New Features in Maple 17: Multivariate Calculus for Students

Maple Student packages are designed explicitly for teaching and learning concepts in mathematics. In Maple 17, the student package for multivariate calculus has been expanded to provide more tools for exploring problems involving lines and planes.

• Work with lines and planes in two and three dimensions.
• Specify objects in a variety of ways, such as defining a line as:
• containing two points,
• containing a point and a direction,
• being the solution of two equations,
• having a parametric representation,
• containing a point and being orthogonal to a plane, or
• being contained in two planes.
• Define lines and planes using specific values or unknown variables.

• Calculate the distance to other objects, relative position, the intersection between multiple objects, and many other values of interest.
• Visualize lines, planes, intersections, and more.

For many applications, the lines and planes that the objects represent will be fully determined. However, Maple supports arbitrary algebraic expressions occurring in the coordinates of the parameters used to define the objects. For example, it can handle the line through the points and .

Examples

 >

We construct two lines; the first contains the point [1, 0, 2]; and the direction `<,>`(2, -2, 1);, the second the points [5, -3, 1]; and [3, -3, 6];. The Line and Plane objects understand lists as points and Vectors as directions.

 >

 >

Let us see if intersects .

 >

 >

The intersection is a point.

 >

 >

We can also find by obtaining equations for both lines (there are two for each) and solving them simultaneously. The default coordinate variables are , , and ; when constructing a line or plane, you can choose different variables.

 >

 >

 >

We can obtain various other representations of a line with the GetRepresentation command.

 >

 >

 >

 >

We construct a third line, parallel to .

 >

 >

What is the relative position of with respect to ?

 >

 >

 >

We can compute the (Euclidean) distance between a pair of lines using the Distance command. Intersecting lines are at distance 0.

 >

 >

 >

The GetPlot command shows a visualization of the line.

 >

In order to combine visualizations, one can use plots:-display. With all features of the visualizations turned on, it is a little crowded, so we turn some of them off.

 >

Let's consider the plane containing and .

 >

What is the relative position of and ?

 >

 >

 >

The distance between and is the same as the distance between and .  This is always the case when is parallel to , which contains , but is not parallel to .

Since and intersect, also intersects the plane containing and .

 >

 >

Now let us consider a family of lines. We let be a line containing the point and the direction , for some values and .

 >

 >

 >

 >

If we can find values for and that make the numerators of both those distances zero, we get a line that intersects both and .

 >

We now let be the particular line with these values for and .

 >

 >

 >

 >