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The ith Stiefel-Whitney class of a real vector bundle (or tangent bundle or a real manifold) is in the ith cohomology group of the base space involved. It is an obstruction ...

A vector norm defined for a vector x=[x_1; x_2; |; x_n], with complex entries by |x|_infty=max_(i)|x_i|. The vector norm |x|_infty of the vector x is implemented in the ...

Two vector bundles are stably equivalent iff isomorphic vector bundles are obtained upon Whitney summing each vector bundle with a trivial vector bundle.

Consider a first-order ODE in the slightly different form p(x,y)dx+q(x,y)dy=0. (1) Such an equation is said to be exact if (partialp)/(partialy)=(partialq)/(partialx). (2) ...

A field K is said to be algebraically closed if every polynomial with coefficients in K has a root in K.

An extension A subset B of a group, ring, module, field, etc., such that A!=B.

Given a vector bundle pi:E->M, its dual bundle is a vector bundle pi^*:E^*->M. The fiber bundle of E^* over a point p in M is the dual vector space to the fiber of E.

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 two-component complex column vector. Spinors can describe both bosons and fermions, while tensors can describe only bosons.

A representation of a group G is a group action of G on a vector space V by invertible linear maps. For example, the group of two elements Z_2={0,1} has a representation phi ...