View source for Directional derivative

{{Short description|Instantaneous rate of change of the function}}
{{refimprove section|date=October 2012|talk=Verifiability of definition}}
{{Calculus |Vector}}

In [[multivariable calculus]], the '''directional derivative''' measures the rate at which a function changes in a particular direction at a given point.{{cn|date=November 2023}}

The directional derivative of a multivariable [[differentiable function|differentiable (scalar) function]] along a given [[vector (mathematics)|vector]] '''v''' at a given point '''x''' intuitively represents the instantaneous rate of change of the function, moving through '''x''' with a direction specified by '''v'''.  

The directional derivative of a [[Scalar field|scalar function]] ''f'' with respect to a vector '''v''' at a point (e.g., position) '''x''' may be denoted by any of the following:
<math display="block">
\begin{aligned}
\nabla_{\mathbf{v}}{f}(\mathbf{x})
&=f'_\mathbf{v}(\mathbf{x})\\
&=D_\mathbf{v}f(\mathbf{x})\\
&=Df(\mathbf{x})(\mathbf{v})\\
&=\partial_\mathbf{v}f(\mathbf{x})\\
&=\mathbf{v}\cdot{\nabla f(\mathbf{x})}\\
&=\mathbf{v}\cdot \frac{\partial f(\mathbf{x})}{\partial\mathbf{x}}.\\
\end{aligned}
</math>

It therefore generalizes the notion of a [[partial derivative]], in which the rate of change is taken along one of the [[Curvilinear coordinates|curvilinear]] [[coordinate curves]], all other coordinates being constant.
The directional derivative is a special case of the [[Gateaux derivative]].

== Definition ==
[[File:Directional derivative contour plot.svg|thumb|275px|A [[contour plot]] of <math>f(x, y)=x^2 + y^2</math>, showing the gradient vector in black, and the unit vector <math>\mathbf{u}</math> scaled by the directional derivative in the direction of <math>\mathbf{u}</math> in orange. The gradient vector is longer because the gradient points in the direction of greatest rate of increase of a function.]]

The ''directional derivative'' of a [[scalar function]] 
<math display="block">f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n)</math>
along a vector
<math display="block">\mathbf{v} = (v_1, \ldots, v_n)</math>
is the [[function (mathematics)|function]] <math>\nabla_{\mathbf{v}}{f}</math> defined by the [[limit (mathematics)|limit]]<ref>{{cite book |author1=R. Wrede |author2=M.R. Spiegel | title=Advanced Calculus|edition=3rd| publisher=Schaum's Outline Series| year=2010 | isbn=978-0-07-162366-7}}</ref>
<math display="block">\nabla_{\mathbf{v}}{f}(\mathbf{x}) = \lim_{h \to 0}{\frac{f(\mathbf{x} + h\mathbf{v}) - f(\mathbf{x})}{h}} = \left.\frac{\mathrm{d}}{\mathrm{d}t}f(\mathbf{x}+t\mathbf{v})\right|_{t=0}.</math>

This definition is valid in a broad range of contexts, for example where the [[Euclidean norm|norm]] of a vector (and hence a unit vector) is undefined.<ref>The applicability extends to functions over spaces without a [[metric (mathematics)|metric]] and to [[differentiable manifold]]s, such as in [[general relativity]].</ref>

=== For differentiable functions ===

If the function ''f'' is [[Differentiable function#Differentiability in higher dimensions|differentiable]] at '''x''', then the directional derivative exists along any unit vector '''v''' at x, and one has

<math display="block">\nabla_{\mathbf{v}}{f}(\mathbf{x}) = \nabla f(\mathbf{x}) \cdot \mathbf{v}</math>

where the <math>\nabla</math> on the right denotes the ''[[gradient]]'', <math>\cdot</math> is the [[dot product]] and '''v''' is a unit vector.<ref>If the dot product is undefined, the [[gradient]] is also undefined; however, for differentiable ''f'', the directional derivative is still defined, and a similar relation exists with the exterior derivative.</ref> This follows from defining a path <math>h(t) = x + tv</math> and using the definition of the derivative as a limit which can be calculated along this path to get:
<math display="block">\begin{align}
0
&=\lim_{t \to 0}\frac {f(x+tv)-f(x)-tDf(x)(v)} t \\
&=\lim_{t \to 0}\frac {f(x+tv)-f(x)} t - Df(x)(v) \\
&=\nabla_v f(x)-Df(x)(v).
\end{align}</math>

Intuitively, the directional derivative of ''f'' at a point '''x''' represents the [[derivative|rate of change]] of ''f'', in the direction of '''v'''.

=== Using only direction of vector ===
[[image:Geometrical interpretation of a directional derivative.svg|thumb|The angle ''α'' between the tangent ''A'' and the horizontal will be maximum if the cutting plane contains the direction of the gradient ''A''.]]
In a [[Euclidean space]], some authors<ref>Thomas, George B. Jr.; and Finney, Ross L. (1979) ''Calculus and Analytic Geometry'', Addison-Wesley Publ. Co., fifth edition, p. 593.</ref> define the directional derivative to be with respect to an arbitrary nonzero vector '''v''' after [[Normalized vector|normalization]], thus being independent of its magnitude and depending only on its direction.<ref>This typically assumes a [[Euclidean space]] – for example, a function of several variables typically has no definition of the magnitude of a vector, and hence of a unit vector.</ref>

This definition gives the rate of increase of {{math|''f''}} per unit of distance moved in the direction given by {{math|'''v'''}}. In this case, one has
<math display="block">\nabla_{\mathbf{v}}{f}(\mathbf{x}) = \lim_{h \to 0}{\frac{f(\mathbf{x} + h\mathbf{v}) - f(\mathbf{x})}{h|\mathbf{v}|}},</math>
or in case ''f'' is differentiable at '''x''',
<math display="block">\nabla_{\mathbf{v}}{f}(\mathbf{x}) = \nabla f(\mathbf{x}) \cdot \frac{\mathbf{v}}{|\mathbf{v}|} .</math>

=== Restriction to a unit vector ===

In the context of a function on a [[Euclidean space]], some texts restrict the vector '''v''' to being a [[unit vector]].  With this restriction, both the above definitions are equivalent.<ref>{{Cite book| title=Calculus : Single and multivariable.|last1=Hughes Hallett|first1=Deborah|author1-link=Deborah Hughes Hallett|last2=McCallum|first2=William G.| author2-link=William G. McCallum|last3=Gleason|first3=Andrew M.|author3-link=Andrew M. Gleason| date=2012-01-01| publisher=John wiley|isbn=9780470888612|pages=780|oclc=828768012}}</ref>

== Properties ==
Many of the familiar properties of the ordinary [[derivative]] hold for the directional derivative.  These include, for any functions ''f'' and ''g'' defined in a [[neighborhood (mathematics)|neighborhood]] of, and [[total derivative|differentiable]] at, '''p''': 

# '''[[Sum rule in differentiation|sum rule]]''': <math display="block">\nabla_{\mathbf{v}} (f + g) = \nabla_{\mathbf{v}} f + \nabla_{\mathbf{v}} g.</math>
# '''[[Constant factor rule in differentiation|constant factor rule]]''': For any constant ''c'', <math display="block">\nabla_{\mathbf{v}} (cf) = c\nabla_{\mathbf{v}} f.</math>
# '''[[product rule]]''' (or '''Leibniz's rule'''): <math display="block">\nabla_{\mathbf{v}} (fg) = g\nabla_{\mathbf{v}} f + f\nabla_{\mathbf{v}} g.</math>
# '''[[chain rule]]''': If ''g'' is differentiable at '''p''' and ''h'' is differentiable at ''g''('''p'''), then <math display="block">\nabla_{\mathbf{v}}(h\circ g)(\mathbf{p}) = h'(g(\mathbf{p})) \nabla_{\mathbf{v}} g (\mathbf{p}).</math>

== In differential geometry ==
{{see also|Tangent space#Tangent vectors as directional derivatives}}

Let {{math|''M''}} be a [[differentiable manifold]] and {{math|'''p'''}} a point of {{math|''M''}}.  Suppose that {{math|''f''}} is a function defined in a neighborhood of {{math|'''p'''}}, and [[total derivative|differentiable]] at {{math|'''p'''}}.  If {{math|'''v'''}} is a [[tangent vector]] to {{math|''M''}} at {{math|'''p'''}}, then the '''directional derivative''' of {{math|''f''}} along {{math|'''v'''}}, denoted variously as {{math|''df''('''v''')}} (see [[Exterior derivative]]), <math>\nabla_{\mathbf{v}} f(\mathbf{p})</math> (see [[Covariant derivative]]), <math>L_{\mathbf{v}} f(\mathbf{p})</math> (see [[Lie derivative]]), or <math>{\mathbf{v}}_{\mathbf{p}}(f)</math> (see {{section link|Tangent space|Definition via derivations}}), can be defined as follows.  Let {{math|''γ'' : [−1, 1] → ''M''}} be a differentiable curve with {{math|1=''γ''(0) = '''p'''}} and {{math|1=''γ''′(0) = '''v'''}}.  Then the directional derivative is defined by
<math display="block">\nabla_{\mathbf{v}} f(\mathbf{p}) = \left.\frac{d}{d\tau} f\circ\gamma(\tau)\right|_{\tau=0}.</math>
This definition can be proven independent of the choice of {{math|''γ''}}, provided {{math|''γ''}} is selected in the prescribed manner so that {{math|1=''γ''(0) = '''p'''}} and {{math|1=''γ''′(0) = '''v'''}}.

===The Lie derivative===
The [[Lie derivative]] of a vector field <math> W^\mu(x)</math> along a vector field <math> V^\mu(x)</math> is given by the difference of two directional derivatives (with vanishing torsion):
<math display="block">\mathcal{L}_V W^\mu=(V\cdot\nabla) W^\mu-(W\cdot\nabla) V^\mu.</math>
In particular, for a scalar field <math> \phi(x)</math>, the Lie derivative reduces to the standard directional derivative:
<math display="block">\mathcal{L}_V \phi=(V\cdot\nabla) \phi.</math>

===The Riemann tensor===
Directional derivatives are often used in introductory derivations of the [[Riemann curvature tensor]]. Consider a curved rectangle with an infinitesimal vector <math>\delta</math> along one edge and <math>\delta'</math> along the other. We translate a covector <math>S</math> along <math>\delta</math> then <math>\delta'</math> and then subtract the translation along <math>\delta'</math> and then <math>\delta</math>. Instead of building the directional derivative using partial derivatives, we use the [[covariant derivative]]. The translation operator for <math>\delta</math> is thus
<math display="block">1+\sum_\nu \delta^\nu D_\nu=1+\delta\cdot D,</math>
and for <math>\delta'</math>,
<math display="block">1+\sum_\mu \delta'^\mu D_\mu=1+\delta'\cdot D.</math>
The difference between the two paths is then
<math display="block">(1+\delta'\cdot D)(1+\delta\cdot D)S^\rho-(1+\delta\cdot D)(1+\delta'\cdot D)S^\rho=\sum_{\mu,\nu}\delta'^\mu \delta^\nu[D_\mu,D_\nu]S_\rho.</math>
It can be argued<ref>{{cite book|last1=Zee|first1=A.|title=Einstein gravity in a nutshell|date=2013|publisher=Princeton University Press|location=Princeton|isbn=9780691145587|page=341}}</ref> that the noncommutativity of the covariant derivatives measures the curvature of the manifold:
<math display="block">[D_\mu,D_\nu]S_\rho=\pm \sum_\sigma R^\sigma{}_{\rho\mu\nu}S_\sigma,</math>
where <math>R</math> is the Riemann curvature tensor and the sign depends on the [[sign convention]] of the author.

== In group theory ==

===Translations===
In the [[Poincaré algebra]], we can define an infinitesimal translation operator '''P''' as
<math display="block">\mathbf{P}=i\nabla.</math>
(the ''i'' ensures that '''P''' is a [[self-adjoint operator]]) For a finite displacement '''λ''', the [[Unitary operator|unitary]] [[Hilbert space]] [[Group representation|representation]] for translations is<ref>{{cite book| last1=Weinberg|first1=Steven|title=The quantum theory of fields|date=1999|publisher=Cambridge Univ. Press| location=Cambridge [u.a.]| isbn=9780521550017|edition=Reprinted (with corr.).|url-access=registration| url=https://archive.org/details/quantumtheoryoff00stev}}</ref> 
<math display="block">U(\boldsymbol{\lambda})=\exp\left(-i\boldsymbol{\lambda}\cdot\mathbf{P}\right).</math>
By using the above definition of the infinitesimal translation operator, we see that the finite translation operator is an exponentiated directional derivative:
<math display="block">U(\boldsymbol{\lambda})=\exp\left(\boldsymbol{\lambda}\cdot\nabla\right).</math>
This is a translation operator in the sense that it acts on multivariable functions ''f''('''x''') as
<math display="block">U(\boldsymbol{\lambda}) f(\mathbf{x})=\exp\left(\boldsymbol{\lambda}\cdot\nabla\right) f(\mathbf{x}) = f(\mathbf{x}+\boldsymbol{\lambda}).</math>

{{math proof|title=Proof of the last equation
|proof=
In standard single-variable calculus, the derivative of a smooth function ''f''(''x'') is defined by (for small ''ε'')
<math display="block">\frac{df}{dx} = \frac{f(x+\varepsilon) - f(x)}{\varepsilon}.</math>
This can be rearranged to find ''f''(''x''+''ε''):
<math display="block">f(x+\varepsilon)=f(x)+\varepsilon \,\frac{df}{dx}=\left(1+\varepsilon\,\frac{d}{dx}\right)f(x).</math>
It follows that <math>[1+\varepsilon\,(d/dx)] </math> is a translation operator. This is instantly generalized<ref>{{cite book |last1=Zee|first1=A.|title=Einstein gravity in a nutshell|date=2013|publisher=Princeton University Press| location=Princeton| isbn=9780691145587}}</ref> to multivariable functions ''f''('''x''')
<math display="block">f(\mathbf{x}+\boldsymbol{\varepsilon}) = \left(1+\boldsymbol{\varepsilon}\cdot\nabla\right) f(\mathbf{x}).</math>
Here <math> \boldsymbol{\varepsilon}\cdot\nabla</math> is the directional derivative along the infinitesimal displacement '''''ε'''''. We have found the infinitesimal version of the translation operator:
<math display="block">U(\boldsymbol{\varepsilon}) = 1 + \boldsymbol{\varepsilon}\cdot\nabla.</math>
It is evident that the group multiplication law<ref>{{cite book | last1=Cahill |first1=Kevin Cahill | title=Physical mathematics | date=2013 | publisher=Cambridge University Press | location=Cambridge | isbn=978-1107005211 | edition=Repr.}}</ref> ''U''(''g'')''U''(''f'')=''U''(''gf'') takes the form
<math display="block">U(\mathbf{a})U(\mathbf{b})=U(\mathbf{a+b}).</math>
So suppose that we take the finite displacement '''''λ''''' and divide it into ''N'' parts (''N''→∞ is implied everywhere), so that '''''λ'''''/''N''='''''ε'''''. In other words,
<math display="block">\boldsymbol{\lambda} = N \boldsymbol{\varepsilon}.</math>
Then by applying ''U''('''''ε''''') ''N'' times, we can construct ''U''('''''λ'''''):
<math display="block">[U(\boldsymbol{\varepsilon})]^N = U(N\boldsymbol{\varepsilon}) = U(\boldsymbol{\lambda}).</math>
We can now plug in our above expression for U('''ε'''):
<math display="block">[U(\boldsymbol{\varepsilon})]^N = \left[1+\boldsymbol{\varepsilon}\cdot\nabla\right]^N = \left[1+\frac{\boldsymbol{\lambda}\cdot\nabla}{N}\right]^N.</math>
Using the identity<ref>{{cite book | first1 = Ron | last1 = Larson | last2 = Edwards | first2 = Bruce H. | title = Calculus of a single variable | date = 2010 | publisher = Brooks/Cole | location =Belmont | isbn = 9780547209982 | edition = 9th}}</ref>
<math display="block">\exp(x)=\left[1+\frac{x}{N}\right]^N,</math>
we have
<math display="block">U(\boldsymbol{\lambda})=\exp\left(\boldsymbol{\lambda}\cdot\nabla\right).</math>
And since {{math|1=''U''('''''ε''''')''f''('''x''') = ''f''('''x'''+'''''ε''''')}} we have
<math display="block">[U(\boldsymbol{\varepsilon})]^N f(\mathbf{x}) = f(\mathbf{x}+N\boldsymbol{\varepsilon}) = f(\mathbf{x}+\boldsymbol{\lambda}) = U(\boldsymbol{\lambda})f(\mathbf{x}) = \exp\left(\boldsymbol{\lambda}\cdot\nabla\right)f(\mathbf{x}),</math>
Q.E.D.

As a technical note, this procedure is only possible because the translation group forms an [[abelian group|Abelian]] [[subgroup]] ([[Cartan subalgebra]]) in the Poincaré algebra. In particular, the group multiplication law ''U''('''a''')''U''('''b''') = ''U''('''a'''+'''b''') should not be taken for granted. We also note that Poincaré is a connected [[Lie group]]. It is a group of transformations ''T''(''ξ'') that are described by a continuous set of real parameters <math>\xi^a</math>. The group multiplication law takes the form
<math display="block">T(\bar{\xi})T(\xi) = T(f(\bar{\xi},\xi)).</math>
Taking <math>\xi^a = 0</math> as the coordinates of the identity, we must have
<math display="block">f^a(\xi,0)=f^a(0,\xi)=\xi^a.</math>
The actual operators on the Hilbert space are represented by unitary operators ''U''(''T''(''ξ'')). In the above notation we suppressed the ''T''; we now write ''U''('''λ''') as ''U''('''P'''('''λ''')). For a small neighborhood around the identity, the [[power series]] representation 
<math display="block">U(T(\xi))=1+i\sum_a\xi^a t_a+\frac{1}{2}\sum_{b,c}\xi^b\xi^c t_{bc}+\cdots</math>
is quite good. Suppose that U(T(ξ)) form a non-projective representation, i.e.,
<math display="block">U(T(\bar{\xi}))U(T(\xi))=U(T(f(\bar{\xi},\xi))).</math>
The expansion of f to second power is
<math display="block">f^a(\bar{\xi},\xi)=\xi^a+\bar{\xi}^a+\sum_{b,c}f^{abc}\bar{\xi}^b\xi^c.</math>
After expanding the representation multiplication equation and equating coefficients, we have the nontrivial condition
<math display="block">t_{bc}=-t_b t_c-i\sum_a f^{abc}t_a.</math>
Since <math> t_{ab}</math> is by definition symmetric in its indices, we have the standard [[Lie algebra]] commutator:
<math display="block">[t_b, t_c]=i\sum_a(-f^{abc}+f^{acb})t_a=i\sum_a C^{abc}t_a,</math>
with ''C'' the [[structure constant]]. The generators for translations are partial derivative operators, which commute:
<math display="block">\left[\frac{\partial}{\partial x^b},\frac{\partial }{\partial x^c}\right]=0.</math>
This implies that the structure constants vanish and thus the quadratic coefficients in the f expansion vanish as well. This means that ''f'' is simply additive:
<math display="block">f^a_\text{abelian}(\bar{\xi},\xi)=\xi^a+\bar{\xi}^a,</math>
and thus for abelian groups,
<math display="block">U(T(\bar{\xi}))U(T(\xi))=U(T(\bar{\xi}+\xi)).</math>
Q.E.D.
}}

===Rotations===
The [[rotation operator (quantum mechanics)|rotation operator]] also contains a directional derivative. The rotation operator for an angle '''''θ''''', i.e. by an amount ''θ'' = |'''''θ'''''| about an axis parallel to <math> \hat{\theta} = \boldsymbol{\theta}/\theta</math> is
<math display="block">U(R(\mathbf{\theta}))=\exp(-i\mathbf{\theta}\cdot\mathbf{L}).</math>
Here '''L''' is the vector operator that generates [[SO(3)]]:
<math display="block">\mathbf{L}=\begin{pmatrix}
 0& 0 & 0\\ 
 0& 0 & 1\\ 
 0& -1 & 0
\end{pmatrix}\mathbf{i}+\begin{pmatrix}
0 &0  & -1\\ 
 0& 0 &0 \\ 
1 & 0 & 0
\end{pmatrix}\mathbf{j}+\begin{pmatrix}
 0&1  &0 \\ 
 -1&0  &0 \\ 
0 & 0 & 0
\end{pmatrix}\mathbf{k}.</math>
It may be shown geometrically that an infinitesimal right-handed rotation changes the position vector '''x''' by
<math display="block">\mathbf{x}\rightarrow \mathbf{x}-\delta\boldsymbol{\theta}\times\mathbf{x}.</math>
So we would expect under infinitesimal rotation:
<math display="block">U(R(\delta\boldsymbol{\theta})) f(\mathbf{x}) = f(\mathbf{x}-\delta\boldsymbol{\theta}\times\mathbf{x})=f(\mathbf{x})-(\delta\boldsymbol{\theta}\times\mathbf{x})\cdot\nabla f.</math>
It follows that 
<math display="block">U(R(\delta\mathbf{\theta}))=1-(\delta\mathbf{\theta}\times\mathbf{x})\cdot\nabla.</math>
Following the same exponentiation procedure as above, we arrive at the rotation operator in the position basis, which is an exponentiated directional derivative:<ref>{{cite book|last1=Shankar|first1=R. | title=Principles of quantum mechanics | date=1994|publisher=Kluwer Academic / Plenum|location=New York|isbn=9780306447907|page=318|edition=2nd}}</ref>
<math display="block">U(R(\mathbf{\theta}))=\exp(-(\mathbf{\theta}\times\mathbf{x})\cdot\nabla).</math>

== Normal derivative ==

A '''normal derivative''' is a directional derivative taken in the direction normal (that is, [[orthogonal]]) to some surface in space, or more generally along a [[normal vector]] field orthogonal to some [[hypersurface]]. See for example [[Neumann boundary condition]]. If the normal direction is denoted by <math>\mathbf{n}</math>, then the normal derivative of a function ''f'' is sometimes denoted as <math display="inline">\frac{ \partial f}{\partial \mathbf{n}}</math>. In other notations,
<math display="block">\frac{ \partial f}{\partial \mathbf{n}} = \nabla f(\mathbf{x}) \cdot \mathbf{n} = \nabla_{\mathbf{n}}{f}(\mathbf{x}) = \frac{\partial f}{\partial \mathbf{x}} \cdot \mathbf{n} = Df(\mathbf{x})[\mathbf{n}].</math>

== In the continuum mechanics of solids ==

Several important results in continuum mechanics require the derivatives of vectors with respect to vectors and of [[tensors]] with respect to vectors and tensors.<ref name=Marsden00>J. E. Marsden and T. J. R. Hughes, 2000, ''Mathematical Foundations of Elasticity'', Dover.</ref>  The '''directional directive''' provides a systematic way of finding these derivatives.

{{excerpt|Tensor derivative (continuum mechanics)|Derivatives with respect to vectors and second-order tensors|subsections=y}}

==See also==

* {{annotated link|Del in cylindrical and spherical coordinates}}
* {{annotated link|Differential form}}
* {{annotated link|Ehresmann connection}}
* {{annotated link|Fréchet derivative}}
* {{annotated link|Gateaux derivative}}
* {{annotated link|Generalizations of the derivative}}
* {{annotated link | Semi-differentiability}}
* {{annotated link|Hadamard derivative}}
* {{annotated link|Lie derivative}}
* {{annotated link|Material derivative}}
* {{annotated link|Structure tensor}}
* {{annotated link|Tensor derivative (continuum mechanics)}}
* {{annotated link|Total derivative}}


== Notes ==
{{reflist|2}}

== References ==
*{{cite book | first=F. B. | last=Hildebrand | title=Advanced Calculus for Applications| publisher=Prentice Hall | year=1976 | isbn=0-13-011189-9 }}
*{{cite book |author1=K.F. Riley |author2=M.P. Hobson |author3=S.J. Bence | title=Mathematical methods for physics and engineering|url=https://archive.org/details/mathematicalmeth00rile |url-access=registration | publisher=Cambridge University Press| year=2010 | isbn=978-0-521-86153-3}}
*{{cite journal |first=A. |last=Shapiro |title=On concepts of directional differentiability |journal=Journal of Optimization Theory and Applications |volume=66 |issue= 3|pages=477–487 |year=1990 |doi=10.1007/BF00940933 |s2cid=120253580 }}

== External links ==
{{Commons category inline|Directional derivative}}
*[http://mathworld.wolfram.com/DirectionalDerivative.html Directional derivatives] at [[MathWorld]].
*[http://planetmath.org/directionalderivative Directional derivative] at [[PlanetMath]].

{{Calculus topics}}

[[Category:Differential calculus]]
[[Category:Differential geometry]]
[[Category:Generalizations of the derivative]]
[[Category:Multivariable calculus]]
[[Category:Scalars]]
[[Category:Rates]]