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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.

Every Lie algebra L is isomorphic to a subalgebra of some Lie algebra A^-, where the associative algebra A may be taken to be the linear operators over a vector space V.

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 Hilbert basis for the vector space of square summable sequences (a_n)=a_1, a_2, ... is given by the standard basis e_i, where e_i=delta_(in), with delta_(in) the Kronecker ...

A ket |psi> is a vector living in a dual vector space to that containing bras <psi|. Bras and kets are commonly encountered in quantum mechanics. Bras and kets can be ...

A standard basis, also called a natural basis, is a special orthonormal vector basis in which each basis vector has a single nonzero entry with value 1. In n-dimensional ...

A divergenceless vector field, also called a solenoidal field, is a vector field for which del ·F=0. Therefore, there exists a G such that F=del xG. Furthermore, F can be ...

A polynomial function of the elements of a vector x can be uniquely decomposed into a sum of harmonic polynomials times powers of |x|.