In mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams. The five lemma is not only valid for abelian categories but also works in the category of groups, for example. The five lemma can be thought of as a combination of two other theorems, the four lemmas, which are dual to each other. == Statements == Consider the following commutative diagram in any abelian category (such as the category of abelian groups or the category of vector spaces over a given field) or in the category...

In mathematics, a volume form on a differentiable manifold is a top-dimensional form (i.e., a differential form of top degree). Thus on a manifold M of dimension n, a volume form is an n-form, a section of the line bundle Ωn(M) = ⋀n(T∗M). A manifold admits a nowhere-vanishing volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a density. A volume form provides a means to define the integral of a function...