Tangent Plane and Surface Normal

Recall from Sec. 9.7 that the tangent vectors of all the curves on a surface S through a point P of S form a plane, called the tangent plane of S at P (Fig. 244). Exceptions are points where S has an edge or a cusp (like a cone), so that S cannot have a tangent plane at such a point.

Furthermore, a vector perpendicular to the tangent plane is called a normal vector of S at P. Now since S can be given by

(u, v) in (2), the new idea is that we get a curve C on S by taking a pair of differentiable functions

(t),

whose derivatives u r du >dt and v r dv >dt are continuous. Then C has the position

v (t)) . By differentiation and the use of the chain rule (Sec. 9.6) we obtain a tangent vector of C on S

vector r

Hence the partial derivatives r u and r v at P are tangential to S at P. We assume that they are linearly independent, which geometrically means that the curves

const and const on S intersect at P at a nonzero angle. Then r u and r v span the tangent plane of S at P. Hence their cross product gives a normal vector N of S at P.

u ⴛr v 0 .

The corresponding unit normal vector n of S at P is (Fig. 244)

Fig. 244. Tangent plane and normal vector

CHAP. 10 Vector Integral Calculus. Integral Theorems

Also, if S is represented by g (x, y, z) ⫽ 0, then, by Theorem 2 in Sec. 9.7,

n⫽

grad g.

ƒ grad g ƒ

A surface S is called a smooth surface if its surface normal depends continuously on the points of S. S is called piecewise smooth if 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.

THEOREM 1 Tangent Plane and Surface Normal

If a surface S is given by (2) with continuous r u ⫽ 0r>0u and r v ⫽ 0r>0v satisfying (4) at every point of S, then S has, at every point P, a unique tangent plane passing through P and spanned by r u and r v , 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.)

EXAMPLE 4 Unit Normal Vector of a Sphere

From (5*) we find that the sphere g (x, y, z) ⫽ x 2 ⫹y 2 ⫹z 2 ⫺a 2 ⫽ 0 has the unit normal vector

n (x, y, z) ⫽ x c , , ⫽ x ad j⫹ a a a i⫹ a a k .

We see that n has the direction of the position vector [x, y, z] of the corresponding point. Is it obvious that this must be the case?

EXAMPLE 5 Unit Normal Vector of a Cone

At the apex of the cone g (x, y, z) ⫽ ⫺z ⫹ 2x 2 ⫹y 2 ⫽ 0 in Example 3, the unit normal vector n becomes

undetermined because from (5*) we get

y ) 22(x

We are now ready to discuss surface integrals and their applications, beginning in the next section.

PROBLEM SET10.5

1–8 PARAMETRIC SURFACE REPRESENTATION

3. Cone r (u, v) ⫽ [u cos v, u sin v, cu]

Familiarize yourself with parametric representations of

4. Elliptic cylinder r (u, v) ⫽ [a cos v,

b sin v, u]

important surfaces by deriving a representation (1), by

5. Paraboloid of revolution r (u, v) ⫽ [u cos v,

u sin v,

finding the parameter curves (curves and const

u⫽

v⫽ const ) of the surface and a normal vector N⫽r u ⴛr v 6. Helicoid Explain r (u, v) ⫽ [u cos v, u sin v, v]. the of the surface. Show the details of your work.

name.

1. xy -plane (thus r (u, v) ⫽ (u, v) u i ⫹ vj ; similarly in 7. Ellipsoid r (u, v) ⫽ [a cos v cos u, b cos v sin u,

Probs. 2–8).

c sin v]

2. xy -plane in polar coordinates r (u, v) ⫽ [u cos v, u sin v]

8. Hyperbolic paraboloid r (u, v) ⫽ [au cosh v,

(thus u ⫽ r, v ⫽ u)

bu sinh v , u 2 ]

SEC. 10.6 Surface Integrals

443 9. CAS EXPERIMENT. Graphing Surfaces, Depen-

DERIVE A PARAMETRIC

dence on a , b, c. Graph the surfaces in Probs. 3–8. In

REPRESENTATION

Prob. 6 generalize the surface by introducing parame- Find a normal vector. The answer gives one representation; ters a, b. Then find out in Probs. 4 and 6–8 how the

there are many. Sketch the surface and parameter curves. shape of the surfaces depends on a, b, c.

14. Plane 4x ⫹ 3y ⫹ 2z ⫽ 12

10. Orthogonal parameter curves u⫽ const and 15. Cylinder of revolution (x ⫺ 2) 2 ⫹ ( y⫹ 1) 2 ⫽ 25

v ⫽ 0. 16. Ellipsoid x ⫹y ⫹ 9 z ⫽ 1 Give examples. Prove it.

v⫽ const on r (u, v) occur if and only if r

2 2 1 u 2 •r

17. Sphere x 2 ⫹ ( y⫹ 2.8) 2 ⫹ (z ⫺ 3.2) 2 ⫽ 2.25

18. Elliptic cone z ⫽ 2x 2 ⫹ 4y 11. Satisfying (4). 2

~ Represent the paraboloid in Prob. 5 so ~ that and N (0, 0) ⫽ 0 show N .

19. Hyperbolic cylinder x 2 ⫺y 2 ⫽ 1 20. PROJECT. Tangent Planes T(P) will be less

12. Condition (4). Find the points in Probs. 1–8 at which important in our work, but you should know how to (4) N⫽0 does not hold. Indicate whether this results

represent them.

from the shape of the surface or from the choice of the (a) If then S : r(u, v), T (P): (r* ⫺ r

r v )⫽0 representation.

(a scalar triple product) or

r *( p , q) ⫽ r( P ) ⫹ pr u ( P ) ⫹ qr ( P ).

(b) If then S : g(x, y, z) ⫽ 0, g⫽z⫺f (x, y) ⫽ 0 can be written (f u ⫽ 0f>0u, etc.)

13. Representation z⫽f (x, y). Show that z⫽f (x, y) or

T ( P ): (r* ⫺ r( P )) ⴢ ⵜg ⫽ 0. (c) If then S :z⫽f (x, y),

Interpret (a)⫺(c) geometrically. Give two examples for

(a), two for (b), and two for (c).

Tangent Plane and Surface Normal