Functions of Several Variables

Now let's try some multivariate functions.

$\begin{maximasession} f(x,y) := (x^2 - y^2)^2; \maximaoutput* \i1. f(x,y) := (x^... ...1. f\left(x , y\right)\mathbin{:=}\left(x^2-y^2\right)^2 \\ \end{maximasession}$

Let's take a look at a plot.

$\begin{maximasession} load(draw); draw3d(explicit(f(x,y), x, -3, 3, y, -3, 3)); ... ...eft[ \mathrm{gr3d}\left(\mathrm{explicit}\right) \right] \\ \end{maximasession}$

See Figure BLANK.

**Figure 8:** A plain surface plot of a function of two variables
$\includegraphics{plot3.eps}$

The important part is to enclose the function with explicit(). We can get a fancier, colored plot if we put enhanced3d = true at the beginning of the function call.

$\begin{maximasession} draw3d(enhanced3d = true, explicit(f(x,y), x, -3, 3, y, -3... ...eft[ \mathrm{gr3d}\left(\mathrm{explicit}\right) \right] \\ \end{maximasession}$

**Figure 9:** An enhanced surface plot of a function of two variables
$\includegraphics{plot4.eps}$

We can see level curves (also known as a contour map) of the function f with the following:

$\begin{maximasession} draw3d(explicit(f(x,y), x, -5, 5, y, -5, 5), contour_lev... ...eft[ \mathrm{gr3d}\left(\mathrm{explicit}\right) \right] \\ \end{maximasession}$

An alternative is to use

$\begin{maximasession} contour_plot(f(x,y), [x, -5, 5], [y, -5, 5] ); \maximaoutp... ...(x,y), [x, -5, 5], [y, -5, 5] ); \\ \o6. \mathbf{false} \\ \end{maximasession}$

**Figure 10:** A contour plot, showing level curves of .
$\includegraphics{plot5.eps}$

The contour map is a 2D plot. If we raise the contour lines up to the plot surface then the lines are more precisely called horizontal traces. Here is a plot of these.

$\begin{maximasession} draw3d(enhanced3d = true, explicit(f(x,y), x, -3, 3, y, -... ...eft[ \mathrm{gr3d}\left(\mathrm{explicit}\right) \right] \\ \end{maximasession}$

See Figure 6. We do surfacehide = true so that we can see the traces better.

**Figure 11:** Another form of contour plot which shows horizontal traces on
$\includegraphics{plot6.eps}$

Subsections

G. Jay Kerns 2009-12-01