Directional derivative

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In multivariable calculus, the directional derivative measures the rate at which a function changes in a particular direction at a given point.Template:Cn

The directional derivative of a multivariable differentiable (scalar) function along a given 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 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 coordinate curves, all other coordinates being constant. The directional derivative is a special case of the Gateaux derivative.

DefinitionEdit

File:Directional derivative contour plot.svg

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 <math>\nabla_{\mathbf{v}}{f}</math> defined by the limit<ref>Template:Cite book</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 norm of a vector (and hence a unit vector) is undefined.<ref>The applicability extends to functions over spaces without a metric and to differentiable manifolds, such as in general relativity.</ref>

For differentiable functionsEdit

If the function f is 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 rate of change of f, in the direction of v.

Using only direction of vectorEdit

File:Geometrical interpretation of a directional derivative.svg

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 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 Template:Math per unit of distance moved in the direction given by Template:Math. 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 vectorEdit

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>Template:Cite book</ref>

PropertiesEdit

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 of, and differentiable at, p:

sum rule: <math display="block">\nabla_{\mathbf{v}} (f + g) = \nabla_{\mathbf{v}} f + \nabla_{\mathbf{v}} g.</math>
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 geometryEdit

Template:See also

Let Template:Math be a differentiable manifold and Template:Math a point of Template:Math. Suppose that Template:Math is a function defined in a neighborhood of Template:Math, and differentiable at Template:Math. If Template:Math is a tangent vector to Template:Math at Template:Math, then the directional derivative of Template:Math along Template:Math, denoted variously as Template:Math (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 Template:Section link), can be defined as follows. Let Template:Math be a differentiable curve with Template:Math and Template:Math. 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 Template:Math, provided Template:Math is selected in the prescribed manner so that Template:Math and Template:Math.

The Lie derivativeEdit

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 tensorEdit

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>Template:Cite book</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 theoryEdit

TranslationsEdit

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 Hilbert space representation for translations is<ref>Template:Cite book</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>

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RotationsEdit

The 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>Template:Cite book</ref> <math display="block">U(R(\mathbf{\theta}))=\exp(-(\mathbf{\theta}\times\mathbf{x})\cdot\nabla).</math>

Normal derivativeEdit

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 solidsEdit

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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