directional derivativeDirectionalDerivative2001-11-14 13:55:382005-04-16 17:49:03Definitionderivative vectordirectional derivativepartial derivative% this is the default PlanetMath preamble. as your knowledge
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\newtheorem{thm}{Theorem}
\newtheorem{defn}{Definition}
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\newtheorem{lemma}{Lemma}
\newtheorem{cor}{Corollary}Let $U$ be an open set in $\R^n$ and $f\colon U\to \C$ is a differentiable
function. If $u\in U$ and $v\in \sR^n$, then the
\emph{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,
\begin{eqnarray*}
(D_v f)(u) &=& \lim_{h\to 0} \frac{ f(u+ hv) - f(u)}{h} \\
&=& Df(u)\cdot v,
\end{eqnarray*}
where $Df$ is the Jacobian matrix of $f$.
\subsubsection*{Properties}
Let $u\in U$.
\begin{enumerate}
\item $D_v f$ 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, $D_0 f=0$.
\item If $f$ is twice differentiable and $v,w\in \R^n$, then
\begin{eqnarray*}
D_v D_w f(u) &=& \frac{\partial^2}{\partial s\partial t} f(u+sv + tw) \Big|_{s=0}, \\
&=& v^T\cdot \operatorname{Hess}f(u)\cdot w,
\end{eqnarray*}
where $\operatorname{Hess}$ is the Hessian matrix of $f$.
\end{enumerate}
\subsubsection*{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
\begin{eqnarray*}
\lim_{h\rightarrow 0}\frac{1}{h}\left((1+h)^2 + 3(2+h)^2(3+h) - 37\right)
&=&\lim_{h\rightarrow 0}\frac{1}{h}(3h^3+37h^2+50h)\\
&=&\lim_{h\rightarrow 0}3h^2+37h +50 = 50\end{eqnarray*}