- First fundamental form
differential geometry, the first fundamental form is the inner producton the tangent spaceof a surfacein three-dimensional Euclidean spacewhich is induced canonically from the dot productof R"3". It permits the calculation of curvatureand 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
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 : :
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.
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 elementmay be expressed in terms of the coefficients of the first fundamental form as:.
The classical area element given by can be expressed in terms of the first fundamental form with the assistance of
spherein R"3" may be parametrized as
Differentiating with respect to u and v yields
The coefficients of the first fundamental form may be found by taking the dot product of the
Length of a curve on the sphere
equatorof the sphere is a parametrized curve given by with t ranging from 0 to . The line element may be used to calculate the length of this curve.
Area of a region on the sphere
The area element may be used to calculate the area of the sphere.
Gaussian curvatureof a surface is given by
where "L", "M", and "N" are the coefficients of the
second fundamental form. Theorema egregiumof 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.
Second fundamental form
* [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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