directional derivative

Let $U$ be an open set in ${ℝ}^n$ and $f:U\to ℂ$ is a differentiable function. If $u\in U$ and $v\in {ℝ}^n$, then the directional derivative of $f$ in the direction of $v$ is

$$(D_{v}f)(u)={\frac{d}{ds}f(u+sv)|}_{s=0}.$$

In other words, $(D_{v}f)(u)$ measures how $f$ changes in the direction of $v$ from $u$.

Alternatively,

$(D_{v}f)(u)$ $=$ $\underset{h\to 0}{lim}{\displaystyle \frac{f(u+hv)-f(u)}{h}}$

$=$ $Df(u)\cdot v,$

where $Df$ is the Jacobian matrix of $f$.

Properties

Let $u\in U$.

1. $D_{v}f$ is linear in $v$. If $v,w\in {ℝ}^n$ and $\lambda ,\mu \in ℝ$, then

   $$D_{\lambda v+\mu w}f(u)=\lambda D_{v}f(u)+\mu D_{w}f(u).$$

   In particular, $D_{0}f=0$.

2. If $f$ is twice differentiable and $v,w\in {ℝ}^n$, then

   $D_v{D}_{w}f(u)$ $=$ ${{\displaystyle \frac{{\partial }^2}{\partial s\partial t}}f(u+sv+tw)|}_{s=0},$

   $=$ $v^T\cdot \mathrm{Hess}f(u)\cdot w,$

   where $\mathrm{Hess}$ is the Hessian matrix of $f$.

Example

For example, if $f\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \\ \hfill z\hfill \end{array}\right)=x^2+3y^{2}z$, and we wanted to find the derivative at the point $𝐚=\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 2\hfill \\ \hfill 3\hfill \end{array}\right)$ in the direction $\overrightarrow{v}=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \\ \hfill 1\hfill \end{array}\right]$, our equation would be

$\underset{h\to 0}{lim}{\displaystyle \frac{1}{h}}\left({(1+h)}^2+3{(2+h)}^2(3+h)-37\right)$ $=$ $\underset{h\to 0}{lim}{\displaystyle \frac{1}{h}}(3h^3+37h^2+50h)$

$=$ $\underset{h\to 0}{lim}3h^2+37h+50=50$

Title	directional derivative
Canonical name	DirectionalDerivative
Date of creation	2013-03-22 11:58:37
Last modified on	2013-03-22 11:58:37
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	15
Author	matte (1858)
Entry type	Definition
Classification	msc 26B12
Classification	msc 26B10
Synonym	derivative with respect to a vector
Synonym	partial derivative with respect to a vector
Related topic	PartialDerivative
Related topic	Derivative
Related topic	DerivativeNotation
Related topic	JacobianMatrix
Related topic	Gradient
Related topic	FixedPointsOfNormalFunctions
Related topic	HessianMatrix