Expressions

Create an expression with the names of the variables you want to use.

>	f:= x^2*y;

>	g:= u*v*sin(w);

Be careful: If one of your variables already has a value, you will not get what you expect.

>

t:= Pi:

>	h:= r*sqrt(s)/t;

>	unassign('t');

>	h:= r*sqrt(s)/t;

You can use eval to evaluate an expression at a particular point.

>	eval( f, {x=2,y=3} );

>	eval( h, {r=-1,s=9,t=3} );

>	simplify( % );

Arrow functions (optional)

Arrow functions are an alternative to expressions.  They make some things much easier and other things harder.

Here is a multivariate function defined using arrows.

>	f:= (x,y,z) -> z*exp(x-y);

Evaluating f at a point is easy.

>

f(2,3,5);

Here is how to convert between arrow functions and expressions.

>

f(x,y,z)

Warning, inserted missing semicolon at end of statement, f(x,y,z);

>	unapply( z*exp(x-y), (x,y,z) );

Two variables

In earlier 3D plots of quadric surfaces, we used implicitplot3d to plot a surface defined by an equation F(x,y,z)=0.  For a function of two variables, we can set z=f(x,y), and this can be plotted in the same way.  However, there is a special command for these plots called plot3d that is in some ways more convenient.

For example, if ,

>	plot3d(x*y^2, x=-1..1,y=-2..2);

As usual, the right mouse button over the graphic (when in Maple) brings up a menu of options.

>	plot3d(abs(x)+abs(y), x=-2..2,y=-2..2, style=patch );

>	plot3d(sin(x)*cos(y), x=-2*Pi..2*Pi,y=-2*Pi..2*Pi, axes=boxed);

There is a contourplot function for plotting level curves.

>	contourplot(abs(x)+abs(y), x=-2..2, y=-2..2);

The colors indicate the contour levels.  To see how contours relate to the surface, try

>	contourplot3d( abs(x)+abs(y), x=-2..2, y=-2..2 );

By default, 8 contour levels are chosen automatically. You can give a different number, or specify the levels explicitly.

>	contourplot(sin(x)*cos(y), x=-2*Pi..2*Pi,y=-2*Pi..2*Pi, contours=12);

Observe that contours tend to "surround" a local min or max.  You can specify the grid option to make the curves a little smoother.

>	contourplot(sin(x)*cos(y), x=-2*Pi..2*Pi,y=-2*Pi..2*Pi, contours=[-1,-1/2,0,1/2,1], grid=[50, 50]);

Here are the contours of a saddle.

>	contourplot(x^2-y^2, x=-1..1,y=-1..1, contours=13, grid=[51, 51]);

The signature of a saddle point is the crossing of contours.

Three variables

In this case a level surface is defined by , which is suitable for  implicitplot3d. For example, one level surface of the function is

>	implicitplot3d(sqrt(x^2+y^2+z^2)=1, x=-1..1,y=-1..1,z=-1..1, scaling=constrained);

The level surfaces of a linear function are planes:

>	implicitplot3d(2*x+y-z=1, x=-2..2,y=-2..2,z=-2..2);

Level surfaces can be quite complicated.

>	implicitplot3d(sin(x)*cos(y)*z=1/2, x=-Pi..Pi,y=-Pi..Pi,z=-2..2, grid=[19,19,19]);