| Version 9 |
Version 8 |
| Let $U$ be an open set in $\R^n$ and $f\colon U\to \C$ is a differentiable |
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 |
function. If $u\in U$ and $v\in \sR^n$, then the |
| \emph{directional derivative} of $f$ in the direction of $v$ is |
\emph{directional derivative} of $f$ in the direction of $v$ is |
| $$ |
$$ |
| (D_v f)(u) = \frac{d}{ds} f(u+sv) \Big|_{s=0}. |
(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 $v$ |
In other words, $(D_v f)(u)$ measures how $f$ changes in the direction of $v$ |
| from $u$. |
from $u$. |
|
|
| Alternatively, |
Alternatively, |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| (D_v f)(u) &=& \lim_{h\to 0} \frac{ f(u+ hv) - f(u)}{h} \\ |
(D_v f)(u) &=& \lim_{h\to 0} \frac{ f(u+ hv) - f(u)}{h} \\ |
| &=& Df(u)\cdot v, |
&=& Df(u)\cdot v, |
| \end{eqnarray*} |
\end{eqnarray*} |
| where $Df$ is the Jacobian matrix of $f$. |
where $Df$ is the Jacobian matrix of $f$. |
|
|
| \subsubsection*{Properties} |
\subsubsection*{Properties} |
| Let $u\in U$. |
Let $u\in U$. |
| \begin{enumerate} |
\begin{enumerate} |
| \item $D_v f$ is linear in $v$. If $v, w\in \R$ and $\lambda, \mu \in \R$, |
\item $D_v f$ is linear in $v$. If $v, w\in \R$ and $\lambda, \mu \in \R$, |
| then |
then |
| $$ |
$$ |
| D_{\lambda v+\mu w}f(u) = \lambda D_{v}f(u) +\mu D_{w}f(u). |
D_{\lambda v+\mu w}f(u) = \lambda D_{v}f(u) +\mu D_{w}f(u). |
| $$ |
$$ |
| In particular, $D_0 f=0$. |
In particular, $D_0 f=0$. |
| \item If $f$ is twice differentiable and $v,w\in \R$, then |
\item If $f$ is twice differentiable and $v,w\in \R$, then |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| D_v D_w f(u) &=& \frac{\partial^2}{\partial s\partial t} f(u+sv + tw) \Big|_{s=0}, \\ |
D_v D_w f(u) &=& \frac{\partial^2}{\partial s\partial t} f(u+sv + tw) \Big|_{s=0}, \\ |
|
&=& v^T\cdot \mbox{Hess}f(u)\cdot w,
|
&=& v\cdot \mbox{Hess}f(u)\cdot w,
|
| \end{eqnarray*} |
\end{eqnarray*} |
| \end{enumerate} |
\end{enumerate} |
|
|
| \subsubsection*{Example} |
\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 |
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*} |
\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}\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}\frac{1}{h}(3h^3+37h^2+50h)\\ |
| &=&\lim_{h\rightarrow 0}3h^2+37h +50 = 50\end{eqnarray*} |
&=&\lim_{h\rightarrow 0}3h^2+37h +50 = 50\end{eqnarray*} |
