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directional derivative (Definition)

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

$\displaystyle (D_v f)(u) = \frac{d}{ds} f(u+sv) \Big\vert _{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. $ D_v f$ is linear in $ v$. If $ v, w\in \mathbbmss{R}^n$ and $ \lambda, \mu \in \mathbbmss{R}$, then
    $\displaystyle 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 \mathbbmss{R}^n$, then
    $\displaystyle D_v D_w f(u)$ $\displaystyle =$ $\displaystyle \frac{\partial^2}{\partial s\partial t} f(u+sv + tw) \Big\vert _{s=0},$  
      $\displaystyle =$ $\displaystyle v^T\cdot \operatorname{Hess}f(u)\cdot w,$  

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



"directional derivative" is owned by matte. [ full author list (3) | owner history (4) ]
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See Also: partial derivative, derivative, derivative notation, Jacobian matrix, gradient, fixed points of normal functions, Hessian matrix

Other names:  derivative with respect to a vector, partial derivative with respect to a vector
Keywords:  derivative vector, directional derivative, partial derivative

Attachments:
derivation of directional derivative (Derivation) by apmc
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Cross-references: equation, point, derivative, Hessian matrix, differentiable, Jacobian matrix, measures, differentiable function, open set
There are 9 references to this entry.

This is version 11 of directional derivative, born on 2001-11-14, modified 2005-04-16.
Object id is 847, canonical name is DirectionalDerivative.
Accessed 14772 times total.

Classification:
AMS MSC: 26B10 (Real functions :: Functions of several variables :: Implicit function theorems, Jacobians, transformations with several variables)
 26B12 (Real functions :: Functions of several variables :: Calculus of vector functions)
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