Directional derivative

In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P, in the direction of V. It therefore generalizes the notion of a partial derivative, in which the direction is always taken parallel to one of the coordinate axes.

The directional derivative is a special case of the Gâteaux derivative.

Definition

The directional derivative of a scalar function $f(vec\{x\})\; =\; f(x\_1,\; x\_2,\; ldots,\; x\_n)$ along a vector $vec\{v\}\; =\; (v\_1,\; ldots,\; v\_n)$ is the function defined by the limit

$nabla\_\{vec\{v\}\}\{f\}(vec\{x\})\; =\; lim\_\{h\; rightarrow\; 0\}\{frac\{f(vec\{x\}\; +\; hvec\{v\})\; -\; f(vec\{x\})\}\{h\}\}.$

Sometimes authors write D_v instead of $nabla\_v$. If the function $f$ is differentiable at $vec\{x\}$, then the directional derivative exists along any vector $vec\{v\},$ and one has

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

where the $nabla$ on the right denotes the gradient and $cdot$ is the Euclidean inner product. At any point $vec\{x\}$, the directional derivative of $f$ intuitively represents the rate of change in $f$ along $vec\{v\}$ at the point $vec\{x\}$. Usually directions are taken to be normalized, so $vec\{v\}$ is a unit vector, although the definition above works for arbitrary (even zero) vectors.

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:

• The sum rule: $nabla\_v\; (f\; +\; g)\; =\; nabla\_v\; f\; +\; nabla\_v\; g$
• The constant factor rule: For any constant c, $nabla\_v\; (cf)\; =\; cnabla\_v\; f$
• The product rule (or Leibniz rule): $nabla\_v\; (fg)\; =\; gnabla\_v\; f\; +\; fnabla\_v\; g$
• The chain rule: If g is differentiable at p and h is differentiable at g(p), then

$nabla\_v\; hcirc\; g\; (p)\; =\; h\text{'}(g(p))\; nabla\_v\; g\; (p)$

In differential geometry

Let M be a differentiable manifold and p a point of M. Suppose that f is 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 $nabla\_v\; f(p)$ (see covariant derivative), $L\_v\; f(p)$ (see Lie derivative), or $v\_p(f)$ (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

$nabla\_v\; f(p)\; =\; left.frac\{d\}\{dtau\}\; fcircgamma(tau)right|\_\{tau=0\}$

This definition can be proven independent of the choice of γ, provided γ is selected in the prescribed manner so that γ'(0) = v.

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 $vec\{n\}$, then the directional derivative of a function ƒ is sometimes denoted as $frac\{\; partial\; f\}\{partial\; n\}$.

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 directive provides a systematic way of finding these derivatives.

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.

Derivatives of scalar valued functions of vectors

Let $f(mathbf\{v\})$ be a real valued function of the vector $mathbf\{v\}$. Then the derivative of $f(mathbf\{v\})$ with respect to $mathbf\{v\}$ (or at $mathbf\{v\}$) in the direction $mathbf\{u\}$ is the vector defined as

$$

frac{partial f}{partial mathbf{v}}cdotmathbf{u} = Df(mathbf{v})[mathbf{u}] = left[frac{d }{d alpha}~f(mathbf{v} + alpha~mathbf{u})right]_{alpha = 0} for all vectors $mathbf\{u\}$.

Properties:

1) If $f(mathbf\{v\})\; =\; f\_1(mathbf\{v\})\; +\; f\_2(mathbf\{v\})$ then $frac\{partial\; f\}\{partial\; mathbf\{v\}\}cdotmathbf\{u\}\; =\; left(frac\{partial\; f\_1\}\{partial\; mathbf\{v\}\}\; +\; frac\{partial\; f\_2\}\{partial\; mathbf\{v\}\}right)cdotmathbf\{u\}$

2) If $f(mathbf\{v\})\; =\; f\_1(mathbf\{v\})~\; f\_2(mathbf\{v\})$ then $frac\{partial\; f\}\{partial\; mathbf\{v\}\}cdotmathbf\{u\}\; =\; left(frac\{partial\; f\_1\}\{partial\; mathbf\{v\}\}cdotmathbf\{u\}right)~f\_2(mathbf\{v\})\; +\; f\_1(mathbf\{v\})~left(frac\{partial\; f\_2\}\{partial\; mathbf\{v\}\}cdotmathbf\{u\}\; right)$

3) If $f(mathbf\{v\})\; =\; f\_1(f\_2(mathbf\{v\}))$ then $frac\{partial\; f\}\{partial\; mathbf\{v\}\}cdotmathbf\{u\}\; =\; frac\{partial\; f\_1\}\{partial\; f\_2\}~frac\{partial\; f\_2\}\{partial\; mathbf\{v\}\}cdotmathbf\{u\}$

Derivatives of vector valued functions of vectors

Let $mathbf\{f\}(mathbf\{v\})$ be a vector valued function of the vector $mathbf\{v\}$. Then the derivative of $mathbf\{f\}(mathbf\{v\})$ with respect to $mathbf\{v\}$ (or at $mathbf\{v\}$) in the direction $mathbf\{u\}$ is the second order tensor defined as

$$

frac{partial mathbf{f}}{partial mathbf{v}}cdotmathbf{u} = Dmathbf{f}(mathbf{v})[mathbf{u}] = left[frac{d }{d alpha}~mathbf{f}(mathbf{v} + alpha~mathbf{u})right]_{alpha = 0} for all vectors $mathbf\{u\}$.

Properties:

1) If $mathbf\{f\}(mathbf\{v\})\; =\; mathbf\{f\}\_1(mathbf\{v\})\; +\; mathbf\{f\}\_2(mathbf\{v\})$ then $frac\{partial\; mathbf\{f\}\}\{partial\; mathbf\{v\}\}cdotmathbf\{u\}\; =\; left(frac\{partial\; mathbf\{f\}\_1\}\{partial\; mathbf\{v\}\}\; +\; frac\{partial\; mathbf\{f\}\_2\}\{partial\; mathbf\{v\}\}right)cdotmathbf\{u\}$

2) If $mathbf\{f\}(mathbf\{v\})\; =\; mathbf\{f\}\_1(mathbf\{v\})timesmathbf\{f\}\_2(mathbf\{v\})$ then $frac\{partial\; mathbf\{f\}\}\{partial\; mathbf\{v\}\}cdotmathbf\{u\}\; =\; left(frac\{partial\; mathbf\{f\}\_1\}\{partial\; mathbf\{v\}\}cdotmathbf\{u\}right)timesmathbf\{f\}\_2(mathbf\{v\})\; +\; mathbf\{f\}\_1(mathbf\{v\})timesleft(frac\{partial\; mathbf\{f\}\_2\}\{partial\; mathbf\{v\}\}cdotmathbf\{u\}\; right)$

3) If $mathbf\{f\}(mathbf\{v\})\; =\; mathbf\{f\}\_1(mathbf\{f\}\_2(mathbf\{v\}))$ then $frac\{partial\; mathbf\{f\}\}\{partial\; mathbf\{v\}\}cdotmathbf\{u\}\; =\; frac\{partial\; mathbf\{f\}\_1\}\{partial\; mathbf\{f\}\_2\}cdotleft(frac\{partial\; mathbf\{f\}\_2\}\{partial\; mathbf\{v\}\}cdotmathbf\{u\}\; right)$

Derivatives of scalar valued functions of second-order tensors

Let $f(boldsymbol\{S\})$ be a real valued function of the second order tensor $boldsymbol\{S\}$. Then the derivative of $f(boldsymbol\{S\})$ with respect to $boldsymbol\{S\}$ (or at $boldsymbol\{S\}$) in the direction $boldsymbol\{T\}$ is the second order tensor defined as

$$

frac{partial f}{partial boldsymbol{S}}:boldsymbol{T} = Df(boldsymbol{S})[boldsymbol{T}] = left[frac{d }{d alpha}~f(boldsymbol{S} + alpha~boldsymbol{T})right]_{alpha = 0} for all second order tensors $boldsymbol\{T\}$.

Properties:

1) If $f(boldsymbol\{S\})\; =\; f\_1(boldsymbol\{S\})\; +\; f\_2(boldsymbol\{S\})$ then $frac\{partial\; f\}\{partial\; boldsymbol\{S\}\}:boldsymbol\{T\}\; =\; left(frac\{partial\; f\_1\}\{partial\; boldsymbol\{S\}\}\; +\; frac\{partial\; f\_2\}\{partial\; boldsymbol\{S\}\}right):boldsymbol\{T\}$

2) If $f(boldsymbol\{S\})\; =\; f\_1(boldsymbol\{S\})~\; f\_2(boldsymbol\{S\})$ then $frac\{partial\; f\}\{partial\; boldsymbol\{S\}\}:boldsymbol\{T\}\; =\; left(frac\{partial\; f\_1\}\{partial\; boldsymbol\{S\}\}:boldsymbol\{T\}right)~f\_2(boldsymbol\{S\})\; +\; f\_1(boldsymbol\{S\})~left(frac\{partial\; f\_2\}\{partial\; boldsymbol\{S\}\}:boldsymbol\{T\}\; right)$

3) If $f(boldsymbol\{S\})\; =\; f\_1(f\_2(boldsymbol\{S\}))$ then $frac\{partial\; f\}\{partial\; boldsymbol\{S\}\}:boldsymbol\{T\}\; =\; frac\{partial\; f\_1\}\{partial\; f\_2\}~left(frac\{partial\; f\_2\}\{partial\; boldsymbol\{S\}\}:boldsymbol\{T\}\; right)$

Derivatives of tensor valued functions of second-order tensors

Let $boldsymbol\{F\}(boldsymbol\{S\})$ be a second order tensor valued function of the second order tensor $boldsymbol\{S\}$. Then the derivative of $boldsymbol\{F\}(boldsymbol\{S\})$ with respect to $boldsymbol\{S\}$ (or at $boldsymbol\{S\}$) in the direction $boldsymbol\{T\}$ is the fourth order tensor defined as

$$

frac{partial boldsymbol{F}}{partial boldsymbol{S}}:boldsymbol{T} = Dboldsymbol{F}(boldsymbol{S})[boldsymbol{T}] = left[frac{d }{d alpha}~boldsymbol{F}(boldsymbol{S} + alpha~boldsymbol{T})right]_{alpha = 0} for all second order tensors $boldsymbol\{T\}$.

Properties:

1) If $boldsymbol\{F\}(boldsymbol\{S\})\; =\; boldsymbol\{F\}\_1(boldsymbol\{S\})\; +\; boldsymbol\{F\}\_2(boldsymbol\{S\})$ then $frac\{partial\; boldsymbol\{F\}\}\{partial\; boldsymbol\{S\}\}:boldsymbol\{T\}\; =\; left(frac\{partial\; boldsymbol\{F\}\_1\}\{partial\; boldsymbol\{S\}\}\; +\; frac\{partial\; boldsymbol\{F\}\_2\}\{partial\; boldsymbol\{S\}\}right):boldsymbol\{T\}$

2) If $boldsymbol\{F\}(boldsymbol\{S\})\; =\; boldsymbol\{F\}\_1(boldsymbol\{S\})cdotboldsymbol\{F\}\_2(boldsymbol\{S\})$ then $frac\{partial\; boldsymbol\{F\}\}\{partial\; boldsymbol\{S\}\}:boldsymbol\{T\}\; =\; left(frac\{partial\; boldsymbol\{F\}\_1\}\{partial\; boldsymbol\{S\}\}:boldsymbol\{T\}right)cdotboldsymbol\{F\}\_2(boldsymbol\{S\})\; +\; boldsymbol\{F\}\_1(boldsymbol\{S\})cdotleft(frac\{partial\; boldsymbol\{F\}\_2\}\{partial\; boldsymbol\{S\}\}:boldsymbol\{T\}\; right)$

3) If $boldsymbol\{F\}(boldsymbol\{S\})\; =\; boldsymbol\{F\}\_1(boldsymbol\{F\}\_2(boldsymbol\{S\}))$ then $frac\{partial\; boldsymbol\{F\}\}\{partial\; boldsymbol\{S\}\}:boldsymbol\{T\}\; =\; frac\{partial\; boldsymbol\{F\}\_1\}\{partial\; boldsymbol\{F\}\_2\}:left(frac\{partial\; boldsymbol\{F\}\_2\}\{partial\; boldsymbol\{S\}\}:boldsymbol\{T\}\; right)$

4) If $f(boldsymbol\{S\})\; =\; f\_1(boldsymbol\{F\}\_2(boldsymbol\{S\}))$ then $frac\{partial\; f\}\{partial\; boldsymbol\{S\}\}:boldsymbol\{T\}\; =\; frac\{partial\; f\_1\}\{partial\; boldsymbol\{F\}\_2\}:left(frac\{partial\; boldsymbol\{F\}\_2\}\{partial\; boldsymbol\{S\}\}:boldsymbol\{T\}\; right)$

References

See also

• Lie derivative
• Differential form
• Structure tensor
• Tensor derivative (continuum mechanics)