Case III. Complex conjugate roots are of minor practical importance, and we discuss the derivation of real solutions from complex ones just in terms of a typical example
10.5 Surfaces for Surface Integrals
Whereas, with line integrals, we integrate over curves in space (Secs. 10.1, 10.2), with surface integrals we integrate over surfaces in space. Each curve in space is represented by a parametric equation (Secs. 9.5, 10.1). This suggests that we should also find parametric representations for the surfaces in space. This is indeed one of the goals of this section. The surfaces considered are cylinders, spheres, cones, and others. The second goal is to learn about surface normals. Both goals prepare us for Sec. 10.6 on surface integrals. Note that for simplicity, we shall say “surface” also for a portion of a surface.
Representation of Surfaces
Representations of a surface Sin xyz-space are (1)
For example, or represents a
hemisphere of radius aand center 0.
Now for curves Cin line integrals, it was more practical and gave greater flexibility to use a parametric representation where This is a mapping of the interval located on the t-axis, onto the curve C (actually a portion of it) in xyz-space. It maps every t in that interval onto the point of C with position vector See Fig. 241A.
Similarly, for surfaces Sin surface integrals, it will often be more practical to use a parametricrepresentation. Surfaces are two-dimensional. Hence we need twoparameters, r(t).
a⬉t⬉b,
a⬉t⬉b.
r⫽r(t),
x2⫹y2⫹z2⫺a2⫽0 (z⭌0) z⫽ ⫹2a2⫺x2⫺y2
z⫽f(x, y) or g(x, y, z)⫽0.
z x y
r(t) r(u,v)
Curve C in space
z x y
a b
t
(t-axis)
v
u
R
(uv-plane) Surface S in space
(A) Curve (B) Surface
Fig. 241. Parametric representations of a curve and a surface
which we call uand v. Thus a parametric representationof a surface Sin space is of the form
(2)
where (u, v) varies in some region Rof the uv-plane. This mapping (2) maps every point (u, v) in Ronto the point of Swith position vector r(u, v). See Fig. 241B.
E X A M P L E 1 Parametric Representation of a Cylinder
The circular cylinder has radius a, height 2, and the z-axis as axis. A parametric representation is
(Fig. 242).
The components of rare The parameters u, vvary in the rectangle in the uv-plane. The curves are vertical straight lines. The curves are parallel circles. The point Pin Fig. 242 corresponds to u⫽p>3⫽60°, v⫽0.7. 䊏
v⫽const u⫽const
2p, ⫺1⬉v⬉1
R: 0⬉u⬉ x⫽a cos u, y⫽a sin u, z⫽v.
r(u, v)⫽[a cos u, a sin u, v]⫽a cos ui⫹a sin uj⫹vk x2⫹y2⫽a2, ⫺1⬉z⬉1,
r(u, v)⫽[x(u, v), y(u, v), z(u, v)]⫽x(u, v)i⫹y(u, v)j⫹z(u, v)k
(v = 1)
(v = 0)
(v = –1) v
u P
x y z
u v
P z
x y
Fig. 242. Parametric representation Fig. 243. Parametric representation
of a cylinder of a sphere
E X A M P L E 2 Parametric Representation of a Sphere
A sphere can be represented in the form (3)
where the parameters u, v vary in the rectangle Rin the uv-plane given by the inequalities The components of rare
The curves and are the “meridians” and “parallels” on S(see Fig. 243). This representation is used ingeographyfor measuring the latitude and longitude of points on the globe.
Another parametric representation of the sphere also used in mathematics is (3*)
where 0⬉u⬉2p, 0⬉v⬉p. 䊏
r(u, v)⫽a cos u sin vi⫹a sin u sin vj⫹a cos vk v⫽const
u⫽const
x⫽a cos v cos u, y⫽a cos v sin u, z⫽a sin v.
⫺p>2⬉v⬉p>2.
0⬉u⬉2p, r(u, v)⫽a cos v cos ui⫹a cos v sin uj⫹a sin vk
x2⫹y2⫹z2⫽a2
E X A M P L E 3 Parametric Representation of a Cone
A circular cone can be represented by
in components The parameters vary in the rectangle Check that as it should be. What are the curves and ?
Tangent Plane and Surface Normal
Recall from Sec. 9.7 that the tangent vectors of all the curves on a surface Sthrough a point Pof Sform a plane, called the tangent planeof Sat P(Fig. 244). Exceptions are points where Shas an edge or a cusp (like a cone), so that Scannot have a tangent plane at such a point.
Furthermore, a vector perpendicular to the tangent plane is called a normal vectorof Sat P.
Now since Scan be given by in (2), the new idea is that we get a curve C on Sby taking a pair of differentiable functions
whose derivatives and are continuous. Then C has the position vector . By differentiation and the use of the chain rule (Sec. 9.6) we obtain a tangent vector of Con S
Hence the partial derivatives and at P are tangential to S at P.We assume that they are linearly independent, which geometrically means that the curves and on Sintersect at Pat a nonzero angle. Then and span the tangent plane of Sat P. Hence their cross product gives a normal vector Nof Sat P.
(4)
The corresponding unit normal vector nof Sat Pis (Fig. 244)
(5) n⫽ 1
ƒNƒ
N⫽ 1 ƒruⴛrvƒ
ruⴛrv. N⫽ruⴛrv⫽0.
rv
ru v⫽const
u⫽const rv
ru
~rr(t)⫽ dr
~ dt ⫽ 0r
0u ur⫹ 0r
0v vr.
~(t)r ⫽r(u(t), v(t))
vr⫽dv>dt
ur ⫽du>dt
u⫽u(t), v⫽v(t) r⫽r(u, v)
䊏
v⫽const u⫽const
x2⫹y2⫽z2,
R: 0⬉u⬉H, 0⬉v⬉2p. x⫽u cos v, y⫽u sin v, z⫽u.
r(u, v)⫽[u cos v, u sin v, u]⫽u cos vi⫹u sin vj⫹uk, z⫽2x2⫹y2, 0⬉t⬉H
n
rv ru
P
S
Fig. 244. Tangent plane and normal vector
Also, if Sis represented by then, by Theorem 2 in Sec. 9.7, (5*)
A surface Sis called a smooth surfaceif its surface normal depends continuously on the points of S.
Sis called piecewise smoothif it consists of finitely many smooth portions.
For instance, a sphere is smooth, and the surface of a cube is piecewise smooth (explain!). We can now summarize our discussion as follows.
T H E O R E M 1 Tangent Plane and Surface Normal
If a surface S is given by(2) with continuous and satisfying (4) at every point of S, then S has, at every point P, a unique tangent plane passing through P and spanned by and and a unique normal whose direction depends continuously on the points of S. A normal vector is given by(4) and the corresponding unit normal vector by(5). (See Fig. 244.)
E X A M P L E 4 Unit Normal Vector of a Sphere
From we find that the sphere has the unit normal vector
We see that nhas the direction of the position vector [x, y, z] of the corresponding point. Is it obvious that this must be the case?
E X A M P L E 5 Unit Normal Vector of a Cone
At the apex of the cone in Example 3, the unit normal vector nbecomes undetermined because from we get
We are now ready to discuss surface integrals and their applications, beginning in the next section.
䊏
n⫽c x
22(x2⫹y2) ,
y 22(x2⫹y2)
, ⫺1 12d ⫽ 1
12a x 2x2⫹y2
i⫹ y
2x2⫹y2 j⫺kb. (5*)
g(x, y, z)⫽ ⫺z⫹2x2⫹y2⫽0
䊏
n(x, y, z)⫽cx a, y
a, z ad ⫽x
a i⫹y a j⫹z
a k.
g(x, y, z)⫽x2⫹y2⫹z2⫺a2⫽0 (5*)
rv, ru
rv⫽ 0r>0v ru⫽ 0r>0u
n⫽ 1 ƒgrad gƒ
grad g.
g(x, y, z)⫽0,
1–8 PARAMETRIC SURFACE REPRESENTATION Familiarize yourself with parametric representations of important surfaces by deriving a representation (1), by finding the parameter curves (curves and
) of the surface and a normal vector of the surface. Show the details of your work.
1. xy-plane (thus similarly in
Probs. 2–8).
2. xy-plane in polar coordinates (thus u⫽r, v⫽u)
u sin v]
r(u, v)⫽[u cos v, ui⫹vj;
r(u, v)⫽(u, v)
N⫽ruⴛrv
v⫽const
u⫽const
3. Cone
4. Elliptic cylinder 5. Paraboloid of revolution
6. Helicoid Explain the
name.
7. Ellipsoid
8. Hyperbolic paraboloid u2] v, bu sinh
r(u, v)⫽[au cosh v, c sin v]
b cos v sin u, r(u, v)⫽[a cos v cos u,
r(u, v)⫽[u cos v, u sin v, v].
u2]
r(u, v)⫽[u cos v, u sin v, r(u, v)⫽[a cos v, b sin v, u]
r(u, v)⫽[u cos v, u sin v, cu]
P R O B L E M S E T 1 0 . 5
9. CAS EXPERIMENT. Graphing Surfaces, Depen- dence on a,b,c.Graph the surfaces in Probs. 3–8. In Prob. 6 generalize the surface by introducing parame- ters a, b. Then find out in Probs. 4 and 6–8 how the shape of the surfaces depends on a, b, c.
10. Orthogonal parameter curves and on occur if and only if
Give examples. Prove it.
11. Satisfying (4). Represent the paraboloid in Prob. 5 so
that and show
12. Condition (4). Find the points in Probs. 1–8 at which (4) does not hold. Indicate whether this results from the shape of the surface or from the choice of the representation.
13. Representation Show that or
can be written etc.)
(6) and
N⫽grad g⫽[⫺fu, ⫺fv, 1].
r(u, v)⫽[u, v, f(u, v)]
( fu⫽0f>0u, g⫽z⫺f(x, y)⫽0
z⫽f(x, y) z⫽f(x, y).
N⫽0
N~ . N~
(0, 0)⫽0
ru•rv⫽0.
r(u, v) v⫽const
u⫽const
14–19 DERIVE A PARAMETRIC REPRESENTATION
Find a normal vector. The answer gives onerepresentation;
there are many. Sketch the surface and parameter curves.
14. Plane
15. Cylinder of revolution 16. Ellipsoid
17. Sphere 18. Elliptic cone 19. Hyperbolic cylinder
20. PROJECT. Tangent Planes T(P) will be less important in our work, but you should know how to represent them.
(a) If then
(a scalar triple product) or
(b) If then
(c) If then
Interpret (a)⫺(c) geometrically. Give two examples for (a), two for (b), and two for (c).
T(P): z*⫺z⫽(x*⫺x)fx(P)⫹(y*⫺y)fy(P).
S: z⫽f(x, y),
T(P): (r*⫺r(P))ⴢⵜg⫽0.
S: g(x, y, z)⫽0,
r*(p, q)⫽r(P)⫹pru(P)⫹qrv(P).
T(P): (r*⫺r ru rv)⫽0 S: r(u, v),
x2⫺y2⫽1 z⫽ 2x2⫹4y2
x2⫹(y⫹2.8)2⫹(z⫺3.2)2⫽2.25 x2⫹y2⫹19 z2⫽1
(x⫺2)2⫹(y⫹1)2⫽25 4x⫹3y⫹2z⫽12