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The Weingarten equations express the derivatives of the normal vector to a surface using derivatives of the position vector. Let x:U->R^3 be a regular patch, then the shape ...

A manifold is a topological space that is locally Euclidean (i.e., around every point, there is a neighborhood that is topologically the same as the open unit ball in R^n). ...

On a Riemannian manifold M, tangent vectors can be moved along a path by parallel transport, which preserves vector addition and scalar multiplication. So a closed loop at a ...

In the plane, there are 17 lattice groups, eight of which are pure translation. In R^3, there are 32 point groups and 230 space groups. In R^4, there are 4783 space lattice ...

The free module of rank n over a nonzero unit ring R, usually denoted R^n, is the set of all sequences {a_1,a_2,...,a_n} that can be formed by picking n (not necessarily ...

A topological space is locally connected at the point x if every neighborhood of x contains a connected open neighborhood. It is called locally connected if it is locally ...

A "pointwise-bounded" family of continuous linear operators from a Banach space to a normed space is "uniformly bounded." Symbolically, if sup||T_i(x)|| is finite for each x ...

An embedding is a representation of a topological object, manifold, graph, field, etc. in a certain space in such a way that its connectivity or algebraic properties are ...

Gauge theory studies principal bundle connections, called gauge fields, on a principal bundle. These connections correspond to fields, in physics, such as an electromagnetic ...

int_a^b(del f)·ds=f(b)-f(a), where del is the gradient, and the integral is a line integral. It is this relationship which makes the definition of a scalar potential function ...