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A matrix that has undergone Gaussian elimination is said to be in row echelon form or, more properly, "reduced echelon form" or "row-reduced echelon form." Such a matrix has ...

A differential k-form can be integrated on an n-dimensional manifold. The basic example is an n-form alpha in the open unit ball in R^n. Since alpha is a top-dimensional ...

In an exterior algebra ^ V, a top-dimensional form has degree n where n=dimV. Any form of higher degree must be zero. For example, if V=R^4 then alpha=e_1 ^ e_2 ^ e_3 ^ e_4 ...

A linear real-valued function omega^1 of vectors v such that omega^1(v)|->R. Vectors (i.e., contravariant vectors or "kets" |psi>) and one-forms (i.e., covariant vectors or ...

The kernel of a symmetric bilinear form Q:V×V-->R is the set Ker(Q)={v in V|Q(v,w)=0 for all w in V}.

The differential forms on C^n decompose into forms of type (p,q), sometimes called (p,q)-forms. For example, on C, the exterior algebra decomposes into four types: ^ C = ^ ^0 ...

The Killing form is an inner product on a finite dimensional Lie algebra g defined by B(X,Y)=Tr(ad(X)ad(Y)) (1) in the adjoint representation, where ad(X) is the adjoint ...

Given a differential p-form q in the exterior algebra ^ ^pV^*, its envelope is the smallest subspace W such that q is in the subspace ^ ^pW^* subset ^ ^pV^*. Alternatively, W ...

Also known as the total curvature. The linear element of the indicatrix ds_P=sqrt(ds_T^2+ds_B^2).

For K a given knot in S^3, choose a Seifert surface M^2 in S^3 for K and a bicollar M^^×[-1,1] in S^3-K. If x in H_1(M^^) is represented by a 1-cycle in M^^, let x^+ denote ...