Directional Derivatives and the Gradient

Remember to do load(vect) if you haven't already during the session. By default Maxima assumes that the coordinate system is rectangular (cartesian) in the variables , , and . If you do not want that then you need to change it by means of the scalefactors command.

Given the function of two variables, $f(x,y)=\mathrm{e}^{x^2} \sin y$ .

$\begin{maximasession} f(x,y) := exp(x^2) * sin(y); \maximaoutput* \i1. f(x,y) :=... ...\\ \o1. f\left(x , y\right)\mathbin{:=}\exp x^2 \sin y \\ \end{maximasession}$

We change to the correct coordinate system.

$\begin{maximasession} load(vect); scalefactors([x,y]); \maximaoutput* \i2. load(... ...{}} \\ \i3. scalefactors([x,y]); \\ \o3. \mathbf{done} \\ \end{maximasession}$

Next we find the gradient.

$\begin{maximasession} gdf: grad(f(x,y)); ev(express(gdf), diff); define(gdf(x,y)... ...=}\left[ 2 x e^{x^2} \sin y , e^{x^2} \cos y \right] \\ \end{maximasession}$

The directional derivative of at the point in the direction of the vector $\mathbf{v} = \langle 3,4 \rangle$ is

$\begin{maximasession} v: [3,4]; (gdf(1,2) . v)/sqrt(v . v); ev(%, diff); float(%... ...over{5}} \\ \i10. float(%); \\ \o10. 2.061108499400332 \\ \end{maximasession}$

We know from our theory that the directional derivative is maximized when $\mathbf{v}$ points in the direction of the gradient, and the maximum value is the length of the gradient vector. Let's see just how big that is.

$\begin{maximasession} sqrt(gdf(1,2) . gdf(1,2)); float(ev(%, diff)); \maximaoutp... ...\ \i12. float(ev(%, diff)); \\ \o12. 5.071228088168654 \\ \end{maximasession}$

G. Jay Kerns 2009-12-01