# Directional derivative

Short description: Instantaneous rate of change of the function

A directional derivative is a concept in multivariable calculus that measures the rate at which a function changes in a particular direction at a given point.[citation needed]

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 velocity 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: $\displaystyle{ \nabla_{\mathbf{v}}{f}(\mathbf{x})=f'_\mathbf{v}(\mathbf{x})=D_\mathbf{v}f(\mathbf{x})=Df(\mathbf{x})(\mathbf{v})=\partial_\mathbf{v}f(\mathbf{x})=\mathbf{v}\cdot{\nabla f(\mathbf{x})}=\mathbf{v}\cdot \frac{\partial f(\mathbf{x})}{\partial\mathbf{x}}. }$

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.

## Definition

A contour plot of $\displaystyle{ f(x, y)=x^2 + y^2 }$, showing the gradient vector in black, and the unit vector $\displaystyle{ \mathbf{u} }$ scaled by the directional derivative in the direction of $\displaystyle{ \mathbf{u} }$ 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 $\displaystyle{ f(\mathbf{x}) = f(x_1, x_2, \ldots, x_n) }$ along a vector $\displaystyle{ \mathbf{v} = (v_1, \ldots, v_n) }$ is the function $\displaystyle{ \nabla_{\mathbf{v}}{f} }$ defined by the limit[1] $\displaystyle{ \nabla_{\mathbf{v}}{f}(\mathbf{x}) = \lim_{h \to 0}{\frac{f(\mathbf{x} + h\mathbf{v}) - f(\mathbf{x})}{h}}. }$

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.[2]

### For differentiable functions

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

$\displaystyle{ \nabla_{\mathbf{v}}{f}(\mathbf{x}) = \nabla f(\mathbf{x}) \cdot \mathbf{v} }$

where the $\displaystyle{ \nabla }$ on the right denotes the gradient, $\displaystyle{ \cdot }$ is the dot product and v is a unit vector.[3] This follows from defining a path $\displaystyle{ h(t) = x + tv }$ and using the definition of the derivative as a limit which can be calculated along this path to get: \displaystyle{ \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} }

Intuitively, the directional derivative of f at a point x represents the rate of change of f, in the direction of v with respect to time, when moving past x.

### Using only direction of vector

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. In a Euclidean space, some authors[4] 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.[5]

This definition gives the rate of increase of f per unit of distance moved in the direction given by v. In this case, one has $\displaystyle{ \nabla_{\mathbf{v}}{f}(\mathbf{x}) = \lim_{h \to 0}{\frac{f(\mathbf{x} + h\mathbf{v}) - f(\mathbf{x})}{h|\mathbf{v}|}}, }$ or in case f is differentiable at x, $\displaystyle{ \nabla_{\mathbf{v}}{f}(\mathbf{x}) = \nabla f(\mathbf{x}) \cdot \frac{\mathbf{v}}{|\mathbf{v}|} . }$

### 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.[6]

## 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: $\displaystyle{ \nabla_{\mathbf{v}} (f + g) = \nabla_{\mathbf{v}} f + \nabla_{\mathbf{v}} g. }$
2. constant factor rule: For any constant c, $\displaystyle{ \nabla_{\mathbf{v}} (cf) = c\nabla_{\mathbf{v}} f. }$
3. product rule (or Leibniz's rule): $\displaystyle{ \nabla_{\mathbf{v}} (fg) = g\nabla_{\mathbf{v}} f + f\nabla_{\mathbf{v}} g. }$
4. chain rule: If g is differentiable at p and h is differentiable at g(p), then $\displaystyle{ \nabla_{\mathbf{v}}(h\circ g)(\mathbf{p}) = h'(g(\mathbf{p})) \nabla_{\mathbf{v}} g (\mathbf{p}). }$

## In differential geometry

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 differentiable at p. If v is a tangent vector to M at p, then the directional derivative of f along v, denoted variously as df(v) (see Exterior derivative), $\displaystyle{ \nabla_{\mathbf{v}} f(\mathbf{p}) }$ (see Covariant derivative), $\displaystyle{ L_{\mathbf{v}} f(\mathbf{p}) }$ (see Lie derivative), or $\displaystyle{ {\mathbf{v}}_{\mathbf{p}}(f) }$ (see 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 $\displaystyle{ \nabla_{\mathbf{v}} f(\mathbf{p}) = \left.\frac{d}{d\tau} f\circ\gamma(\tau)\right|_{\tau=0}. }$ This definition can be proven independent of the choice of γ, provided γ is selected in the prescribed manner so that γ(0) = p and γ′(0) = v.

### The Lie derivative

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

### The Riemann tensor

Directional derivatives are often used in introductory derivations of the Riemann curvature tensor. Consider a curved rectangle with an infinitesimal vector $\displaystyle{ \delta }$ along one edge and $\displaystyle{ \delta' }$ along the other. We translate a covector $\displaystyle{ S }$ along $\displaystyle{ \delta }$ then $\displaystyle{ \delta' }$ and then subtract the translation along $\displaystyle{ \delta' }$ and then $\displaystyle{ \delta }$. Instead of building the directional derivative using partial derivatives, we use the covariant derivative. The translation operator for $\displaystyle{ \delta }$ is thus $\displaystyle{ 1+\sum_\nu \delta^\nu D_\nu=1+\delta\cdot D, }$ and for $\displaystyle{ \delta' }$, $\displaystyle{ 1+\sum_\mu \delta'^\mu D_\mu=1+\delta'\cdot D. }$ The difference between the two paths is then $\displaystyle{ (1+\delta'\cdot D)(1+\delta\cdot D)S^\rho-(1+\delta\cdot D)(1+\delta'\cdot D)S^\rho=\sum_{\mu,\nu}\delta'^\mu \delta^\nu[D_\mu,D_\nu]S_\rho. }$ It can be argued[7] that the noncommutativity of the covariant derivatives measures the curvature of the manifold: $\displaystyle{ [D_\mu,D_\nu]S_\rho=\pm \sum_\sigma R^\sigma{}_{\rho\mu\nu}S_\sigma, }$ where $\displaystyle{ R }$ 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 $\displaystyle{ \mathbf{P}=i\nabla. }$ (the i ensures that P is a self-adjoint operator) For a finite displacement λ, the unitary Hilbert space representation for translations is[8] $\displaystyle{ U(\boldsymbol{\lambda})=\exp\left(-i\boldsymbol{\lambda}\cdot\mathbf{P}\right). }$ By using the above definition of the infinitesimal translation operator, we see that the finite translation operator is an exponentiated directional derivative: $\displaystyle{ U(\boldsymbol{\lambda})=\exp\left(\boldsymbol{\lambda}\cdot\nabla\right). }$ This is a translation operator in the sense that it acts on multivariable functions f(x) as $\displaystyle{ U(\boldsymbol{\lambda}) f(\mathbf{x})=\exp\left(\boldsymbol{\lambda}\cdot\nabla\right) f(\mathbf{x}) = f(\mathbf{x}+\boldsymbol{\lambda}). }$

### 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 $\displaystyle{ \hat{\theta} = \boldsymbol{\theta}/\theta }$ is $\displaystyle{ U(R(\mathbf{\theta}))=\exp(-i\mathbf{\theta}\cdot\mathbf{L}). }$ Here L is the vector operator that generates SO(3): $\displaystyle{ \mathbf{L}=\begin{pmatrix} 0& 0 & 0\\ 0& 0 & 1\\ 0& -1 & 0 \end{pmatrix}\mathbf{i}+\begin{pmatrix} 0 &0 & -1\\ 0& 0 &0 \\ 1 & 0 & 0 \end{pmatrix}\mathbf{j}+\begin{pmatrix} 0&1 &0 \\ -1&0 &0 \\ 0 & 0 & 0 \end{pmatrix}\mathbf{k}. }$ It may be shown geometrically that an infinitesimal right-handed rotation changes the position vector x by $\displaystyle{ \mathbf{x}\rightarrow \mathbf{x}-\delta\boldsymbol{\theta}\times\mathbf{x}. }$ So we would expect under infinitesimal rotation: $\displaystyle{ U(R(\delta\boldsymbol{\theta})) f(\mathbf{x}) = f(\mathbf{x}-\delta\boldsymbol{\theta}\times\mathbf{x})=f(\mathbf{x})-(\delta\boldsymbol{\theta}\times\mathbf{x})\cdot\nabla f. }$ It follows that $\displaystyle{ U(R(\delta\mathbf{\theta}))=1-(\delta\mathbf{\theta}\times\mathbf{x})\cdot\nabla. }$ Following the same exponentiation procedure as above, we arrive at the rotation operator in the position basis, which is an exponentiated directional derivative:[12] $\displaystyle{ U(R(\mathbf{\theta}))=\exp(-(\mathbf{\theta}\times\mathbf{x})\cdot\nabla). }$

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

## 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.[13] The directional directive provides a systematic way of finding these derivatives.

## Notes

1. R. Wrede; M.R. Spiegel (2010). Advanced Calculus (3rd ed.). Schaum's Outline Series. ISBN 978-0-07-162366-7.
2. The applicability extends to functions over spaces without a metric and to differentiable manifolds, such as in general relativity.
3. 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.
4. Thomas, George B. Jr.; and Finney, Ross L. (1979) Calculus and Analytic Geometry, Addison-Wesley Publ. Co., fifth edition, p. 593.
5. 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.
6. Hughes Hallett, Deborah; McCallum, William G.; Gleason, Andrew M. (2012-01-01). Calculus : Single and multivariable.. John wiley. pp. 780. ISBN 9780470888612. OCLC 828768012.
7. Zee, A. (2013). Einstein gravity in a nutshell. Princeton: Princeton University Press. p. 341. ISBN 9780691145587.
8. Weinberg, Steven (1999). The quantum theory of fields (Reprinted (with corr.). ed.). Cambridge [u.a.]: Cambridge Univ. Press. ISBN 9780521550017.
9. Zee, A. (2013). Einstein gravity in a nutshell. Princeton: Princeton University Press. ISBN 9780691145587.
10. Cahill, Kevin Cahill (2013). Physical mathematics (Repr. ed.). Cambridge: Cambridge University Press. ISBN 978-1107005211.
11. Larson, Ron; Edwards, Bruce H. (2010). Calculus of a single variable (9th ed.). Belmont: Brooks/Cole. ISBN 9780547209982.
12. Shankar, R. (1994). Principles of quantum mechanics (2nd ed.). New York: Kluwer Academic / Plenum. p. 318. ISBN 9780306447907.
13. J. E. Marsden and T. J. R. Hughes, 2000, Mathematical Foundations of Elasticity, Dover.