directional derivative

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

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

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

Alternatively,

 $\displaystyle(D_{v}f)(u)$ $\displaystyle=$ $\displaystyle\lim_{h\to 0}\frac{f(u+hv)-f(u)}{h}$ $\displaystyle=$ $\displaystyle Df(u)\cdot v,$

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

Properties

Let $u\in U$.

1. 1.

$D_{v}f$ is linear in $v$. If $v,w\in\mathbbmss{R}^{n}$ and $\lambda,\mu\in\mathbbmss{R}$, 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. 2.

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

 $\displaystyle D_{v}D_{w}f(u)$ $\displaystyle=$ $\displaystyle\frac{\partial^{2}}{\partial s\partial t}f(u+sv+tw)\Big{|}_{s=0},$ $\displaystyle=$ $\displaystyle v^{T}\cdot\operatorname{Hess}f(u)\cdot w,$

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

Example

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

 $\displaystyle\lim_{h\rightarrow 0}\frac{1}{h}\left((1+h)^{2}+3(2+h)^{2}(3+h)-3% 7\right)$ $\displaystyle=$ $\displaystyle\lim_{h\rightarrow 0}\frac{1}{h}(3h^{3}+37h^{2}+50h)$ $\displaystyle=$ $\displaystyle\lim_{h\rightarrow 0}3h^{2}+37h+50=50$
 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