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On a differentiable manifold, the **exterior derivative** extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. It allows for a natural, metric-independent generalization of Stokes' theorem, Gauss's theorem, and Green's theorem from vector calculus.

- Definition
- In terms of axioms
- In terms of local coordinates
- In terms of invariant formula
- Examples
- Stokes' theorem on manifolds
- Further properties
- Closed and exact forms
- de Rham cohomology
- Naturality
- Exterior derivative in vector calculus
- Gradient
- Divergence
- Curl
- Invariant formulations of operators in vector calculus
- See also
- Notes
- References
- External links

If a differential *k*-form is thought of as measuring the flux through an infinitesimal *k*-parallelotope at each point of the manifold, then its exterior derivative can be thought of as measuring the net flux through the boundary of a (*k* + 1)-parallelotope at each point.

The exterior derivative of a differential form of degree *k* (also differential *k*-form, or just *k*-form for brevity here) is a differential form of degree *k* + 1.

If *f* is a smooth function (a 0-form), then the exterior derivative of *f* is the differential of *f* . That is, *df* is the unique 1-form such that for every smooth vector field *X*, *df* (*X*) = *d*_{X} *f* , where *d*_{X} *f* is the directional derivative of *f* in the direction of *X*.

The exterior product of differential forms (denoted with the same symbol ∧) is defined as their pointwise exterior product.

There are a variety of equivalent definitions of the exterior derivative of a general *k*-form.

The exterior derivative is defined to be the unique ℝ-linear mapping from *k*-forms to (*k* + 1)-forms that has the following properties:

*df*is the differential of*f*for a 0-form*f*.*d*(*df*) = 0 for a 0-form*f*.*d*(*α*∧*β*) =*dα*∧*β*+ (−1)^{p}(*α*∧*dβ*) where α is a*p*-form. That is to say,*d*is an antiderivation of degree 1 on the exterior algebra of differential forms.

The second defining property holds in more generality: *d*(*dα*) = 0 for any *k*-form α; more succinctly, *d*^{2} = 0. The third defining property implies as a special case that if *f* is a function and α a is *k*-form, then *d*( *fα*) = *d*( *f* ∧ *α*) = *df* ∧ *α* + *f* ∧ *dα* because a function is a 0-form, and scalar multiplication and the exterior product are equivalent when one of the arguments is a scalar.^{[ citation needed ]}

Alternatively, one can work entirely in a local coordinate system (*x*^{1}, ..., *x*^{n}). The coordinate differentials *dx*^{1}, ..., *dx*^{n} form a basis of the space of one-forms, each associated with a coordinate. Given a multi-index *I* = (*i*_{1}, ..., *i*_{k}) with 1 ≤ *i*_{p} ≤ *n* for 1 ≤ *p* ≤ *k* (and denoting *dx*^{i1} ∧ ... ∧ *dx*^{ik} with an abuse of notation *dx*^{I}), the exterior derivative of a (simple) *k*-form

over ℝ^{n} is defined as

(using the Einstein summation convention). The definition of the exterior derivative is extended linearly to a general *k*-form

where each of the components of the multi-index *I* run over all the values in {1, ..., *n*}. Note that whenever *i* equals one of the components of the multi-index *I* then *dx*^{i} ∧ *dx*^{I} = 0 (see * Exterior product *).

The definition of the exterior derivative in local coordinates follows from the preceding definition in terms of axioms. Indeed, with the *k*-form *φ* as defined above,

Here, we have interpreted *g* as a 0-form, and then applied the properties of the exterior derivative.

This result extends directly to the general *k*-form *ω* as

In particular, for a 1-form *ω*, the components of *dω* in local coordinates are

*Caution*: There are two conventions regarding the meaning of . Most current authors^{[ citation needed ]} have the convention that

while in older text like Kobayashi and Nomizu or Helgason

Alternatively, an explicit formula can be given^{[ citation needed ]} for the exterior derivative of a *k*-form *ω*, when paired with *k* + 1 arbitrary smooth vector fields *V*_{0},*V*_{1}, ..., *V*_{k}:

where [*V _{i}*,

In particular, when *ω* is a 1-form we have that *dω*(*X*, *Y*) = *d*_{X}(*ω*(*Y*)) − *d*_{Y}(*ω*(*X*)) − *ω*([*X*, *Y*]).

**Note:** With the conventions of e.g., Kobayashi–Nomizu and Helgason the formula differs by a factor of 1/*k* + 1:

**Example 1.** Consider *σ* = *u* *dx*^{1} ∧ *dx*^{2} over a 1-form basis *dx*^{1}, ..., *dx*^{n} for a scalar field *u*. The exterior derivative is:

The last formula, where summation starts at *i* = 3, follows easily from the properties of the exterior product. Namely, *dx*^{i} ∧ *dx*^{i} = 0.

**Example 2.** Let *σ* = *u* *dx* + *v* *dy* be a 1-form defined over ℝ^{2}. By applying the above formula to each term (consider *x*^{1} = *x* and *x*^{2} = *y*) we have the following sum,

If *M* is a compact smooth orientable *n*-dimensional manifold with boundary, and *ω* is an (*n* − 1)-form on *M*, then the generalized form of Stokes' theorem states that:

Intuitively, if one thinks of *M* as being divided into infinitesimal regions, and one adds the flux through the boundaries of all the regions, the interior boundaries all cancel out, leaving the total flux through the boundary of *M*.

A *k*-form *ω* is called *closed* if *dω* = 0; closed forms are the kernel of *d*. *ω* is called *exact* if *ω* = *dα* for some (*k* − 1)-form *α*; exact forms are the image of *d*. Because *d*^{2} = 0, every exact form is closed. The Poincaré lemma states that in a contractible region, the converse is true.

Because the exterior derivative *d* has the property that *d*^{2} = 0, it can be used as the differential (coboundary) to define de Rham cohomology on a manifold. The *k*-th de Rham cohomology (group) is the vector space of closed *k*-forms modulo the exact *k*-forms; as noted in the previous section, the Poincaré lemma states that these vector spaces are trivial for a contractible region, for *k* > 0. For smooth manifolds, integration of forms gives a natural homomorphism from the de Rham cohomology to the singular cohomology over ℝ. The theorem of de Rham shows that this map is actually an isomorphism, a far-reaching generalization of the Poincaré lemma. As suggested by the generalized Stokes' theorem, the exterior derivative is the "dual" of the boundary map on singular simplices.

The exterior derivative is natural in the technical sense: if *f* : *M* → *N* is a smooth map and Ω^{k} is the contravariant smooth functor that assigns to each manifold the space of *k*-forms on the manifold, then the following diagram commutes

so *d*( *f*^{∗}*ω*) = *f*^{∗}*dω*, where *f*^{∗} denotes the pullback of *f* . This follows from that *f*^{∗}*ω*(·), by definition, is *ω*( *f*_{∗}(·)), *f*_{∗} being the pushforward of *f* . Thus *d* is a natural transformation from Ω^{k} to Ω^{k+1}.

Most vector calculus operators are special cases of, or have close relationships to, the notion of exterior differentiation.

A smooth function *f* : *M* → ℝ on a real differentiable manifold *M* is a 0-form. The exterior derivative of this 0-form is the 1-form *df*.

When an inner product ⟨·,·⟩ is defined, the gradient ∇*f* of a function *f* is defined as the unique vector in *V* such that its inner product with any element of *V* is the directional derivative of *f* along the vector, that is such that

That is,

where ♯ denotes the musical isomorphism ♯ : *V*^{∗} → *V* mentioned earlier that is induced by the inner product.

The 1-form *df* is a section of the cotangent bundle, that gives a local linear approximation to *f* in the cotangent space at each point.

A vector field *V* = (*v*_{1}, *v*_{2}, ... *v _{n}*) on ℝ

where denotes the omission of that element.

(For instance, when *n* = 3, i.e. in three-dimensional space, the 2-form *ω _{V}* is locally the scalar triple product with

The exterior derivative of this (*n* − 1)-form is the *n*-form

A vector field *V* on ℝ^{n} also has a corresponding 1-form

Locally, *η _{V}* is the dot product with

When *n* = 3, in three-dimensional space, the exterior derivative of the 1-form *η _{V}* is the 2-form

The standard vector calculus operators can be generalized for any pseudo-Riemannian manifold, and written in coordinate-free notation as follows:

where ⋆ is the Hodge star operator, ♭ and ♯ are the musical isomorphisms, *f* is a scalar field and *F* is a vector field.

Note that the expression for curl requires ♯ to act on ⋆*d*(*F*^{♭}), which is a form of degree *n* − 2. A natural generalization of ♯ to *k*-forms of arbitrary degree allows this expression to make sense for any *n*.

In vector calculus, the **curl** is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation. The curl of a field is formally defined as the circulation density at each point of the field.

In differential geometry, a subject of mathematics, a **symplectic manifold** is a smooth manifold, , equipped with a closed nondegenerate differential 2-form , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system.

In vector calculus and differential geometry, the **generalized Stokes theorem**, also called the **Stokes–Cartan theorem**, is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. It is a generalization of Isaac Newton's fundamental theorem of calculus that relates two-dimensional line integrals to three-dimensional surface integrals.

In the mathematical fields of differential geometry and tensor calculus, **differential forms** are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.

In mathematics, the **exterior product** or **wedge product** of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors and , denoted by , is called a bivector and lives in a space called the *exterior square*, a vector space that is distinct from the original space of vectors. The magnitude of can be interpreted as the area of the parallelogram with sides and , which in three dimensions can also be computed using the cross product of the two vectors. More generally, all parallel plane surfaces with the same orientation and area have the same bivector as a measure of their oriented area. Like the cross product, the exterior product is anticommutative, meaning that for all vectors and , but, unlike the cross product, the exterior product is associative.

In mathematics, a **differential operator** is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function.

In differential geometry, the **Lie derivative**, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field, along the flow defined by another vector field. This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold.

In mathematics, the **Hodge star operator** or **Hodge star** is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the algebra produces the **Hodge dual** of the element. This map was introduced by W. V. D. Hodge.

In mathematics, and especially differential geometry and gauge theory, a **connection** on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. The most common case is that of a **linear connection** on a vector bundle, for which the notion of parallel transport must be linear. A linear connection is equivalently specified by a *covariant derivative*, an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, the Levi-Civita connection on the tangent bundle of a pseudo-Riemannian manifold, which gives a standard way to differentiate vector fields. Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear.

In mathematics, especially vector calculus and differential topology, a **closed form** is a differential form *α* whose exterior derivative is zero, and an **exact form** is a differential form, *α*, that is the exterior derivative of another differential form *β*. Thus, an *exact* form is in the *image* of *d*, and a *closed* form is in the *kernel* of *d*.

In abstract algebra and multilinear algebra, a **multilinear form** on a vector space over a field is a map

In contexts including complex manifolds and algebraic geometry, a **logarithmic** differential form is a meromorphic differential form with poles of a certain kind. The concept was introduced by Deligne.

In mathematics, a **volume form** on a differentiable manifold is a top-dimensional form. Thus on a manifold of dimension , a volume form is an -form, a section of the line bundle . 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.

In mathematics, a **complex differential form** is a differential form on a manifold which is permitted to have complex coefficients.

In differential geometry, the notion of **torsion** is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the Frenet–Serret formulas, for instance, quantifies the twist of a curve about its tangent vector as the curve evolves. In the geometry of surfaces, the *geodesic torsion* describes how a surface twists about a curve on the surface. The companion notion of curvature measures how moving frames "roll" along a curve "without twisting".

In mathematics, a **metric connection** is a connection in a vector bundle *E* equipped with a bundle metric; that is, a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported along any curve. This is equivalent to:

In mathematics, a **holomorphic vector bundle** is a complex vector bundle over a complex manifold X such that the total space E is a complex manifold and the projection map π : *E* → *X* is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A **holomorphic line bundle** is a rank one holomorphic vector bundle.

In mathematics, **geometric calculus** extends the geometric algebra to include differentiation and integration. The formalism is powerful and can be shown to encompass other mathematical theories including differential geometry and differential forms.

In differential geometry, the **integration along fibers** of a *k*-form yields a -form where *m* is the dimension of the fiber, via "integration".

This article summarizes several identities in exterior calculus.

- Cartan, Élie (1899). "Sur certaines expressions différentielles et le problème de Pfaff".
*Annales Scientifiques de l'École Normale Supérieure*. Série 3 (in French). Paris: Gauthier-Villars.**16**: 239–332. ISSN 0012-9593. JFM 30.0313.04 . Retrieved 2 Feb 2016. - Conlon, Lawrence (2001).
*Differentiable manifolds*. Basel, Switzerland: Birkhäuser. p. 239. ISBN 0-8176-4134-3. - Darling, R. W. R. (1994).
*Differential forms and connections*. Cambridge, UK: Cambridge University Press. p. 35. ISBN 0-521-46800-0. - Flanders, Harley (1989).
*Differential forms with applications to the physical sciences*. New York: Dover Publications. p. 20. ISBN 0-486-66169-5. - Loomis, Lynn H.; Sternberg, Shlomo (1989).
*Advanced Calculus*. Boston: Jones and Bartlett. pp. 304–473 (ch. 7–11). ISBN 0-486-66169-5. - Ramanan, S. (2005).
*Global calculus*. Providence, Rhode Island: American Mathematical Society. p. 54. ISBN 0-8218-3702-8. - Spivak, Michael (1971).
*Calculus on Manifolds*. Boulder, Colorado: Westview Press. ISBN 9780805390216. - Warner, Frank W. (1983),
*Foundations of differentiable manifolds and Lie groups*, Graduate Texts in Mathematics,**94**, Springer, ISBN 0-387-90894-3

- "The derivative isn't what you think it is".
*Aleph Zero*. November 3, 2020 – via YouTube.

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