unit sphere.

$\int _{0}^{\pi }\int _{0}^{2\pi }{\sqrt {EG-F^{2))}\ du\,dv=\int _{0}^{\pi }\int _{0}^{2\pi }\sin v\,du\,dv=2\pi \left[-\cos v\right]_{0}^{\pi }=4\pi$

##
Gaussian curvature

The Gaussian curvature of a surface is given by

$K={\frac {\det \mathrm {I\!I} _{p)){\det \mathrm {I} _{p))}={\frac {LN-M^{2)){EG-F^{2))},$

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.