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{{Short description|Inner product of a surface in 3D, induced by the dot product}}
In [[Differential geometry|differential geometry]], the '''first fundamental form''' is the inner product on the [[Tangent space|tangent space]] of a surface in three-dimensional [[Euclidean space]] which is induced [[Canonical form|canonically]] from the [[Dot product|dot product]] of {{math|'''R'''<sup>3</sup>}}.  It permits the calculation of [[Curvature|curvature]] and metric properties of a surface such as length and area in a manner consistent with the [[Ambient space|ambient space]].  The first fundamental form is denoted by the Roman numeral {{math|I}},
<math display="block">\mathrm{I}(x,y)= \langle x,y \rangle.</math>

== Definition ==

Let {{math|''X''(''u'', ''v'')}} be a [[Parametric surface|parametric surface]].  Then the inner product of two [[Tangent vector|tangent vector]]s is
<math display="block">
\begin{align}
& {} \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{align}
</math>
where {{mvar|E}}, {{mvar|F}}, and {{mvar|G}} are the '''coefficients of the first fundamental form'''.

The first fundamental form may be represented as a [[Symmetric matrix|symmetric matrix]].

<math display="block">\mathrm{I}(x,y) = x^\mathsf{T}
\begin{bmatrix}
E & F \\
F & G
\end{bmatrix}y
</math>

==Further notation==
When the first fundamental form is written with only one argument, it denotes the inner product of that vector with itself.
<math display="block">\mathrm{I}(v)= \langle v,v \rangle = |v|^2</math>

The first fundamental form is often written in the modern notation of the [[Physics:Metric tensor|metric tensor]].  The coefficients may then be written as {{mvar|g<sub>ij</sub>}}: 
<math display="block"> \left(g_{ij}\right) = \begin{pmatrix}
g_{11} & g_{12} \\
g_{21} & g_{22}
\end{pmatrix} =\begin{pmatrix}
E & F \\
F & G
\end{pmatrix}</math>

The components of this tensor are calculated as the scalar product of tangent vectors {{math|''X''<sub>1</sub>}} and {{math|''X''<sub>2</sub>}}:
<math display="block">g_{ij} = \langle X_i, X_j \rangle  </math>
for {{math|1=''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|line element]] {{math|''ds''}} may be expressed in terms of the coefficients of the first fundamental form as
<math display="block">ds^2 = E\,du^2+2F\,du\,dv+G\,dv^2 \,.</math> 

The classical area element given by {{math|1=''dA'' = {{abs|''X<sub>u</sub>'' × ''X<sub>v</sub>''}} ''du'' ''dv''}} can be expressed in terms of the first fundamental form with the assistance of [[Lagrange's identity]],
<math display="block">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.</math>

===Example: curve on a sphere===
A spherical curve on the [[Unit sphere|unit sphere]] in {{math|'''R'''<sup>3</sup>}} may be parametrized as
<math display="block">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].</math>

Differentiating {{math|''X''(''u'',''v'')}} with respect to {{mvar|u}} and {{mvar|v}} yields
<math display="block">\begin{align}
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{align}</math>

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

<math display="block">\begin{align}
E &= X_u \cdot X_u = \sin^2 v \\
F &= X_u \cdot X_v = 0 \\
G &= X_v \cdot X_v = 1
\end{align}</math>

so:
<math display="block"> \begin{bmatrix}E & F \\F & G\end{bmatrix} =\begin{bmatrix} \sin^2 v & 0 \\0 & 1\end{bmatrix}.</math>

====Length of a curve on the sphere====

The [[Astronomy:Equator|equator]] of the unit sphere is a parametrized curve given by
<math display="block">(u(t),v(t))=(t,\tfrac{\pi}{2})</math>
with {{mvar|t}} ranging from 0 to 2{{pi}}.  The line element may be used to calculate the length of this curve.

<math display="block">\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</math>

====Area of a region on the sphere====

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

<math display="block">\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</math>

==Gaussian curvature==

The [[Gaussian curvature]] of a surface is given by 
<math display="block"> K = \frac{\det \mathrm{I\!I}_p}{\det \mathrm{I}_p} = \frac{ LN-M^2}{EG-F^2 }, </math>
where {{mvar|L}}, {{mvar|M}}, and {{mvar|N}} are the coefficients of the [[Second fundamental form|second fundamental form]].

Theorema egregium of [[Biography:Carl Friedrich Gauss|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 {{mvar|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 [[Gaussian curvature#Alternative formulas|Brioschi formula]].

==See also==
*[[Physics:Metric tensor|Metric tensor]]
*[[Second fundamental form]]
*[[Third fundamental form]]
*[[Tautological one-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] -->

{{curvature}}

[[Category:Differential geometry of surfaces]]
[[Category:Differential geometry]]
[[Category:Surfaces]]

{{Sourceattribution|First fundamental form}}