# 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.

### Line element

**elements**

The line element

### 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**

### Differential geometry of surfaces

**surfaceshape operatorsmooth surface**

### Euclidean space

**EuclideanspaceEuclidean vector space**

### Dot product

**scalar productdotinner product**

### 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 plane0°The 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.

### Tautological one-form

**canonical one-formCanonical symplectic formLiouville form**