First Fundamental Form

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## Definition

## Further notation

## Calculating lengths and areas

### Example: curve on a sphere

#### Length of a curve on the sphere

#### Area of a region on the sphere

## Gaussian curvature

## See also

## External links

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

First Fundamental Form

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,

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

where E, F, and G are the **coefficients of the first fundamental form**.

The first fundamental form may be represented as a symmetric matrix.

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

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

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

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

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

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

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

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

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

so:

The equator of the sphere is a parametrized curve given by

with t ranging from 0 to 2?. The line element may be used to calculate the length of this curve.

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

The Gaussian curvature of a surface is given by

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.

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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