Scratching the Surface

Purpose

The purpose of this lab is to teach you how to use Maple to formulate and compute areas of surfaces in 3.

Background

We have seen how double and triple integrals can be used to compute areas of plane regions, volumes, and centers of mass. This lab deals with using double integrals to compute areas of surfaces. You might have that sinking feeling that surface areas are like arc length, but worse, and you would be right! Fortunately, Maple can help a lot with visualization and doing the actual integrals.

To begin with, consider a surface defined parametrically by a vector function , where s and t lie in some domain D. For example, the unit sphere, , has the parametric representation in spherical coordinates

with and . Note that any surface defined by can be represented parametrically as

The problem of computing the surface area via integration is really to find the differential element of surface area dS. This turns out to be a little involved, but the development is a typical calculus exercise: first approximate a small element of surface area, then take a limit as the size of the element goes to zero.

For the first step in the development, we need to recall the tangent plane at a point on a surface. Remember that this is the plane that touches the surface at and contains all vectors tangent to the surface at . Fortunately, to generate the plane, we only need the two tangent vectors

and

both evaluated at the point . The tangent plane to our surface at some particular point , is just the linear approximation

In an earlier lab, the Maple procedure TanPlane for computing the tangent plane of a surface defined by an equation was introduced. With the use of the powerful Maple map function, TanPlane can be used to generate tangent planes for parametric surfaces as well. The following example shows how to apply this to the unit sphere, computing the tangent plane at the point , (the North Pole) and plotting the sphere and the tangent plane at . Note the method for defining the sphere with a list, and the use of the map function.

  > with(CalcP):

  > with(plots):

  > sphere := (s,t) -> [cos(s)*sin(t),sin(s)*sin(t),cos(t)] ;

  > map(TanPlane,sphere(s,t),s=0,t=0);

  > p1 := plot3d(map(TanPlane,sphere(s,t),s=Pi/4,t=Pi/4),s=0..Pi/2,t=0..Pi/2):

  > p2 := plot3d(sphere(s,t),s=0..2*Pi,t=0..Pi):

  > display3d(p1,p2);

The splitting up into two plots was done deliberately, so you can use different ranges for s and t in the two plots. To see what this does, change the definition of p1 to

  > p1 := plot3d(map(TanPlane,sphere(s,t),s=Pi/4,t=Pi/4),s=Pi/4..Pi/4+0.2,t=Pi/4..Pi/4+0.2):

reissue the display3d command, and you should see only a small piece of the tangent plane.

In the derivation of dS, we approximate a small area with the area of a small piece of the tangent plane that we get by using the linear approximation to map a small rectangle in the s-t plane with sides of length and . The linear approximation maps this rectangle into a parallelogram, whose sides are the vectors and . This parallelogram is what you saw in the Maple example above. From earlier in calculus we know that the area of this parallelogram can be computed by taking the cross product of these two vectors, giving

Taking limits, we obtain

which gives a formula for the differential element of surface area. So, if our surface is defined on a region D in the s-t plane, the area A is given by

In any given case, computing dS is clearly not an easy task. In the special case that the surface is defined by for and , the formula for the surface area reduces to

The similarity to the case of arc length should be evident, and even in the case of relatively simple surfaces, the area often cannot be computed analytically.

In the case of a parametrically defined surface, Maple and the functions for vector calculus from the CalcP package can be useful. The following example shows how to use VDiff, VMag, and the linalg procedures dotprod and crossprod to compute the area of a sphere. If you've forgotten what any of these procedures do, look at the help screens.

  > with(linalg):

Warning: new definition for   norm

Warning: new definition for   trace

  > t1 := VDiff(sphere(s,t),t);

  > t2 := VDiff(sphere(s,t),s);

  > VMag(crossprod(t1,t2));

  > int(int(sin(t),s=0..2*Pi),t=0..Pi);

Note that VMag puts an extra absolute value around the result and that Maple doesn't automatically replace with . An alternative is the command

  > sqrt(simplify(dotprod(crossprod(t1,t2),crossprod(t1,t2))));

but Maple is not good about using trig identities and, furthermore, often makes mistakes choosing branches of the square root function, as shown below. You might want to experiment with peeling off the sqrt and simplify commands, to see why they were added.

  > int(int(sqrt(1-cos(t)^2),s=0..2*Pi),t=0..Pi);

Exercises

1. Find the tangent plane to the surface at the point .
2. Use Maple to verify that in the special case of a surface . (Hint - define your surface by and let Maple do the hard work.)
3. Find the surface area of above the rectangle , .
4. Find the surface area of the part of that lies above z=0. (Hint - parametrize the surface by polar coordinates .)
5. Find the surface area of the torus defined parametrically by , , .