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The term "gradient" has several meanings in mathematics. The simplest is as a synonym for slope. The more general gradient, called simply "the" gradient in vector analysis, ...

A line bundle is a special case of a vector bundle in which the fiber is either R, in the case of a real line bundle, or C, in the case of a complex line bundle.

dN^^+kappa_idr=0, where N^^ is the unit normal vector and kappa_i is one of the two principal curvatures.

A function f(x_1,...,x_n) of one or more variables whose range is one-dimensional, as compared to a vector function, whose range is three-dimensional (or, in general, ...

There does not exist an everywhere nonzero tangent vector field on the 2-sphere S^2. This implies that somewhere on the surface of the Earth, there is a point with zero ...

A topological space is second countable if it has a countable topological basis.

A complete metric is a metric in which every Cauchy sequence is convergent. A topological space with a complete metric is called a complete metric space.

A contravariant tensor is a tensor having specific transformation properties (cf., a covariant tensor). To examine the transformation properties of a contravariant tensor, ...

Let V be an n-dimensional linear space over a field K, and let Q be a quadratic form on V. A Clifford algebra is then defined over T(V)/I(Q), where T(V) is the tensor algebra ...

The total number of contravariant and covariant indices of a tensor. The rank R of a tensor is independent of the number of dimensions N of the underlying space. An intuitive ...