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An alternating multilinear form on a real vector space V is a multilinear form F:V tensor ... tensor V->R (1) such that ...

A symplectic form on a smooth manifold M is a smooth closed 2-form omega on M which is nondegenerate such that at every point m, the alternating bilinear form omega_m on the ...

The qubit |psi>=a|0>+b|1> can be represented as a point (theta,phi) on a unit sphere called the Bloch sphere. Define the angles theta and phi by letting a=cos(theta/2) and ...

The Frobenius norm, sometimes also called the Euclidean norm (a term unfortunately also used for the vector L^2-norm), is matrix norm of an m×n matrix A defined as the square ...

The dot product can be defined for two vectors X and Y by X·Y=|X||Y|costheta, (1) where theta is the angle between the vectors and |X| is the norm. It follows immediately ...

Let g be a finite-dimensional Lie algebra over some field k. A subalgebra h of g is called a Cartan subalgebra if it is nilpotent and equal to its normalizer, which is the ...

A extension ring (or ring extension) of a ring R is any ring S of which R is a subring. For example, the field of rational numbers Q and the ring of Gaussian integers Z[i] ...

The group algebra K[G], where K is a field and G a group with the operation *, is the set of all linear combinations of finitely many elements of G with coefficients in K, ...

If A and B are commutative unit rings, and A is a subring of B, then A is called integrally closed in B if every element of B which is integral over A belongs to A; in other ...

Multivariable Calculus