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{{Calculus |Vector}}


In [[mathematics]], the '''directional derivative''' of a multivariate [[differentiable 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 velocity specified by '''v'''. It therefore generalizes the notion of a [[partial derivative]], in which the rate of change is taken along one of the [[Curvilinear coordinates|coordinate curves]], all other coordinates being constant.
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The directional derivative is a special case of the [[Gâteaux derivative]].
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[[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 blue, and the unit vector <math>\bold{u}</math> scaled by the directional derivative in the direction of <math>\bold{u}</math> in orange. The gradient vector is longer because gradients points in the direction of greatest rate of increase of a function.]]
 
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=== Generally applicable definition ===
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The directional derivative of a [[scalar function]]
:<math>f(\bold{x}) = f(x_1, x_2, \ldots, x_n)</math>
along a vector
:<math>\bold{v} = (v_1, \ldots, v_n)</math>
is the [[function (mathematics)|function]] defined by the [[limit (mathematics)|limit]]<ref>{{cite book | author=R. Wrede, M.R. Spiegel| title=Advanced Calculus|edition=3rd edition| publisher=Schaum's Outline Series| year=2010 | isbn=978-0-07-162366-7}}</ref>
:<math>\nabla_{\bold{v}}{f}(\bold{x}) = \lim_{h \rightarrow 0}{\frac{f(\bold{x} + h\bold{v}) - f(\bold{x})}{h}}.</math>


If the function ''f'' is [[Differentiable function#Differentiability in higher dimensions|differentiable]] at '''x''', then the directional derivative exists along any vector '''v''', and one has
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:<math>\nabla_{\bold{v}}{f}(\bold{x}) = \nabla f(\bold{x}) \cdot \bold{v}</math>
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where the <math>\nabla</math> on the right denotes the [[gradient]] and <math>\cdot</math> is the [[dot product]].<ref>Technically, the gradient ∇''f'' is a [[covector]], and the "dot product" is the action of this covector on the vector '''v'''.</ref> At any point '''x''', the directional derivative of ''f'' intuitively represents the [[derivative|rate of change]] of <math>f</math> with respect to time when it is moving at a speed and direction given by '''v''' at the point '''x'''.
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=== Variation using only direction of vector ===
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[[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.]]
Some authors define the directional derivative to be with respect to the vector '''v''' after [[Normalized vector|normalization]], thus ignoring its magnitude. In this case, one has
:<math>\nabla_{\bold{v}}{f}(\bold{x}) = \lim_{h \rightarrow 0}{\frac{f(\bold{x} + h\bold{v}) - f(\bold{x})}{h|\bold{v}|}},</math>
or in case ''f'' is differentiable at '''x''',
:<math>\nabla_{\bold{v}}{f}(\bold{x}) = \nabla f(\bold{x}) \cdot \frac{\bold{v}}{|\bold{v}|} .</math>
This definition has some disadvantages: its applicability is limited to when the [[Norm (mathematics)|norm]] of a vector is defined and nonzero. It is incompatible with notation used in some other areas of mathematics, physics and engineering, but should be used when what is wanted is the rate of increase in ''f'' per unit distance.


=== Restriction to unit vector ===
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Some authors restrict the definition of the directional derivative to being with respect to a [[unit vector]]. With this restriction, the two definitions above become the same.
 
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== Notation ==
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Directional derivatives can be also denoted by:
:<math>\nabla_{\bold{v}}{f}(\bold{x}) \sim \frac{\partial{f(\bold{x})}}{\partial{v}} \sim f'_\mathbf{v}(\bold{x}) \sim D_\bold{v}f(\bold{x}) \sim \mathbf{v}\cdot{\nabla f(\bold{x})} \sim \bold{v}\cdot \frac{\partial f(\bold{x})}{\partial\bold{x}} </math>
where ''v'' is a parameterization of a curve to which '''v''' is tangent and which determines its magnitude.


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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''':  
 
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|1=  '''The [[Sum rule in differentiation|sum rule]]:'''
 
:<math>\nabla_{\bold{v}} (f + g) = \nabla_{\bold{v}} f + \nabla_{\bold{v}} g</math>
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|2=  '''The [[Constant factor rule in differentiation|constant factor rule]]:''' For any constant ''c'',
 
:<math>\nabla_{\bold{v}} (cf) = c\nabla_{\bold{v}} f</math>
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|3=  '''The [[product rule]] (or [[Leibniz rule]]):'''
 
:<math>\nabla_{\bold{v}} (fg) = g\nabla_{\bold{v}} f + f\nabla_{\bold{v}} g</math>
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|4=  '''The [[chain rule]]:''' If ''g'' is differentiable at '''p''' and ''h'' is differentiable at ''g''('''p'''), then
 
:<math>\nabla_{\bold{v}}(h\circ g)(\bold{p}) = h'(g(\bold{p})) \nabla_{\bold{v}} g (\bold{p})</math>
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}}
 
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== In differential geometry ==
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{{see also|Tangent space#Tangent vectors as directional derivatives}}


Let ''M'' be a [[differentiable manifold]] and '''p''' a point of ''M''. Suppose that ''f'' is a function defined in a neighborhood of '''p''', and [[total derivative|differentiable]] at '''p'''. If '''v''' is a [[tangent vector]] to ''M'' at '''p''', then the '''directional derivative''' of ''f'' along '''v''', denoted variously as <math>\nabla_{\bold{v}} f(\bold{p})</math> (see [[covariant derivative]]), <math>L_{\bold{v}} f(\bold{p})</math> (see [[Lie derivative]]), or <math>{\bold{v}}_{\bold{p}}(f)</math> (see [[Tangent space#Definition via derivations|Tangent space §Definition via derivations]]), can be defined as follows. Let γ : [−1,1] → ''M'' be a differentiable curve with γ(0) = '''p''' and γ&prime;(0) = '''v'''. Then the directional derivative is defined by
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:<math>\nabla_{\bold{v}} f(\bold{p}) = \left.\frac{d}{d\tau} f\circ\gamma(\tau)\right|_{\tau=0}</math>
 
This definition can be proven independent of the choice of γ, provided γ is selected in the prescribed manner so that γ&prime;(0) = '''v'''.
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== Normal derivative ==
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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>\bold{n}</math>, then the directional derivative of a function ''f'' is sometimes denoted as <math>\frac{ \partial f}{\partial n}</math>. In other notations
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:<math>\frac{ \partial f}{\partial n} = \nabla f(\bold{x}) \cdot \bold{n} = \nabla_{\bold{n}}{f}(\bold{x}) = \frac{\partial f}{\partial \bold{x}}\cdot\bold{n} = Df(\bold{x})[\bold{n}] </math>
 
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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.
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The definitions of directional derivatives for various situations are given below. It is assumed that the functions are sufficiently smooth that derivatives can be taken.
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Let <math>f(\mathbf{v})</math> be a real valued function of the vector <math>\mathbf{v}</math>. Then the derivative of <math>f(\mathbf{v})</math> with respect to <math>\mathbf{v}</math> (or at <math>\mathbf{v}</math>) in the direction <math>\mathbf{u}</math> is defined as
 
:<math>
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   \frac{\partial f}{\partial \mathbf{v}}\cdot\mathbf{u} = Df(\mathbf{v})[\mathbf{u}]  
 
    = \left[\frac{d }{d \alpha}~f(\mathbf{v} + \alpha~\mathbf{u})\right]_{\alpha = 0}
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</math>
    
for all vectors <math>\mathbf{u}</math>.
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|1= If <math>f(\mathbf{v}) = f_1(\mathbf{v}) + f_2(\mathbf{v})</math> then <math>
  \frac{\partial f}{\partial \mathbf{v}}\cdot\mathbf{u} = \left(\frac{\partial f_1}{\partial \mathbf{v}} + \frac{\partial f_2}{\partial \mathbf{v}}\right)\cdot\mathbf{u}
</math>


|2= If <math>f(\mathbf{v}) = f_1(\mathbf{v})~ f_2(\mathbf{v})</math> then <math>
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   \frac{\partial f}{\partial \mathbf{v}}\cdot\mathbf{u} = \left(\frac{\partial f_1}{\partial \mathbf{v}}\cdot\mathbf{u}\right)~f_2(\mathbf{v}) + f_1(\mathbf{v})~\left(\frac{\partial f_2}{\partial \mathbf{v}}\cdot\mathbf{u} \right)
 
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|3= If <math>f(\mathbf{v}) = f_1(f_2(\mathbf{v}))</math> then <math>
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  \frac{\partial f}{\partial \mathbf{v}}\cdot\mathbf{u} = \frac{\partial f_1}{\partial f_2}~\frac{\partial f_2}{\partial \mathbf{v}}\cdot\mathbf{u}
</math>
}}


===Derivatives of vector valued functions of vectors===
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Let <math>\mathbf{f}(\mathbf{v})</math> be a vector valued function of the vector <math>\mathbf{v}</math>. Then the derivative of <math>\mathbf{f}(\mathbf{v})</math> with respect to <math>\mathbf{v}</math> (or at <math>\mathbf{v}</math>) in the direction <math>\mathbf{u}</math> is the '''second-order tensor''' defined as
 
:<math>
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   \frac{\partial \mathbf{f}}{\partial \mathbf{v}}\cdot\mathbf{u} = D\mathbf{f}(\mathbf{v})[\mathbf{u}]  
    
    = \left[\frac{d }{d \alpha}~\mathbf{f}(\mathbf{v} + \alpha~\mathbf{u})\right]_{\alpha = 0}
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</math>
 
for all vectors <math>\mathbf{u}</math>.
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{{ordered list
|1= If <math>\mathbf{f}(\mathbf{v}) = \mathbf{f}_1(\mathbf{v}) + \mathbf{f}_2(\mathbf{v})</math> then <math>
  \frac{\partial \mathbf{f}}{\partial \mathbf{v}}\cdot\mathbf{u} = \left(\frac{\partial \mathbf{f}_1}{\partial \mathbf{v}} + \frac{\partial \mathbf{f}_2}{\partial \mathbf{v}}\right)\cdot\mathbf{u}
</math>


|2= If <math>\mathbf{f}(\mathbf{v}) = \mathbf{f}_1(\mathbf{v})\times\mathbf{f}_2(\mathbf{v})</math> then <math>
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   \frac{\partial \mathbf{f}}{\partial \mathbf{v}}\cdot\mathbf{u} = \left(\frac{\partial \mathbf{f}_1}{\partial \mathbf{v}}\cdot\mathbf{u}\right)\times\mathbf{f}_2(\mathbf{v}) + \mathbf{f}_1(\mathbf{v})\times\left(\frac{\partial \mathbf{f}_2}{\partial \mathbf{v}}\cdot\mathbf{u} \right)
 
  </math>
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|3= If <math>\mathbf{f}(\mathbf{v}) = \mathbf{f}_1(\mathbf{f}_2(\mathbf{v}))</math> then <math>
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  \frac{\partial \mathbf{f}}{\partial \mathbf{v}}\cdot\mathbf{u} = \frac{\partial \mathbf{f}_1}{\partial \mathbf{f}_2}\cdot\left(\frac{\partial \mathbf{f}_2}{\partial \mathbf{v}}\cdot\mathbf{u} \right)
</math>
}}


===Derivatives of scalar valued functions of second-order tensors===
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Let <math>f(\boldsymbol{S})</math> be a real valued function of the second order tensor <math>\boldsymbol{S}</math>. Then the derivative of <math>f(\boldsymbol{S})</math> with respect to <math>\boldsymbol{S}</math> (or at <math>\boldsymbol{S}</math>) in the direction
 
<math>\boldsymbol{T}</math> is the ''' second order tensor''' defined as
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:<math>
 
   \frac{\partial f}{\partial \boldsymbol{S}}:\boldsymbol{T} = Df(\boldsymbol{S})[\boldsymbol{T}]  
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    = \left[\frac{d }{d \alpha}~f(\boldsymbol{S} + \alpha\boldsymbol{T})\right]_{\alpha = 0}
    
</math>
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for all second order tensors <math>\boldsymbol{T}</math>.
 
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{{ordered list
|1= If <math>f(\boldsymbol{S}) = f_1(\boldsymbol{S}) + f_2(\boldsymbol{S})</math> then <math> \frac{\partial f}{\partial \boldsymbol{S}}:\boldsymbol{T} = \left(\frac{\partial f_1}{\partial \boldsymbol{S}} + \frac{\partial f_2}{\partial \boldsymbol{S}}\right):\boldsymbol{T} </math>


|2= If <math>f(\boldsymbol{S}) = f_1(\boldsymbol{S})~ f_2(\boldsymbol{S})</math> then <math> \frac{\partial f}{\partial \boldsymbol{S}}:\boldsymbol{T} = \left(\frac{\partial f_1}{\partial \boldsymbol{S}}:\boldsymbol{T}\right)~f_2(\boldsymbol{S}) + f_1(\boldsymbol{S})~\left(\frac{\partial f_2}{\partial \boldsymbol{S}}:\boldsymbol{T} \right) </math>
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|3= If <math>f(\boldsymbol{S}) = f_1(f_2(\boldsymbol{S}))</math> then <math> \frac{\partial f}{\partial \boldsymbol{S}}:\boldsymbol{T} = \frac{\partial f_1}{\partial f_2}~\left(\frac{\partial f_2}{\partial \boldsymbol{S}}:\boldsymbol{T} \right) </math>
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===Derivatives of tensor valued functions of second-order tensors===
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Let <math>\boldsymbol{F}(\boldsymbol{S})</math> be a second order tensor valued function of the second order tensor <math>\boldsymbol{S}</math>Then the derivative of <math>\boldsymbol{F}(\boldsymbol{S})</math> with respect to <math>\boldsymbol{S}</math>  
 
(or at <math>\boldsymbol{S}</math>) in the direction <math>\boldsymbol{T}</math> is the ''' fourth order tensor''' defined as
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   \frac{\partial \boldsymbol{F}}{\partial \boldsymbol{S}}:\boldsymbol{T} = D\boldsymbol{F}(\boldsymbol{S})[\boldsymbol{T}]  
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    = \left[\frac{d }{d \alpha}~\boldsymbol{F}(\boldsymbol{S} + \alpha\boldsymbol{T})\right]_{\alpha = 0}
 
</math>
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{{ordered list
|1= If <math>\boldsymbol{F}(\boldsymbol{S}) = \boldsymbol{F}_1(\boldsymbol{S}) + \boldsymbol{F}_2(\boldsymbol{S})</math> then <math> \frac{\partial \boldsymbol{F}}{\partial \boldsymbol{S}}:\boldsymbol{T} = \left(\frac{\partial \boldsymbol{F}_1}{\partial \boldsymbol{S}} + \frac{\partial \boldsymbol{F}_2}{\partial \boldsymbol{S}}\right):\boldsymbol{T} </math>


|2= If <math>\boldsymbol{F}(\boldsymbol{S}) = \boldsymbol{F}_1(\boldsymbol{S})\cdot\boldsymbol{F}_2(\boldsymbol{S})</math> then <math> \frac{\partial \boldsymbol{F}}{\partial \boldsymbol{S}}:\boldsymbol{T} \left(\frac{\partial \boldsymbol{F}_1}{\partial \boldsymbol{S}}:\boldsymbol{T}\right)\cdot\boldsymbol{F}_2(\boldsymbol{S}) + \boldsymbol{F}_1(\boldsymbol{S})\cdot\left(\frac{\partial \boldsymbol{F}_2}{\partial \boldsymbol{S}}:\boldsymbol{T} \right) </math>
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== See also ==
== Nike Shoes Nz  000 ==
* [[Fréchet derivative]]
* [[Gâteaux derivative]]
* [[Derivative (generalizations)]]
* [[Lie derivative]]
* [[Differential form]]
* [[Structure tensor]]
* [[Tensor derivative (continuum mechanics)]]
* [[Del in cylindrical and spherical coordinates]]


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== Ugg Boots New Zealand Suppliers Produce ==
*{{cite book | first=F. B. | last=Hildebrand | title=Advanced Calculus for Applications| publisher=Prentice Hall | year=1976 | isbn=0-13-011189-9 }}
*{{cite book | author=K.F. Riley, M.P. Hobson, S.J. Bence| title=Mathematical methods for physics and engineering| publisher=Cambridge University Press| year=2010 | isbn=978-0-521-86153-3}}


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