This article summarizes several identities in exterior calculus, a mathematical notation used in differential geometry.
[1][2][3][4][5] The following summarizes short definitions and notations that are used in this article.
denote the tangent bundle and cotangent bundle, respectively, of the smooth manifold
Sections of the tangent bundles, also known as vector fields, are typically denoted as
Sections of the cotangent bundle, also known as differential 1-forms (or covector fields), are typically denoted as
We generally omit the subscript when it is clear from the context.
,[6] The Lie bracket of sections
defines a tangent map from
is a smooth map, then the pull-back of a
that effectively substitutes the first input of a
induces duality mappings between vector fields and one-forms: these are the musical isomorphisms flat
corresponds to the unique one-form
corresponds to the unique vector field
-vector fields through For an n-manifold M, the Hodge star operator
It can be defined in terms of an oriented frame
, orthonormal with respect to the given metric tensor
, is a Dirac operator studied in Clifford analysis.
that is continuous and nonzero everywhere on M. On an orientable manifold
the canonical choice of a volume form given a metric tensor
ordered to match the orientation.
and a unit normal vector
we can also define an area form
A generalization of the metric tensor, the symmetric bilinear form between two
is defined by In the case of a Riemannian manifold, each is an inner product (i.e. is positive-definite).
through Cartan's magic formula for a given section
and its exterior derivative operators
are in the same cohomology class if their difference is an exact form i.e. A closed surface of genus
α , β , γ ,
with unit normal vector
such that[citation needed] If a boundaryless manifold