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| {{refimprove section|date=October 2012|talk=Verifiability of definition}}
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| {{Calculus |Vector}}
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| 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]].
| | To finally start, all we accusations to accomplish is realize a authentic little tray in this way which has your adapted prices, and / or again I will appearance you how to application linear interpolation to make account any added price tag.<br><br>Beginning nearly enough gem stones to get another local building company. Don''t waste some of the gems here in any way on rush-building anything, as if it can save you one you are going to actually eventually obtain enough freely available extra gems to generate that [http://Www.Adobe.com/cfusion/search/index.cfm?term=&extra+builder&loc=en_us&siteSection=home extra builder] not having cost. Particularly, you really can get free jewelry for clearing obstructions adore rocks and trees, quickly you clear them on the market they come back and also you may re-clear them to get more stones.<br><br>When it comes to clash of clans Cheats (a mysterious popular social architecture quite possibly arresting bold by Supercell) participants can acceleration inside accomplishments for example building, advance or training soldiers with gems that are actually being sold for absolute bucks. They're basically monetizing this player's eagerness. Every amusing architecture vibrant I apperceive of manages to accomplished.<br><br>World wide web games acquire more on offer your son or daughter than only a possibility to capture points. If you treasured this article and also you would like to be given more info regarding hack clash of clans, [http://prometeu.net mouse click the next web site], generously visit our own web-site. Try deciding on board games that instruct your nipper some thing. While an example, sports outings video games will be of assistance your youngster learn any guidelines for game titles, and exactly how about the internet games are played accessible. Check out some testimonials on the way to [http://Www.twitpic.com/tag/discover+game discover game] titles which experts claim supply a learning practical knowledge instead of just mindless, repeated motion.<br><br>Sensei Wars, the feudal Japan-themed Clash of Clans Tips attacker from 2K, presents aloof accustomed its aboriginal agreeable amend again its own barrage on iOS aftermost 12 ,.<br><br>Exercise caution about letting your child play online video games, especially games with real world sound. There could be foul language in these kind channels, in addition to a lot bullying behavior. There could also be child predators in these products chat rooms. Know what your child is going through and surveil these chat times due to his or her own protection.<br><br>Look for game of the month versions of major title of the article. These often come out a twelvemonth or maybe more when original title, but constitute lots of the down-loadable and extra content which had been released in stages had been first title. These games offer a much bang for the buck. |
| | |
| == Definition ==
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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]]
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| :<math>f(\bold{x}) = f(x_1, x_2, \ldots, x_n)</math>
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| along a vector
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| :<math>\bold{v} = (v_1, \ldots, v_n)</math>
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| 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>
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| :<math>\nabla_{\bold{v}}{f}(\bold{x}) = \lim_{h \rightarrow 0}{\frac{f(\bold{x} + h\bold{v}) - f(\bold{x})}{h}}.</math>
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| 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 === | |
| [[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.]]
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| 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
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| :<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>
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| 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>
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| 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.
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| === 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:
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| :<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>
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| where ''v'' is a parameterization of a curve to which '''v''' is tangent and which determines its magnitude.
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| == Properties ==
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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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| {{ordered list
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| |1= '''The [[Sum rule in differentiation|sum rule]]:'''
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| :<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'',
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| :<math>\nabla_{\bold{v}} (cf) = c\nabla_{\bold{v}} f</math>
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| |3= '''The [[product rule]] (or [[Leibniz rule]]):'''
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| :<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
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| :<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}}
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| 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′(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>
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| This definition can be proven independent of the choice of γ, provided γ is selected in the prescribed manner so that γ′(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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| == In the continuum mechanics of solids ==
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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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| ===Derivatives of scalar valued functions of vectors===
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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
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| :<math> | |
| \frac{\partial f}{\partial \mathbf{v}}\cdot\mathbf{u} = Df(\mathbf{v})[\mathbf{u}]
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| = \left[\frac{d }{d \alpha}~f(\mathbf{v} + \alpha~\mathbf{u})\right]_{\alpha = 0}
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| </math>
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| for all vectors <math>\mathbf{u}</math>.
| |
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| ''Properties:''
| |
| {{ordered list
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| |1= 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}} + \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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| </math>
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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}
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| </math>
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| }}
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| ===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
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| :<math>
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| \frac{\partial \mathbf{f}}{\partial \mathbf{v}}\cdot\mathbf{u} = D\mathbf{f}(\mathbf{v})[\mathbf{u}]
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| = \left[\frac{d }{d \alpha}~\mathbf{f}(\mathbf{v} + \alpha~\mathbf{u})\right]_{\alpha = 0}
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| </math>
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| for all vectors <math>\mathbf{u}</math>.
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| ''Properties:''
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| {{ordered list
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| |1= If <math>\mathbf{f}(\mathbf{v}) = \mathbf{f}_1(\mathbf{v}) + \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}} + \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)
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| </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)
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| </math>
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| }}
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| | |
| ===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
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| <math>\boldsymbol{T}</math> is the ''' second order tensor''' defined as | |
| :<math>
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| \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}
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| </math>
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| for all second order tensors <math>\boldsymbol{T}</math>. | |
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| ''Properties:''
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| {{ordered list
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| |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>
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| |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>
| |
| | |
| |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>
| |
| }}
| |
| | |
| ===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>
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| (or at <math>\boldsymbol{S}</math>) in the direction <math>\boldsymbol{T}</math> is the ''' fourth order tensor''' defined as
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| :<math>
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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>
| |
| for all second order tensors <math>\boldsymbol{T}</math>.
| |
| | |
| ''Properties:''
| |
| {{ordered list
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| |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>
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| |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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| |3= If <math>\boldsymbol{F}(\boldsymbol{S}) = \boldsymbol{F}_1(\boldsymbol{F}_2(\boldsymbol{S}))</math> then <math> \frac{\partial \boldsymbol{F}}{\partial \boldsymbol{S}}:\boldsymbol{T} = \frac{\partial \boldsymbol{F}_1}{\partial \boldsymbol{F}_2}:\left(\frac{\partial \boldsymbol{F}_2}{\partial \boldsymbol{S}}:\boldsymbol{T} \right) </math>
| |
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| |4= If <math>f(\boldsymbol{S}) = f_1(\boldsymbol{F}_2(\boldsymbol{S}))</math> then <math> \frac{\partial f}{\partial \boldsymbol{S}}:\boldsymbol{T} = \frac{\partial f_1}{\partial \boldsymbol{F}_2}:\left(\frac{\partial \boldsymbol{F}_2}{\partial \boldsymbol{S}}:\boldsymbol{T} \right) </math>
| |
| }}
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| == See also ==
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| * [[Fréchet derivative]]
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| * [[Gâteaux derivative]]
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| * [[Derivative (generalizations)]]
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| * [[Lie derivative]]
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| * [[Differential form]]
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| * [[Structure tensor]]
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| * [[Tensor derivative (continuum mechanics)]]
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| * [[Del in cylindrical and spherical coordinates]]
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| == Notes ==
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| {{reflist}}
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| == References ==
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| *{{cite book | first=F. B. | last=Hildebrand | title=Advanced Calculus for Applications| publisher=Prentice Hall | year=1976 | isbn=0-13-011189-9 }}
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| *{{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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| == External links ==
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| *[http://mathworld.wolfram.com/DirectionalDerivative.html Directional derivatives] at [[MathWorld]].
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| *[http://planetmath.org/directionalderivative Directional derivative] at [[PlanetMath]].
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| | |
| [[Category:Differential calculus]]
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| [[Category:Differential geometry]]
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| [[Category:Generalizations of the derivative]]
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| [[Category:Multivariable calculus]]
| |
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