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Directional derivative
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== 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 ''Ξ±'' 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]] β 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>
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