First fundamental form

firstfirst quadratic forms
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 ofwikipedia
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Parametric surface

Curvature of parametric surfacesparameterizeparametric
be a parametric surface.
The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization.

Canonical form

canonicalnormal formcanonically
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
In the study of manifolds in three dimensions, one has the first fundamental form, the second fundamental form and the third fundamental form.

Metric tensor

metricmetricsround metric
The first fundamental form is often written in the modern notation of the metric tensor.
is called the first fundamental form of M.

Curvature

curvednegative curvatureextrinsic curvature
. 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.
in terms of the coefficients of the first and second fundamental forms as

Second fundamental form

extrinsic curvaturesecondshape tensor
where L, M, and N are the coefficients of the second fundamental form.
Together with the first fundamental form, it serves to define extrinsic invariants of the surface, its principal curvatures.

Gaussian curvature

Gauss curvaturecurvatureLiebmann's theorem
The Gaussian curvature of a surface is given by
In fact, it can be found given the full knowledge of the first fundamental form and expressed via the first fundamental form and its partial derivatives of first and second order.

Third fundamental form

The third fundamental form is expressible entirely in terms of the first fundamental form and second fundamental form.

Differential geometry

differentialdifferential geometerdifferential geometry and topology
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

Inner product space

inner productinner-product spaceinner products
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

Tangent space

tangent planetangenttangent vector
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

Differential geometry of surfaces

surfaceshape operatorsmooth surface
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

Euclidean space

EuclideanspaceEuclidean vector space
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

Dot product

scalar productdotinner 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

Ambient space

ambient
. 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.

Tangent vector

tangent vectorstangent directionstangent
Then the inner product of two tangent vectors is

Symmetric matrix

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

Lagrange's identity

can be expressed in terms of the first fundamental form with the assistance of Lagrange's identity,

Sphere

sphericalhemisphereglobose
The unit sphere in

Partial derivative

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

Equator

equatorial planeThe Equator
The equator of the sphere is a parametrized curve given by

Theorema Egregium

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.

Carl Friedrich Gauss

GaussCarl GaussCarl Friedrich Gauß
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.