First fundamental form

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.

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.

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

is called the first fundamental form of M.

. 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

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.

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.

The line element

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

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 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 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 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 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

. 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.

Then the inner product of two tangent vectors is

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

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

The unit sphere in

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

The equator of the sphere is a parametrized curve given by

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.

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.