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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; it allows for a natural, metricindependent generalization of Stokes' theorem, Gauss's theorem, and Green's theorem from vector calculus.
If a kform is thought of as measuring the flux through an infinitesimal kparallelepiped, then its exterior derivative can be thought of as measuring the net flux through the boundary of a (k + 1)parallelepiped.
Contents

Definition 1

Axioms for the exterior derivative 1.1

Exterior derivative in local coordinates 1.2

Invariant formula 1.3

Stokes' theorem on manifolds 2

Examples 3

Further properties 4

Closed and exact forms 4.1

de Rham cohomology 4.2

Naturality 4.3

Exterior derivative in vector calculus 5

Gradient 5.1

Divergence 5.2

Curl 5.3

Invariant formulations of grad, curl, div, and Laplacian 5.4

See also 6

References 7
Definition
The exterior derivative of a differential form of degree k is a differential form of degree k + 1.
If f is a smooth function (a 0form), then the exterior derivative of f is the differential of f . That is, df is the unique 1form 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.
There are a variety of equivalent definitions of the exterior derivative of a general kform.
Axioms for the exterior derivative
The exterior derivative is defined to be the unique Rlinear mapping from kforms to (k + 1)forms satisfying the following properties:

df is the differential of f for smooth functions f .

d(df ) = 0 for any smooth function f .

d(α ∧ β) = dα ∧ β + (−1)^{p} (α ∧ dβ) where α is a pform. 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: in fact, d(dα) = 0 for any kform α; more succinctly, d^{2} = 0. The third defining property implies as a special case that if f is a function and α a kform, then d(fα) = d(f ∧ α) = df ∧ α + f ∧ dα because functions are 0forms, and scalar multiplication and the exterior product are equivalent when one of the arguments is a scalar.
Exterior derivative in local coordinates
Alternatively, one can work entirely in a local coordinate system (x^{1}, ..., x^{n}). First, the coordinate differentials dx^{1}, ..., dx^{n} form a basic set of oneforms within the coordinate chart. The formulas in this section rely on the Einstein summation convention. Given a multiindex I = (i_{1}, ..., i_{k}) with 1 ≤ i_{p} ≤ n for 1 ≤ p ≤ k (and an abuse of notation dx^{I}), the exterior derivative of a kform

\omega = f_I \mathrm{d} x^I = f_{i_1,i_2\cdots i_k}\mathrm{d}x^{i_1}\wedge \mathrm{d}x^{i_2}\wedge\cdots\wedge \mathrm{d}x^{i_k}
over R^{n} is defined as

\mathrm{d}{\omega} = \sum_{i=1}^n \frac{\partial f_I}{\partial x^i} \mathrm{d}x^i \wedge \mathrm{d} x^I.
For a general kform

\omega = \sum_I f_I \mathrm{d}x^I,
where the components of the multiindex I run over all the values in {1, ..., n}, the definition of the exterior derivative is extended linearly. Note that whenever i is one of the components of the multiindex I then dx^{i} ∧ dx^{I} = 0 (see wedge product).
The definition of the exterior derivative in local coordinates follows from the preceding definition. Indeed, if ω = f_{I} dx^{i1} ∧ ... ∧ dx^{ik}, then

\begin{align} \mathrm{d}{\omega} &= \mathrm{d} \left (f_I \mathrm{d}x^{i_1} \wedge \cdots \wedge \mathrm{d}x^{i_k} \right ) \\ &= \mathrm{d}f_I \wedge \left (\mathrm{d}x^{i_1} \wedge \cdots \wedge \mathrm{d}x^{i_k} \right ) + f_I \mathrm{d} \left ( \mathrm{d} x^{i_1}\wedge \cdots \wedge \mathrm{d}x^{i_k} \right ) \\ &= \mathrm{d}f_I \wedge \mathrm{d}x^{i_1} \wedge \cdots \wedge \mathrm{d}x^{i_k} + \sum_{p=1}^k (1)^{(p1)} f_I \mathrm{d} x^{i_1} \wedge \cdots \wedge \mathrm{d}x^{i_{p1}} \wedge \mathrm{d}^2x^{i_p} \wedge \mathrm{d}x^{i_{p+1}} \wedge \cdots \wedge\mathrm{d} x^{i_k} \\ &= \mathrm{d}f_I \wedge \mathrm{d}x^{i_1} \wedge \cdots \wedge \mathrm{d}x^{i_k} \\ &= \sum_{i=1}^n \frac{\partial f_I}{\partial x^i} \mathrm{d}x^i \wedge \mathrm{d}x^{i_1} \wedge \cdots \wedge \mathrm{d}x^{i_k} \\ \end{align}
Here, we have interpreted f_{I} as a 0form, and then applied the properties of the exterior derivative.
Invariant formula
Alternatively, an explicit formula can be given for the exterior derivative of a kform ω, when paired with k + 1 arbitrary smooth vector fields V_{0},V_{1}, ..., V_{k}:

\mathrm{d}\omega(V_0,...,V_k) = \sum_i(1)^{i} V_i \left( \omega \left (V_0, \ldots, \hat V_i, \ldots,V_k \right )\right) +\sum_{i
where [V_{i}, V_{j}] denotes the Lie bracket and a hat denotes the omission of that element:

\omega \left (V_0, \ldots, \hat V_i, \ldots,V_k \right ) = \omega \left (V_0, \ldots, V_{i1}, V_{i+1}, \ldots, V_k \right ).
In particular, for 1forms we have: dω(X, Y) = X(ω(Y)) − Y(ω(X)) − ω([X, Y]), where X and Y are vector fields, X(ω(Y)) is the scalar field defined by the vector field X∈Γ(T M) applied as a differential operator ("directional derivative along X") to the scalar field defined by applying ω∈Γ^{*}(T M) as a covector field to the vector field Y∈Γ(T M) and likewise for Y(ω(X)).
Stokes' theorem on manifolds
If M is a compact smooth orientable ndimensional manifold with boundary, and ω is an (n − 1)form on M, then the generalized form of Stokes' theorem states that:

\int_M \mathrm{d}\omega = \int_{\partial{M}} \omega
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.
Examples
Example 1. Consider σ = u dx^{1} ∧ dx^{2} over a 1form basis dx^{1}, ..., dx^{n}. The exterior derivative is:

\begin{align} \mathrm{d} \sigma &= \mathrm{d}(u) \wedge \mathrm{d}x^1 \wedge \mathrm{d}x^2 \\ &= \left(\sum_{i=1}^n \frac{\partial u}{\partial x^i} \mathrm{d}x^i\right) \wedge \mathrm{d}x^1 \wedge \mathrm{d}x^2 \\ &= \sum_{i=3}^n \left( \frac{\partial u}{\partial x^i} \mathrm{d}x^i \wedge \mathrm{d}x^1 \wedge \mathrm{d}x^2 \right ) \end{align}
The last formula follows easily from the properties of the wedge product. Namely, dx^{i} ∧ dx^{i} = 0.
Example 2. Let σ = u dx + v dy be a 1form 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,

\begin{align} \mathrm{d} \sigma &= \left( \sum_{i=1}^2 \frac{\partial u}{\partial x^i} \mathrm{d}x^i \wedge \mathrm{d}x \right) + \left( \sum_{i=1}^2 \frac{\partial v}{\partial x^i} \mathrm{d}x^i \wedge \mathrm{d}y \right) \\ &= \left(\frac{\partial{u}}{\partial{x}} \mathrm{d}x \wedge \mathrm{d}x + \frac{\partial{u}}{\partial{y}} \mathrm{d}y \wedge \mathrm{d}x\right) + \left(\frac{\partial{v}}{\partial{x}} \mathrm{d}x \wedge \mathrm{d}y + \frac{\partial{v}}{\partial{y}} \mathrm{d}y \wedge \mathrm{d}y\right) \\ &= 0  \frac{\partial{u}}{\partial{y}} \mathrm{d}x \wedge \mathrm{d}y + \frac{\partial{v}}{\partial{x}} \mathrm{d}x \wedge \mathrm{d}y + 0 \\ &= \left(\frac{\partial{v}}{\partial{x}}  \frac{\partial{u}}{\partial{y}}\right) \mathrm{d}x \wedge \mathrm{d}y \end{align}
Further properties
Closed and exact forms
A kform 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.
de Rham cohomology
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 kth de Rham cohomology (group) is the vector space of closed kforms modulo the exact kforms; 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 farreaching 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.
Naturality
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 kforms 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}.
Exterior derivative in vector calculus
Most vector calculus operators are special cases of, or have close relationships to, the notion of exterior differentiation.
Gradient
A smooth function f : R^{n} → R is a 0form. The exterior derivative of this 0form is the 1form

\mathrm{d}f = \sum_{i=1}^n \frac{\partial f}{\partial x^i}\, \mathrm{d}x^i = \langle \nabla f,\cdot \rangle.
That is, the form df acts on any vector field V by outputting, at each point, the scalar product of V with the gradient ∇f of f .
The 1form df is a section of the cotangent bundle, that gives a local linear approximation to f in the cotangent space at each point.
Divergence
A vector field V = (v_{1}, v_{2}, ... v_{n}) on R^{n} has a corresponding (n − 1)form

\begin{align} \omega_V &= v_1 \left (\mathrm{d}x^2 \wedge \mathrm{d}x^3 \wedge \cdots \wedge \mathrm{d}x^n \right)  v_2 \left (\mathrm{d}x^1 \wedge \mathrm{d}x^3 \cdots \wedge \mathrm{d}x^n \right ) + \cdots + (1)^{n1}v_n \left (\mathrm{d}x^1 \wedge \cdots \wedge \mathrm{d}x^{n1} \right) \\ &=\sum_{p=1}^n (1)^{(p1)}v_p \left (\mathrm{d}x^1 \wedge \cdots \wedge \mathrm{d}x^{p1} \wedge \widehat{\mathrm{d}x^{p}} \wedge \mathrm{d}x^{p+1} \wedge \cdots \wedge \mathrm{d}x^n \right ) \end{align}
where \widehat{\mathrm{d}x^{p}} denotes the omission of that element.
(For instance, when n = 3, in threedimensional space, the 2form ω_{V} is locally the scalar triple product with V.) The integral of ω_{V} over a hypersurface is the flux of V over that hypersurface.
The exterior derivative of this (n − 1)form is the nform

\mathrm{d} \omega _V = \operatorname{div}(V) \left (\mathrm{d}x^1 \wedge \mathrm{d}x^2 \wedge \cdots \wedge \mathrm{d}x^n \right ).
Curl
A vector field V on R^{n} also has a corresponding 1form

\eta_V = v_1 \mathrm{d}x^1 + v_2 \mathrm{d}x^2 + \cdots + v_n \mathrm{d}x^n.,
Locally, η_{V} is the dot product with V. The integral of η_{V} along a path is the work done against −V along that path.
When n = 3, in threedimensional space, the exterior derivative of the 1form η_{V} is the 2form

\mathrm{d} \eta_V = \omega_{\operatorname{curl}(V)}.
Invariant formulations of grad, curl, div, and Laplacian
On any Riemannian manifold, the standard vector calculus operators can be written in coordinatefree notation as follows:

\begin{array}{rcccl} \operatorname{grad}(f) &=& \nabla f &=& \left( \mathrm{d} f \right)^\sharp \\ \operatorname{div}(F) &=& \nabla \cdot F &=& \star \mathrm{d} \left( \star F^\flat \right) \\ \operatorname{curl}(F) &=& \nabla \times F &=& \left[ \star \left( \mathrm{d} F^\flat \right) \right]^\sharp, \\ \Delta f &=& \nabla^2 f &=& \star \mathrm{d} \left( \star \mathrm{d} f \right) \\ \end{array}
where \star is the Hodge star operator and \flat and \sharp are the musical isomorphisms.
See also
References

Flanders, Harley (1989). Differential forms with applications to the physical sciences. New York: Dover Publications. p. 20.

Ramanan, S. (2005). Global calculus. Providence, Rhode Island: American Mathematical Society. p. 54.

Conlon, Lawrence (2001). Differentiable manifolds. Basel, Switzerland: Birkhäuser. p. 239.

Darling, R. W. R. (1994). Differential forms and connections. Cambridge, UK: Cambridge University Press. p. 35.
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