MATH 261, Spring 2001 Due Date: Name(s):
Project 17.7a: Surface Area and Surface Integrals
Objective
The objective of this project is to use illustrate the computation of surface area and surface integrals.
Narrative
If you have not already done so, read Sections 17.6 and 17.7 of the text.
If f : R3
 R is continuous and r = r(u, v) = x(u, v), y(u, v), z(u, v) , (u, v)  D  R2
, parametrizes the
surface S in R3
then the surface integral
S
f dS =
D
f(r(u, v)) |ru(u, v) × rv(u, v)| dA.
In particular, if F is any vector field in R3
and
n =
ru(u, v) × rv(u, v)
|ru(u, v) × rv(u, v)|
is a unit normal vector to S, then the surface integral
S
F  dS = D
F(r(u, v))  (ru(u, v) × rv(u, v)) dAuv
= D
F(r(u, v)) 
ru(u, v) × rv(u, v))
|ru(u, v) × rv(u, v)|
|ru(u, v) × rv(u, v)|dAuv
= S
F  n dS
is the flux of F through S. (And this integral is of the form S
f dS if we let f = F  n.)
Task
1. a) Type the command lines below into Maple in the order in which they are listed. In doing so, you
will compute the surface area S2 dS = S2 1 dS = D
|r × r| dA of the unit sphere S2
. (See
Example 10, p. 1123 in the text.)
> # Project 17.7a: Surface Area and Surface Integrals
> restart; with(linalg):
> r := vector([sin(phi)*cos(theta),sin(phi)*sin(theta),cos(phi)]);
> r phi := map(diff,r,phi); r theta := map(diff,r,theta);
> cprod := crossprod(r phi,r theta);
> # The surface area
> integrand := sqrt(simplify(cprod[1]^2+cprod[2]^2+cprod[3]^2));
> Int(Int(integrand,phi=0..Pi),theta=0..2*Pi) =
evalf(int(int(integrand,phi=0..Pi),theta=0..2*Pi));
b) Continue by typing the command lines below into Maple. In doing so, you will compute the surface
integral S2 FdS = S2 Fn dS, where F(x, y, z) = z, y, x , over the unit sphere S2
. (See Example
5, p. 1135 in the text.)
> # A surface integral
> F := vector([z,y,x]);
> plot3d([r[1],r[2],r[3]],phi=0..Pi/2,theta=0..Pi/2,axes=normal,
orientation=[30,60],scaling=constrained);
> integrand := subs({x=r[1],y=r[2],z=r[3]},dotprod(F,cprod));
> Int(Int(integrand,phi=0..Pi),theta=0..2*Pi) =
evalf(int(int(simplify(integrand),phi=0..Pi),theta=0..2*Pi));
Do not adjust the options used to create the graphic for part (b).
At this time make a hard-copy of your typed input and Maple's responses. Then, ...
2. On the graphic you produced in part (b) of Task 1, draw the vectors F(x, y) = z, y, x at the following
points:
(x, y, z) = (1, 0, 0), (0, 1, 0), (0, 0, 1),
(0, 1/

2, 1/

2), (1/

2, 0, 1/

2), (1/

2, 1/

2, 0),
(1/

3, 1/

3, 1/

3)
Comments
Surface integrals are also often written in the form
S
P dy dz + Q dx dz + R dx dy.
To relate this form to the one described in the Narrative, observe that if F = P, Q, R is a vector field on
R3
and n = nx, ny, nz is the unit normal to S then
S
F  dS =
S
F  n dS =
S
P, Q, R  nx, ny, nz dS =
S
(Pnx + Qny + Rnz) dS.
Now let us consider S
Rnz dS. If, over the projection Dxy of S into the xy-coordinate plane, S is the
graph of z = f(x, y) then
n =
-fx, -fy, 1
1 + f2
x + f2
y
,
so nz = 1/ 1 + f2
x + f2
y , and dS = 1 + f2
x + f2
y dx dy. Thus
S
Rnz dS =
Dxy
R
1
1 + f2
x + f2
y
1 + f2
x + f2
y dx dy =
Dxy
R dx dy.
Similarly
S
Pnx dS =
Dyz
P dy dz and
S
Qny dS =
Dxz
Q dx dz.
Thus we may write
S
F  dS =
S
P dy dz + Q dx dz + R dx dy.