Exterior calculus identities

This article summarizes several identities in exterior calculus.
Notation.
The following summarizes short definitions and notations that are used in this article.
Manifold.
formula_1, formula_2 are formula_3-dimensional smooth manifolds, where formula_4. That is, differentiable manifolds that can be differentiated enough times for the purposes on this page.
formula_5, formula_6 denote one point on each of the manifolds.
The boundary of a manifold formula_7 is a manifold formula_8, which has dimension formula_9. An orientation on formula_7 induces an orientation on formula_8.
We usually denote a submanifold by formula_12.
Tangent and cotangent bundles.
formula_13, formula_14 denote the tangent bundle and cotangent bundle, respectively, of the smooth manifold formula_1.
formula_16, formula_17 denote the tangent spaces of formula_1, formula_2 at the points formula_20, formula_21, respectively. formula_22 denotes the cotangent space of formula_1 at the point formula_20.
Sections of the tangent bundles, also known as vector fields, are typically denoted as formula_25 such that at a point formula_5 we have formula_27. Sections of the cotangent bundle, also known as differential 1-forms (or covector fields), are typically denoted as formula_28 such that at a point formula_5 we have formula_30. An alternative notation for formula_31 is formula_32.
Differential "k"-forms.
Differential formula_33-forms, which we refer to simply as formula_33-forms here, are differential forms defined on formula_13. We denote the set of all formula_33-forms as formula_37. For formula_38 we usually write formula_39, formula_40, formula_41.
formula_42-forms formula_43 are just scalar functions formula_44 on formula_1. formula_46 denotes the constant formula_42-form equal to formula_48 everywhere.
Omitted elements of a sequence.
When we are given formula_49 inputs formula_50 and a formula_33-form formula_39 we denote omission of the formula_53th entry by writing
Exterior product.
The exterior product is also known as the "wedge product". It is denoted by formula_55. The exterior product of a formula_33-form formula_39 and an formula_58-form formula_40 produce a formula_60-form formula_61. It can be written using the set formula_62 of all permutations formula_63 of formula_64 such that formula_65 as
Directional derivative.
The directional derivative of a 0-form formula_43 along a section formula_68 is a 0-form denoted formula_69
Exterior derivative.
The exterior derivative formula_70 is defined for all formula_71. We generally omit the subscript when it is clear from the context.
For a formula_42-form formula_43 we have formula_74 as the formula_48-form that gives the directional derivative, i.e., for the section formula_76 we have formula_77, the directional derivative of formula_78 along formula_79.
For formula_80,
Lie bracket.
The Lie bracket of sections formula_82 is defined as the unique section formula_83 that satisfies
Tangent maps.
If formula_85 is a smooth map, then formula_86 defines a tangent map from formula_1 to formula_2. It is defined through curves formula_89 on formula_1 with derivative formula_91 such that
Note that formula_93 is a formula_42-form with values in formula_2.
Pull-back.
If formula_85 is a smooth map, then the pull-back of a formula_33-form formula_98 is defined such that for any formula_33-dimensional submanifold formula_100
The pull-back can also be expressed as
Interior product.
Also known as the interior derivative, the interior product given a section formula_103 is a map formula_104 that effectively substitutes the first input of a formula_49-form with formula_106. If formula_107 and formula_108 then
Metric tensor.
Given a nondegenerate bilinear form formula_110 on each formula_16 that is continuous on formula_1, the manifold becomes a pseudo-Riemannian manifold. We denote the metric tensor formula_113, defined pointwise by formula_114. We call formula_115 the signature of the metric. A Riemannian manifold has formula_116, whereas Minkowski space has formula_117.
Musical isomorphisms.
The metric tensor formula_118 induces duality mappings between vector fields and one-forms: these are the musical isomorphisms flat formula_119 and sharp formula_120. A section formula_121 corresponds to the unique one-form formula_122 such that for all sections formula_123, we have:
A one-form formula_125 corresponds to the unique vector field formula_126 such that for all formula_123, we have:
These mappings extend via multilinearity to mappings from formula_33-vector fields to formula_33-forms and formula_33-forms to formula_33-vector fields through
Hodge star.
For an "n"-manifold "M", the Hodge star operator formula_135 is a duality mapping taking a formula_33-form formula_137 to an formula_138-form formula_139.
It can be defined in terms of an oriented frame formula_140 for formula_13, orthonormal with respect to the given metric tensor formula_113:
Co-differential operator.
The co-differential operator formula_144 on an formula_3 dimensional manifold formula_1 is defined by
The Hodge–Dirac operator, formula_148, is a Dirac operator operator studied in Clifford analysis.
Oriented manifold.
An formula_3-dimensional orientable manifold formula_1 is a manifold that can be equipped with a choice of an formula_3-form formula_152 that is continuous and nonzero everywhere on formula_1.
Volume form.
On an orientable manifold formula_1 the canonical choice of a volume form given a metric tensor formula_113 and an orientation is formula_156 for any basis formula_157 ordered to match the orientation.
Area form.
Given a volume form formula_158 and a unit normal vector formula_2 we can also define an area form formula_160 on the 
Bilinear form on "k"-forms.
A generalization of the metric tensor, the symmetric bilinear form between two formula_33-forms formula_162, is defined pointwise on formula_1 by
The formula_165-bilinear form for the space of formula_33-forms formula_37 is defined by
In the case of a Riemannian manifold, each is an inner product (i.e. is positive-definite).
Lie derivative.
We define the Lie derivative formula_169 through Cartan's magic formula for a given section formula_76 as
It describes the change of a formula_33-form along a flow formula_173 associated to the section formula_79.
Laplace–Beltrami operator.
The Laplacian formula_175 is defined as formula_176.
Important definitions.
Definitions on Ω"k"("M").
formula_39 is called...
Cohomology.
The formula_33-th cohomology of a manifold formula_1 and its exterior derivative operators formula_186 is given by
Two closed formula_33-forms formula_162 are in the same cohomology class if their difference is an exact form i.e.
A closed surface of genus formula_113 will have formula_192 generators which are harmonic.
Dirichlet energy.
Given formula_39, its Dirichlet energy is
Exterior calculus identities.
Dimensions.
If formula_272
If formula_277 is a basis, then a basis of formula_37 is
Exterior products.
Let formula_280 and formula_281 be vector fields.
Projection and rejection.
If formula_291, then
Given the boundary formula_298 with unit normal vector formula_2
Hodge decomposition.
If formula_307, formula_308 such that
Poincaré lemma.
If a boundaryless manifold formula_1 has trivial cohomology formula_311, then any closed formula_312 is exact. This is the case if "M" is contractible.
Relations to vector calculus.
Identities in Euclidean 3-space.
Let Euclidean metric formula_313.
We use formula_314 differential operator formula_315