Directional derivative

Template:Short description Template:Refimprove section In multivariable calculus, the directional derivative measures the instantaneous rate at which a function changes along a specified vector through a given point. If the vector is multiplied by a scalar, the corresponding directional derivative is multiplied by the same scalar.

Some elementary texts instead use the phrase "directional derivative in the direction of Template:Math" for the rate of change of a function per unit distance in that direction. In that convention the nonzero vector Template:Math is first normalized to the unit vector $\widehat{𝐯}=𝐯/\Vert 𝐯\Vert$, where the normalized vector is denoted with a circumflex (hat) symbol: $\hat{}$.

The directional derivative of a scalar function f with respect to a vector v may be denoted by any of the following: $$\begin{array}{cc}{\mathrm{\nabla }}_{𝐯}f(𝐱)& =f{\text{'}}_{𝐯}(𝐱)\\ & =D_{𝐯}f(𝐱)\\ & =Df(𝐱)(𝐯)\\ & ={\partial }_{𝐯}f(𝐱)\\ & =\frac{\partial f(𝐱)}{\partial 𝐯}\\ & =𝐯\cdot \mathrm{\nabla }f(𝐱)\\ & =𝐯\cdot \frac{\partial f(𝐱)}{\partial 𝐱}.\end{array}$$

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. In functional analysis, the analogous notion for functions between topological vector spaces is the Gateaux derivative.

Definition

A contour plot of $f(x,y)=x^2+y^2$, showing the gradient vector in black, and the unit vector $𝐮$ scaled by the directional derivative in the direction of $𝐮$ 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 $$f(𝐱)=f(x_1,x_2,\dots ,x_n)$$ along a vector $$𝐯=(v_1,\dots ,v_n)$$ is the function ${\mathrm{\nabla }}_{𝐯}f$ defined by the limit $${\mathrm{\nabla }}_{𝐯}f(𝐱)=\underset{h\to 0}{lim}\frac{f(𝐱+h𝐯)-f(𝐱)}{h}={\frac{\mathrm{d}}{\mathrm{d}t}f(𝐱+t𝐯)|}_{t=0}.$$

This definition is valid in a broad range of contexts, for example, where the norm of a vector is defined. In finite dimensions, it does not depend on the choice of norm, since all norms are equivalent. Its applicability extends to functions on finite-dimensional vector spaces without a metric and to differentiable manifolds, such as in general relativity.

For differentiable functions

If the function f is differentiable at x, then the directional derivative exists along any vector v at x, and one has

$${\mathrm{\nabla }}_{𝐯}f(𝐱)=\mathrm{\nabla }f(𝐱)\cdot 𝐯$$

where the $\mathrm{\nabla }$ on the right denotes the gradient and $\cdot$ is the dot product.

It can be derived by using the property that all directional derivatives at a point make up a single tangent plane which can be defined using partial derivatives. This can be used to find a formula for the gradient vector and an alternative formula for the directional derivative, the latter of which can be rewritten as shown above for convenience.

It also follows from defining a path $𝐡(t)=𝐱+t𝐯$ and using the definition of the derivative as a limit along this path to get $$\begin{array}{cc}{\mathrm{\nabla }}_{𝐯}f(𝐱)-\mathrm{\nabla }f(𝐱)\cdot 𝐯& =\underset{t\to 0}{lim}\frac{f(𝐱+t𝐯)-f(𝐱)}{t}-\mathrm{\nabla }f(𝐱)\cdot 𝐯\\ & =\underset{t\to 0}{lim}\frac{f(𝐱+t𝐯)-f(𝐱)-t\mathrm{\nabla }f(𝐱)\cdot 𝐯}{t}\\ & =0,\end{array}$$ whence $\mathrm{\nabla }f(𝐱)\cdot 𝐯={\mathrm{\nabla }}_{𝐯}f(𝐱)$.

Using only direction of vector

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 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.

This definition gives the rate change of a function Template:Math per unit of distance moved in the direction given by Template:Math. In this case, one has $${\mathrm{\nabla }}_{\widehat{𝐯}}f(𝐱)=\underset{h\to 0}{lim}\frac{f(𝐱+h𝐯)-f(𝐱)}{h\Vert v\Vert },$$ or in case f is differentiable at x, $${\mathrm{\nabla }}_{\widehat{𝐯}}f(𝐱)=\mathrm{\nabla }f(𝐱)\cdot \widehat{𝐯}.$$

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 of, and differentiable at, p:

1. sum rule: $${\mathrm{\nabla }}_{𝐯}(f+g)={\mathrm{\nabla }}_{𝐯}f+{\mathrm{\nabla }}_{𝐯}g.$$
2. constant factor rule: For any constant c, $${\mathrm{\nabla }}_{𝐯}(cf)=c{\mathrm{\nabla }}_{𝐯}f.$$
3. product rule (or Leibniz's rule): $${\mathrm{\nabla }}_{𝐯}(fg)=g{\mathrm{\nabla }}_{𝐯}f+f{\mathrm{\nabla }}_{𝐯}g.$$
4. chain rule: If g is differentiable at p and h is differentiable at g(p), then $${\mathrm{\nabla }}_{𝐯}(h\circ g)(𝐩)=h^{\prime }(g(𝐩)){\mathrm{\nabla }}_{𝐯}g(𝐩).$$

In differential geometry

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), ${\mathrm{\nabla }}_{𝐯}f(𝐩)$ (see Covariant derivative), $L_{𝐯}f(𝐩)$ (see Lie derivative), or ${𝐯}_{𝐩}(f)$ (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 $${\mathrm{\nabla }}_{𝐯}f(𝐩)={\frac{d}{d\tau }f\circ \gamma (\tau )|}_{\tau =0}.$$ 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 derivative

The Lie derivative of a vector field $W^{\mu }(x)$ along a vector field $V^{\mu }(x)$ is given by the difference of two directional derivatives (with vanishing torsion): $${ℒ}_V{W}^{\mu }=(V\cdot \mathrm{\nabla })W^{\mu }-(W\cdot \mathrm{\nabla })V^{\mu }.$$ In particular, for a scalar field $\varphi (x)$, the Lie derivative reduces to the standard directional derivative: $${ℒ}_V\varphi =(V\cdot \mathrm{\nabla })\varphi .$$

The Riemann tensor

Directional derivatives are often used in introductory derivations of the Riemann curvature tensor. Consider a curved rectangle with an infinitesimal vector $\delta$ along one edge and ${\delta }^{\prime }$ along the other. We translate a covector $S$ along $\delta$ then ${\delta }^{\prime }$ and then subtract the translation along ${\delta }^{\prime }$ and then $\delta$. Instead of building the directional derivative using partial derivatives, we use the covariant derivative. The translation operator for $\delta$ is thus $$1+\sum _{\nu }{\delta }^{\nu }{D}_{\nu }=1+\delta \cdot D,$$ and for ${\delta }^{\prime }$, $$1+\sum _{\mu }\delta {\text{'}}^{\mu }{D}_{\mu }=1+{\delta }^{\prime }\cdot D.$$ The difference between the two paths is then $$(1+{\delta }^{\prime }\cdot D)(1+\delta \cdot D)S^{\rho }-(1+\delta \cdot D)(1+{\delta }^{\prime }\cdot D)S^{\rho }=\sum _{\mu ,\nu }\delta {\text{'}}^{\mu }{\delta }^{\nu }[D_{\mu },D_{\nu }]S_{\rho }.$$ It can be argued that the noncommutativity of the covariant derivatives measures the curvature of the manifold: $$[D_{\mu },D_{\nu }]S_{\rho }=\pm \sum _{\sigma }{R}^{\sigma }{}_{\rho \mu \nu }{S}_{\sigma },$$ where $R$ 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 $$𝐏=i\mathrm{\nabla }.$$ (the i ensures that P is a self-adjoint operator) For a finite displacement λ, the unitary Hilbert space representation for translations is $$U(𝝀)=\exp (-i𝝀\cdot 𝐏).$$ By using the above definition of the infinitesimal translation operator, we see that the finite translation operator is an exponentiated directional derivative: $$U(𝝀)=\exp (𝝀\cdot \mathrm{\nabla }).$$ This is a translation operator in the sense that it acts on multivariable functions f(x) as $$U(𝝀)f(𝐱)=\exp (𝝀\cdot \mathrm{\nabla })f(𝐱)=f(𝐱+𝝀).$$

Rotations

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 $\hat{\theta }=𝜽/\theta$ is $$U(R(𝜽))=\exp (-i𝜽\cdot 𝐋).$$ Here L is the vector operator that generates SO(3): $$𝐋=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& -1& 0\end{array}\right)𝐢+\left(\begin{array}{ccc}0& 0& -1\\ 0& 0& 0\\ 1& 0& 0\end{array}\right)𝐣+\left(\begin{array}{ccc}0& 1& 0\\ -1& 0& 0\\ 0& 0& 0\end{array}\right)𝐤.$$ It may be shown geometrically that an infinitesimal right-handed rotation changes the position vector x by $$𝐱\to 𝐱-\delta 𝜽\times 𝐱.$$ So we would expect under infinitesimal rotation: $$U(R(\delta 𝜽))f(𝐱)=f(𝐱-\delta 𝜽\times 𝐱)=f(𝐱)-(\delta 𝜽\times 𝐱)\cdot \mathrm{\nabla }f.$$ It follows that $$U(R(\delta 𝜽))=1-(\delta 𝜽\times 𝐱)\cdot \mathrm{\nabla }.$$ Following the same exponentiation procedure as above, we arrive at the rotation operator in the position basis, which is an exponentiated directional derivative: $$U(R(𝜽))=\exp (-(𝜽\times 𝐱)\cdot \mathrm{\nabla }).$$

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 $𝐧$, then the normal derivative of a function f is sometimes denoted as $\frac{\partial f}{\partial 𝐧}$. In other notations, $$\frac{\partial f}{\partial 𝐧}=\mathrm{\nabla }f(𝐱)\cdot 𝐧={\mathrm{\nabla }}_{𝐧}f(𝐱)=\frac{\partial f}{\partial 𝐱}\cdot 𝐧=Df(𝐱)[𝐧].$$

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. The directional derivative provides a systematic way of finding these derivatives.

See also

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References

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External links

• Directional derivatives at MathWorld.
• Directional derivative at PlanetMath.