exterior derivative#In terms of invariant formula

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

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

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

There are a variety of equivalent definitions of the exterior derivative of a general {{math|k}}-form.

The exterior derivative $d$ is defined to be the unique {{math|ℝ}}-linear mapping from {{math|k}}-forms to {{math|(k + 1)}}-forms that has the following properties:

• The operator $d$ applied to the $0$-form $f$ is the differential $df$ of $f$
• If $\backslash alpha$ and $\backslash beta$ are two $k$-forms, then $d(a\backslash alpha+b\backslash beta)=ad\backslash alpha+bd\backslash beta$ for any field elements $a,b$
• If $\backslash alpha$ is a $k$-form and $\backslash beta$ is an $l$-form, then $d(\backslash alpha\backslash wedge\backslash beta)=d\backslash alpha\backslash wedge\backslash beta+(-1)^k\backslash alpha\backslash wedge\; d\backslash beta$ (graded product rule)
• If $\backslash alpha$ is a $k$-form, then $d(d\backslash alpha)=0$ (Poincaré's lemma)

If $f$ and $g$ are two $0$-forms (functions), then from the third property for the quantity $d(f\backslash wedge\; g)$, which is simply $d(fg)$, the familiar product rule $d(fg)=g\backslash ,df+f\backslash ,dg$ is recovered. The third property can be generalised, for instance, if $\backslash alpha$ is a $k$-form, $\backslash beta$ is an $l$-form and $\backslash gamma$ is an $m$-form, then

:$d(\backslash alpha\backslash wedge\backslash beta\backslash wedge\backslash gamma)=d\backslash alpha\backslash wedge\backslash beta\backslash wedge\backslash gamma+(-1)^k\backslash alpha\backslash wedge\; d\backslash beta\backslash wedge\backslash gamma+(-1)^\{k+l\}\backslash alpha\backslash wedge\backslash beta\backslash wedge\; d\backslash gamma.$

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

: $\backslash varphi\; =\; g\backslash ,dx^I\; =\; g\backslash ,dx^\{i\_1\}\backslash wedge\; dx^\{i\_2\}\backslash wedge\backslash cdots\backslash wedge\; dx^\{i\_k\}$

over {{math|ℝ{{sup|n}}}} is defined as

: $d\{\backslash varphi\}\; =\; dg\backslash wedge\; dx^\{i\_1\}\backslash wedge\; dx^\{i\_2\}\backslash wedge\backslash cdots\backslash wedge\; dx^\{i\_k\}\; =\; \backslash frac\{\backslash partial\; g\}\{\backslash partial\; x^j\}\; \backslash ,\; dx^j\; \backslash wedge\; \backslash ,dx^\{i\_1\}\backslash wedge\; dx^\{i\_2\}\backslash wedge\backslash cdots\backslash wedge\; dx^\{i\_k\}$

(using the Einstein summation convention). The definition of the exterior derivative is extended linearly to a general {{math|k}}-form (which is expressible as a linear combination of basic simple $k$-forms)

: $\backslash omega\; =\; f\_I\; \backslash ,\; dx^I,$

where each of the components of the multi-index {{math|I}} run over all the values in {{math|{1, ..., n}}}. Note that whenever {{math|j}} equals one of the components of the multi-index {{math|I}} then {{math|1=dx{{i sup|j}} ∧ dx{{i sup|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 {{math|k}}-form {{math|φ}} as defined above,

: $\backslash begin\{align\}$

d{\varphi} &= d\left (g\,dx^{i_1} \wedge \cdots \wedge dx^{i_k} \right ) \\

&= dg \wedge \left (dx^{i_1} \wedge \cdots \wedge dx^{i_k} \right ) +

g\,d\left (dx^{i_1}\wedge \cdots \wedge dx^{i_k} \right ) \\

&= dg \wedge dx^{i_1} \wedge \cdots \wedge dx^{i_k} + g \sum_{p=1}^k (-1)^{p-1} \, dx^{i_1}

\wedge \cdots \wedge dx^{i_{p-1}} \wedge d^2x^{i_p} \wedge dx^{i_{p+1}} \wedge \cdots \wedge dx^{i_k} \\

&= dg \wedge dx^{i_1} \wedge \cdots \wedge dx^{i_k} \\

&= \frac{\partial g}{\partial x^i} \, dx^i \wedge dx^{i_1} \wedge \cdots \wedge dx^{i_k} \\

\end{align}

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

This result extends directly to the general {{math|k}}-form {{math|ω}} as

: $d\backslash omega\; =\; \backslash frac\{\backslash partial\; f\_I\}\{\backslash partial\; x^i\}\; \backslash ,\; dx^i\; \backslash wedge\; dx^I\; .$

In particular, for a {{math|1}}-form {{math|ω}}, the components of {{math|dω}} in local coordinates are

: $(d\backslash omega)\_\{ij\}\; =\; \backslash partial\_i\; \backslash omega\_j\; -\; \backslash partial\_j\; \backslash omega\_i.$

Caution: There are two conventions regarding the meaning of $dx^\{i\_1\}\; \backslash wedge\; \backslash cdots\; \backslash wedge\; dx^\{i\_k\}$. Most current authors{{fact|date=April 2020}} have the convention that

: $\backslash left(dx^\{i\_1\}\; \backslash wedge\; \backslash cdots\; \backslash wedge\; dx^\{i\_k\}\backslash right)\; \backslash left(\; \backslash frac\{\backslash partial\}\{\backslash partial\; x^\{i\_1\}\},\; \backslash ldots,\; \backslash frac\{\backslash partial\}\{\backslash partial\; x^\{i\_k\}\}\; \backslash right)\; =\; 1\; .$

while in older texts like Kobayashi and Nomizu or Helgason

: $\backslash left(dx^\{i\_1\}\; \backslash wedge\; \backslash cdots\; \backslash wedge\; dx^\{i\_k\}\backslash right)\; \backslash left(\; \backslash frac\{\backslash partial\}\{\backslash partial\; x^\{i\_1\}\},\; \backslash ldots,\; \backslash frac\{\backslash partial\}\{\backslash partial\; x^\{i\_k\}\}\; \backslash right)\; =\; \backslash frac\{1\}\{k!\}\; .$

Alternatively, an explicit formula can be given Spivak(1970), p 7-18, Th. 13 for the exterior derivative of a {{math|k}}-form {{math|ω}}, when paired with {{math|k + 1}} arbitrary smooth vector fields {{math|V₀, V₁, ..., V_k}}:

:

where {{math|[V_i, V_j]}} denotes the Lie bracket and a hat denotes the omission of that element:

: $\backslash omega\; (V\_0,\; \backslash ldots,\; \backslash widehat\; V\_i,\; \backslash ldots,\; V\_k\; )\; =\; \backslash omega(V\_0,\; \backslash ldots,\; V\_\{i-1\},\; V\_\{i+1\},\; \backslash ldots,\; V\_k\; ).$

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

Note: With the conventions of e.g., Kobayashi–Nomizu and Helgason the formula differs by a factor of {{math|{{sfrac|k + 1}}}}:

:$\backslash begin\{align\}$

d\omega(V_0, \ldots, V_k) ={}

& {1 \over k+1} \sum_i(-1)^i \, V_i ( \omega (V_0, \ldots, \widehat V_i, \ldots,V_k )) \\

& {}+ {1 \over k+1} \sum_{i

\end{align}

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

: $\backslash begin\{align\}$

d\sigma &= du \wedge dx^1 \wedge dx^2 \\

&= \left(\sum_{i=1}^n \frac{\partial u}{\partial x^i} \, dx^i\right) \wedge dx^1 \wedge dx^2 \\

&= \sum_{i=3}^n \left( \frac{\partial u}{\partial x^i} \, dx^i \wedge dx^1 \wedge dx^2 \right )

\end{align}

The last formula, where summation starts at {{math|i {{=}} 3}}, follows easily from the properties of the exterior product. Namely, {{math|1=dx{{i sup|i}} ∧ dx{{i sup|i}} = 0}}.

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

: $\backslash begin\{align\}$

d\sigma

&= \left( \sum_{i=1}^2 \frac{\partial u}{\partial x^i} dx^i \wedge dx \right) + \left( \sum_{i=1}^2 \frac{\partial v}{\partial x^i} \, dx^i \wedge dy \right) \\

&= \left(\frac{\partial{u}}{\partial{x}} \, dx \wedge dx + \frac{\partial{u}}{\partial{y}} \, dy \wedge dx\right) + \left(\frac{\partial{v}}{\partial{x}} \, dx \wedge dy + \frac{\partial{v}}{\partial{y}} \, dy \wedge dy\right) \\

&= 0 - \frac{\partial{u}}{\partial{y}} \, dx \wedge dy + \frac{\partial{v}}{\partial{x}} \, dx \wedge dy + 0 \\

&= \left(\frac{\partial{v}}{\partial{x}} - \frac{\partial{u}}{\partial{y}}\right) \, dx \wedge dy

\end{align}

{{main|Generalized Stokes' theorem}}

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

: $\backslash int\_M\; d\backslash omega\; =\; \backslash int\_\{\backslash partial\{M\}\}\; \backslash omega$

Intuitively, if one thinks of {{math|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 {{math|M}}.

{{main article|Closed and exact forms}}

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

Because the exterior derivative {{math|d}} has the property that {{math|1=d{{i sup|2}} = 0}}, it can be used as the differential (coboundary) to define de Rham cohomology on a manifold. The {{math|k}}-th de Rham cohomology (group) is the vector space of closed {{math|k}}-forms modulo the exact {{math|k}}-forms; as noted in the previous section, the Poincaré lemma states that these vector spaces are trivial for a contractible region, for {{math|k > 0}}. For smooth manifolds, integration of forms gives a natural homomorphism from the de Rham cohomology to the singular cohomology over {{math|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 {{math| f : M → N}} is a smooth map and {{math|Ω{{sup|k}}}} is the contravariant smooth functor that assigns to each manifold the space of {{math|k}}-forms on the manifold, then the following diagram commutes

: none

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

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

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

When an inner product {{math|{{langle}}·,·{{rangle}}}} is defined, the gradient {{math|∇f }} of a function {{math| f }} is defined as the unique vector in {{math|V}} such that its inner product with any element of {{math|V}} is the directional derivative of {{math| f }} along the vector, that is such that

: $\backslash langle\; \backslash nabla\; f,\; \backslash cdot\; \backslash rangle\; =\; df\; =\; \backslash sum\_\{i=1\}^n\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; x^i\}\backslash ,\; dx^i\; .$

That is,

: $\backslash nabla\; f\; =\; (df)^\backslash sharp\; =\; \backslash sum\_\{i=1\}^n\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; x^i\}\backslash ,\; \backslash left(dx^i\backslash right)^\backslash sharp\; ,$

where {{math|{{music|sharp}}}} denotes the musical isomorphism {{math|{{music|sharp}} : V{{sup|∗}} → V}} mentioned earlier that is induced by the inner product.

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

A vector field {{math|1=V = (v₁, v₂, ..., v_n)}} on {{math|ℝ{{sup|n}}}} has a corresponding {{math|(n − 1)}}-form

: $\backslash begin\{align\}$

\omega_V &= v_1 \left (dx^2 \wedge \cdots \wedge dx^n \right) - v_2 \left (dx^1 \wedge dx^3 \wedge \cdots \wedge dx^n \right ) + \cdots + (-1)^{n-1}v_n \left (dx^1 \wedge \cdots \wedge dx^{n-1} \right) \\

&= \sum_{i=1}^n (-1)^{(i-1)}v_i \left (dx^1 \wedge \cdots \wedge dx^{i-1} \wedge \widehat{dx^{i}} \wedge dx^{i+1} \wedge \cdots \wedge dx^n \right )

\end{align}

where $\backslash widehat\{dx^\{i\}\}$ denotes the omission of that element.

(For instance, when {{math|1=n = 3}}, i.e. in three-dimensional space, the {{math|2}}-form {{math|ω_V}} is locally the scalar triple product with {{math|V}}.) The integral of {{math|ω_V}} over a hypersurface is the flux of {{math|V}} over that hypersurface.

The exterior derivative of this {{math|(n − 1)}}-form is the {{math|n}}-form

: $d\backslash omega\; \_V\; =\; \backslash operatorname\{div\}\; V\; \backslash left\; (dx^1\; \backslash wedge\; dx^2\; \backslash wedge\; \backslash cdots\; \backslash wedge\; dx^n\; \backslash right\; ).$

A vector field {{math|V}} on {{math|ℝ{{sup|n}}}} also has a corresponding {{math|1}}-form

: $\backslash eta\_V\; =\; v\_1\; \backslash ,\; dx^1\; +\; v\_2\; \backslash ,\; dx^2\; +\; \backslash cdots\; +\; v\_n\; \backslash ,\; dx^n.$

Locally, {{math|η_V}} is the dot product with {{math|V}}. The integral of {{math|η_V}} along a path is the work done against {{math|−V}} along that path.

When {{math|1=n = 3}}, in three-dimensional space, the exterior derivative of the {{math|1}}-form {{math|η_V}} is the {{math|2}}-form

: $d\backslash eta\_V\; =\; \backslash omega\_\{\backslash operatorname\{curl\}\; V\}.$

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

: $\backslash begin\{array\}\{rcccl\}$

\operatorname{grad} f &\equiv& \nabla f &=& \left( d f \right)^\sharp \\

\operatorname{div} F &\equiv& \nabla \cdot F &=& {\star d {\star} \mathord{\left( F^\flat \right)}} \\

\operatorname{curl} F &\equiv& \nabla \times F &=& \left( {\star} d \mathord{\left( F^\flat \right)} \right)^\sharp \\

\Delta f &\equiv& \nabla^2 f &=& {\star} d {\star} d f \\

& & \nabla^2 F &=& \left(d{\star}d{\star}\mathord{\left(F^{\flat}\right)} - {\star}d{\star}d\mathord{\left(F^{\flat}\right)}\right)^{\sharp} , \\

\end{array}

where {{math|⋆}} is the Hodge star operator, {{math|{{music|flat}}}} and {{math|{{music|sharp}}}} are the musical isomorphisms, {{math| f }} is a scalar field and {{math|F}} is a vector field.

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

{{Div col|colwidth=20em}}

• Exterior covariant derivative
• de Rham complex
• Finite element exterior calculus
• Discrete exterior calculus
• Green's theorem
• Lie derivative
• Stokes' theorem
• Fractal derivative

{{div col end}}

{{Reflist|30em}}

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• {{citation|last=Spivak|first= MIchael| author-link =Michael Spivak|title= A Comprehensive Introduction to Differential Geometry|volume= 1|publisher= Publish or Perish, Inc|year=1970|isbn= 0-914098-00-4| location= Boston, MA}}
• {{citation|last=Warner|first= Frank W.|title= Foundations of differentiable manifolds and Lie groups|series= Graduate Texts in Mathematics|volume= 94|publisher= Springer|year=1983|isbn= 0-387-90894-3}}

• Archived at [https://ghostarchive.org/varchive/youtube/20211211/2ptFnIj71SM Ghostarchive]{{cbignore}} and the [https://web.archive.org/web/20201104033452/https://www.youtube.com/watch?v=2ptFnIj71SM&feature=youtu.be Wayback Machine]{{cbignore}}: {{cite web |title=The derivative isn't what you think it is |work=Aleph Zero |date=November 3, 2020 |via=YouTube |url=https://www.youtube.com/watch?v=2ptFnIj71SM }}{{cbignore}}

{{Manifolds}}

{{Calculus topics}}

{{Tensors|state=collapsed}}

Category:Differential forms

Category:Differential operators

Category:Generalizations of the derivative