View source for Exterior derivative

{{Short description|Operation on differential forms}}
{{no footnotes|date=July 2019}}
{{Calculus |Multivariable}}

On a [[differentiable manifold]], the '''exterior derivative''' extends the concept of the [[pushforward (differential)|differential]] of a function to [[differential form]]s of higher degree. The exterior derivative was first described in its current form by [[Élie Cartan]] in 1899. The resulting calculus, known as [[exterior calculus]], allows for a natural, metric-independent generalization of [[Stokes' theorem]], [[Gauss's theorem]], and [[Green's theorem]] from vector calculus.

If a differential {{math|''k''}}-form is thought of as measuring the [[flux]] through an infinitesimal {{math|''k''}}-[[Parallelepiped#Parallelotope|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 {{math|(''k'' + 1)}}-parallelotope at each point.

== Definition ==
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|&thinsp;''f''&thinsp;}} is a [[Smoothness|smooth function]] (a {{math|0}}-form), then the exterior derivative of {{math|&thinsp;''f''&thinsp;}} is the [[Pushforward (differential)|differential]] of {{math|&thinsp;''f''&thinsp;}}. That is, {{math|''df''&thinsp;}} is the unique [[1-form|{{math|1}}-form]] such that for every smooth [[Vector field#Vector fields on manifolds|vector field]] {{math|''X''}}, {{math|1=''df''&thinsp;(''X'') = ''d''<sub>''X''</sub>&thinsp;''f''&thinsp;}}, where {{math|''d''<sub>''X''</sub>&thinsp;''f''&thinsp;}} is the [[directional derivative]] of {{math|&thinsp;''f''&thinsp;}} 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.

=== In terms of axioms ===
The exterior derivative 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 <math>d</math> applied to the <math>0</math>-form <math>f</math> is the differential <math>df</math> of <math>f</math>
*If <math>\alpha</math> and <math>\beta</math> are two <math>k</math>-forms, then <math>d(a\alpha+b\beta)=ad\alpha+bd\beta</math> for any field elements <math>a,b</math>
*If <math>\alpha</math> is a <math>k</math>-form and <math>\beta</math> is an <math>l</math>-form, then <math>d(\alpha\wedge\beta)=d\alpha\wedge\beta+(-1)^k\alpha\wedge d\beta</math> (''[[Product rule#Derivations in abstract algebra and differential geometry|graded product rule]]'')
*If <math>\alpha</math> is a <math>k</math>-form, then <math>d(d\alpha)=0</math> (Poincaré's lemma) 

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

:<math>d(\alpha\wedge\beta\wedge\gamma)=d\alpha\wedge\beta\wedge\gamma+(-1)^k\alpha\wedge d\beta\wedge\gamma+(-1)^{k+l}\alpha\wedge\beta\wedge d\gamma.</math>

=== In terms of local coordinates ===
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
: <math>\varphi = g\,dx^I = g\,dx^{i_1}\wedge dx^{i_2}\wedge\cdots\wedge dx^{i_k}</math>
over {{math|ℝ{{sup|''n''}}}} is defined as
: <math>d{\varphi} =  dg\wedge dx^{i_1}\wedge dx^{i_2}\wedge\cdots\wedge dx^{i_k} = \frac{\partial g}{\partial x^j} \, dx^j \wedge \,dx^{i_1}\wedge dx^{i_2}\wedge\cdots\wedge dx^{i_k}</math>
(using the [[Einstein notation|Einstein summation convention]]).  The definition of the exterior derivative is extended [[linear]]ly to a general {{math|''k''}}-form (which is expressible as a linear combination of basic simple <math>k</math>-forms)
: <math>\omega = f_I \, dx^I,</math>
where each of the components of the multi-index {{math|''I''}} run over all the values in {{math|{1, ..., ''n''}<nowiki/>}}. 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 [[#In terms of axioms|definition in terms of axioms]]. Indeed, with the {{math|''k''}}-form {{math|''φ''}} as defined above,
: <math>\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}</math>

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
: <math>d\omega = \frac{\partial f_I}{\partial x^i} \, dx^i \wedge dx^I .</math>

In particular, for a {{math|1}}-form {{math|''ω''}}, the components of {{math|''dω''}} in [[Local coordinate system|local coordinates]] are
: <math>(d\omega)_{ij} = \partial_i \omega_j - \partial_j \omega_i. </math>

''Caution'':  There are two conventions regarding the meaning of <math>dx^{i_1} \wedge \cdots \wedge dx^{i_k}</math>. Most current authors{{fact|date=April 2020}} have the convention that 
: <math>\left(dx^{i_1} \wedge \cdots \wedge dx^{i_k}\right) \left( \frac{\partial}{\partial x^{i_1}}, \ldots, \frac{\partial}{\partial x^{i_k}} \right) = 1 .</math>
while in older texts like Kobayashi and Nomizu or Helgason 
: <math>\left(dx^{i_1} \wedge \cdots \wedge dx^{i_k}\right) \left( \frac{\partial}{\partial x^{i_1}}, \ldots, \frac{\partial}{\partial x^{i_k}} \right) = \frac{1}{k!} .</math>

=== In terms of invariant formula ===
Alternatively, an explicit formula can be given <ref> Spivak(1970), p 7-18, Th. 13 </ref> for the exterior derivative of a {{math|''k''}}-form {{math|''ω''}}, when paired with {{math|''k'' + 1}} arbitrary smooth [[vector field]]s {{math|''V''<sub>0</sub>, ''V''<sub>1</sub>, ..., ''V''<sub>''k''</sub>}}:
: <math>d\omega(V_0, \ldots, V_k) = \sum_i(-1)^{i} V_i ( \omega  (V_0, \ldots, \widehat V_i, \ldots,V_k )) + \sum_{i<j}(-1)^{i+j}\omega  ([V_i, V_j], V_0, \ldots, \widehat V_i, \ldots, \widehat V_j, \ldots, V_k )</math>

where {{math|[''V<sub>i</sub>'', ''V<sub>j</sub>'']}} denotes the [[Lie bracket of vector fields|Lie bracket]] and a hat denotes the omission of that element:
: <math>\omega (V_0, \ldots, \widehat V_i, \ldots, V_k ) = \omega(V_0, \ldots, V_{i-1}, V_{i+1}, \ldots, V_k ).</math>

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}}}}:
:<math>\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<j}(-1)^{i+j}\omega([V_i, V_j], V_0, \ldots, \widehat V_i, \ldots, \widehat V_j, \ldots, V_k ).
\end{align}</math>

== Examples ==
'''Example 1.''' Consider {{math|1=''σ'' = ''u''&thinsp;''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:
: <math>\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}</math>

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''&thinsp;''dx'' + ''v''&thinsp;''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
: <math>\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}</math>

== Stokes' theorem on manifolds ==
{{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 [[Generalized Stokes' theorem|the generalized form of Stokes' theorem]] states that
: <math>\int_M d\omega = \int_{\partial{M}} \omega</math>

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''}}.

== Further properties ==

=== Closed and exact forms ===
{{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 (algebra)|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 (mathematics)|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.

=== de Rham cohomology ===
Because the exterior derivative {{math|''d''}} has the property that {{math|1=''d''{{i sup|2}} = 0}}, it can be used as the [[Cochain complex|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 manifold]]s, 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 [[Chain complex#Formal definition|boundary map]] on singular simplices.

=== Naturality ===
The exterior derivative is natural in the technical sense: if {{math|&thinsp;''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
: [[Image:Exteriorderivnatural.png|none]]
so {{math|1=''d''(&thinsp;''f''{{i sup|∗}}''ω'') = &thinsp;''f''{{i sup|∗}}''dω''}}, where {{math|&thinsp;''f''{{i sup|∗}}}} denotes the [[pullback (differential geometry)|pullback]] of {{math|&thinsp;''f''&thinsp;}}. This follows from that {{math|&thinsp;''f''{{i sup|∗}}''ω''(·)}}, by definition, is {{math|''ω''(&thinsp;''f''<sub>∗</sub>(·))}}, {{math|&thinsp;''f''<sub>∗</sub>}} being the [[Pushforward (differential)|pushforward]] of {{math|&thinsp;''f''&thinsp;}}. Thus {{math|''d''}} is a [[natural transformation]] from {{math|Ω{{sup|''k''}}}} to {{math|Ω{{sup|''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]] {{math|&thinsp;''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''&thinsp;}} of a function {{math|&thinsp;''f''&thinsp;}} 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|&thinsp;''f''&thinsp;}} along the vector, that is such that
: <math>\langle \nabla f, \cdot \rangle = df = \sum_{i=1}^n \frac{\partial f}{\partial x^i}\, dx^i .</math>

That is,
: <math>\nabla f = (df)^\sharp = \sum_{i=1}^n \frac{\partial f}{\partial x^i}\, \left(dx^i\right)^\sharp ,</math>
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''&thinsp;}} is a section of the [[cotangent bundle]], that gives a local linear approximation to {{math|&thinsp;''f''&thinsp;}} in the cotangent space at each point.

=== Divergence ===
A vector field {{math|1=''V'' = (''v''<sub>1</sub>, ''v''<sub>2</sub>, ..., ''v<sub>n</sub>'')}} on {{math|ℝ{{sup|''n''}}}} has a corresponding {{math|(''n'' − 1)}}-form
: <math>\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}</math>
where <math>\widehat{dx^{i}}</math> 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<sub>V</sub>''}} is locally the [[scalar triple product]] with {{math|''V''}}.)  The integral of {{math|''ω<sub>V</sub>''}} 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
: <math>d\omega _V = \operatorname{div} V \left (dx^1 \wedge dx^2 \wedge \cdots \wedge dx^n \right ).</math>

=== Curl ===
A vector field {{math|''V''}} on {{math|ℝ{{sup|''n''}}}} also has a corresponding {{math|1}}-form
: <math>\eta_V = v_1 \, dx^1 + v_2 \, dx^2 + \cdots + v_n \, dx^n.</math>

Locally, {{math|''η<sub>V</sub>''}} is the [[dot product]] with {{math|''V''}}. The integral of {{math|''η<sub>V</sub>''}} along a path is the [[Mechanical work|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<sub>V</sub>''}} is the {{math|2}}-form
: <math>d\eta_V = \omega_{\operatorname{curl} V}.</math>

=== Invariant formulations of operators in vector calculus ===
The standard [[vector calculus]] operators can be generalized for any [[pseudo-Riemannian manifold]], and written in coordinate-free notation as follows:
: <math>\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}</math>
where {{math|⋆}} is the [[Hodge dual|Hodge star operator]], {{math|{{music|flat}}}} and {{math|{{music|sharp}}}} are the [[musical isomorphism]]s, {{math|&thinsp;''f''&thinsp;}} 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''}}.

== See also ==
{{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}}

== Notes ==
<!--- See [[Wikipedia:Footnotes]] on how to create references using <ref></ref> tags which will then appear here automatically -->
{{Reflist|30em}}

== References ==
* {{cite journal | last =Cartan | first =Élie | author-link =Élie Cartan
 | title =Sur certaines expressions différentielles et le problème de Pfaff
 | journal =Annales Scientifiques de l'École Normale Supérieure |series=Série 3
 | volume =16 | pages =239–332 | publisher =Gauthier-Villars | location =Paris | date =1899 | doi =10.24033/asens.467 | language =fr
 | url =http://www.numdam.org/item?id=ASENS_1899_3_16__239_0
 | issn =0012-9593 | jfm =30.0313.04 | access-date = 2 Feb 2016| doi-access =free }}
* {{cite book |author=Conlon, Lawrence |title=Differentiable manifolds |publisher=Birkhäuser |location=Basel, Switzerland |year=2001 |pages= 239 |isbn=0-8176-4134-3 }}
* {{cite book |author=Darling, R. W. R. |title=Differential forms and connections |publisher=Cambridge University Press |location=Cambridge, UK |year=1994 |pages=35 |isbn=0-521-46800-0 }}
* {{cite book |author=Flanders, Harley |title=Differential forms with applications to the physical sciences |publisher=Dover Publications |location=New York |year=1989 |pages=20 |isbn=0-486-66169-5 }}
* {{cite book|url=https://archive.org/details/LoomisL.H.SternbergS.AdvancedCalculusRevisedEditionJonesAndBartlett|title=Advanced Calculus|first=Lynn H.|last2=Sternberg|first2=Shlomo|publisher=Jones and Bartlett|year=1989|isbn=0-486-66169-5|location=Boston|pages=[https://archive.org/details/LoomisL.H.SternbergS.AdvancedCalculusRevisedEditionJonesAndBartlett/page/n313 304]–473 (ch. 7–11)|author=Loomis}}
* {{cite book |author=Ramanan, S. |title=Global calculus |publisher=American Mathematical Society |location=Providence, Rhode Island |year=2005 |pages=54 |isbn=0-8218-3702-8 }}
* {{cite book | last =Spivak | first =Michael | author-link =Michael Spivak | title =[[Calculus on Manifolds (book)|Calculus on Manifolds]]
 | publisher =Westview Press | date =1971 | location =Boulder, Colorado | isbn =9780805390216 }}
* {{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}}

== External links ==
* 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]]