VECTOR CALCULUS

1. 16 VECTOR CALCULUS

2. VECTOR CALCULUS 16.7 Surface Integrals • In this section, we will learn about: • Integration of different types of surfaces.

3. SURFACE INTEGRALS • The relationship between surface integrals and surface area is much the same as the relationship between line integrals and arc length.

4. SURFACE INTEGRALS • Suppose f is a function of three variables whose domain includes a surface S. • We will define the surface integral of f over S such that, in the case where f(x, y, z) = 1, the value of the surface integral is equal to the surface area of S.

5. SURFACE INTEGRALS • We start with parametric surfaces. • Then, we deal with the special case where S is the graph of a function of two variables.

6. PARAMETRIC SURFACES • Suppose a surface S has a vector equation • r(u, v) = x(u, v) i + y(u, v) j + z(u, v) k • (u, v) D

7. PARAMETRIC SURFACES • We first assume that the parameter domain D is a rectangle and we divide it into subrectangles Rijwith dimensions ∆u and ∆v.

8. PARAMETRIC SURFACES • Then, the surface Sis divided into corresponding patches Sij.

9. PARAMETRIC SURFACES • We evaluate f at a point Pij* in each patch, multiply by the area ∆Sijof the patch, and form the Riemann sum

10. SURFACE INTEGRAL Equation 1 • Then, we take the limit as the number of patches increases and define the surface integral of f over the surface S as:

11. SURFACE INTEGRALS • Notice the analogy with: • The definition of a line integral (Definition 2 in Section 16.2) • The definition of a double integral (Definition 5 in Section 15.1)

12. SURFACE INTEGRALS • To evaluate the surface integral in Equation 1, we approximate the patch area ∆Sij by the area of an approximating parallelogram in the tangent plane.

13. SURFACE INTEGRALS • In our discussion of surface area in Section 16.6, we made the approximation ∆Sij≈ |rux rv|∆u ∆v • where:are the tangent vectors at a corner of Sij.

14. SURFACE INTEGRALS Formula 2 • If the components are continuous and ru and rv are nonzero and nonparallel in the interior of D, it can be shown from Definition 1—even when D is not a rectangle—that:

15. SURFACE INTEGRALS • This should be compared with the formula for a line integral: • Observe also that:

16. SURFACE INTEGRALS • Formula 2 allows us to compute a surface integral by converting it into a double integral over the parameter domain D. • When using this formula, remember that f(r(u, v) is evaluated by writing x =x(u, v), y =y(u, v), z =z(u, v) in the formula for f(x, y, z)

17. SURFACE INTEGRALS Example 1 • Compute the surface integral , where S is the unit sphere x2 + y2 + z2 = 1.

18. SURFACE INTEGRALS Example 1 • As in Example 4 in Section 16.6, we use the parametric representation x = sin Φ cos θ,y = sin Φ sin θ,z = cos Φ0 ≤ Φ ≤ π,0≤ θ ≤ 2π • That is, r(Φ,θ) = sin Φcos θi + sin Φ sin θj + cos Φk

19. SURFACE INTEGRALS Example 1 • As in Example 10 in Section 16.6, we can compute that: • |rΦxrθ| = sin Φ

20. SURFACE INTEGRALS Example 1 • Therefore, by Formula 2,

21. SURFACE INTEGRALS Example 1

22. APPLICATIONS • Surface integrals have applications similar to those for the integrals we have previously considered.

23. APPLICATIONS • For example, suppose a thin sheet (say, of aluminum foil) has: • The shape of a surface S. • The density (mass per unit area) at the point (x, y, z) as ρ(x, y, z).

24. MASS • Then, the total mass of the sheet is:

25. CENTER OF MASS • The center of mass is: where

26. MOMENTS OF INERTIA • Moments of inertia can also be defined as before. • See Exercise 39.

27. GRAPHS • Any surface S with equation z =g(x, y) can be regarded as a parametric surface with parametric equations • x =x y =y z =g(x, y) • So, we have:

28. GRAPHS Equation 3 • Thus, • and

29. GRAPHS Formula 4 • Therefore, in this case, Formula 2 becomes:

30. GRAPHS • Similar formulas apply when it is more convenient to project S onto the yz-plane or xy-plane.

31. GRAPHS • For instance, if S is a surface with equation y =h(x, z) and D is its projection on the xz-plane, then

32. GRAPHS Example 2 • Evaluate where S is the surface z =x +y2, 0 ≤ x ≤ 1, 0 ≤ y ≤ 2

33. GRAPHS Example 2 • So, Formula 4 gives:

34. GRAPHS • If S is a piecewise-smooth surface—a finite union of smooth surfaces S1, S2, . . . , Sn that intersect only along their boundaries—then the surface integral of f over S is defined by:

35. GRAPHS Example 3 • Evaluate , where S is the surface whose: • Sides S1 are given by the cylinder x2 + y2 = 1. • Bottom S2 is the disk x2 + y2≤ 1 in the plane z = 0. • Top S3 is the part of the plane z = 1 + x that lies above S2.

36. GRAPHS Example 3 • The surface S is shown. • We have changed the usual position of the axes to get a better look at S.

37. GRAPHS Example 3 • For S1, we use θand z as parameters (Example 5 in Section 16.6) and write its parametric equations as: • x = cos θy = sin θz =zwhere: • 0 ≤θ≤ 2π • 0 ≤z ≤ 1 + x = 1 + cos θ

38. GRAPHS Example 3 • Therefore, • and

39. GRAPHS Example 3 • Thus, the surface integral over S1 is:

40. GRAPHS Example 3 • Since S2 lies in the plane z = 0, we have:

41. GRAPHS Example 3 • S3 lies above the unit disk D and is part of the plane z = 1 + x. • So, taking g(x, y) = 1 + x in Formula 4 and converting to polar coordinates, we have the following result.

42. GRAPHS Example 3

43. GRAPHS Example 3 • Therefore,

44. ORIENTED SURFACES • To define surface integrals of vector fields, we need to rule out nonorientable surfaces such as the Möbius strip shown. • It is named after the German geometer August Möbius (1790–1868).

45. MOBIUS STRIP • You can construct one for yourself by: • Taking a long rectangular strip of paper. • Giving it a half-twist. • Taping the short edges together.

46. MOBIUS STRIP • If an ant were to crawl along the Möbius strip starting at a point P, it would end up on the “other side” of the strip—that is, with its upper side pointing in the opposite direction.

47. MOBIUS STRIP • Then, if it continued to crawl in the same direction, it would end up back at the same point P without ever having crossed an edge. • If you have constructed a Möbius strip, try drawing a pencil line down the middle.

48. MOBIUS STRIP • Therefore, a Möbius strip really has only one side. • You can graph the Möbius strip using the parametric equations in Exercise 32 in Section 16.6.

49. ORIENTED SURFACES • From now on, we consider only orientable (two-sided) surfaces.

50. ORIENTED SURFACES • We start with a surface S that has a tangent plane at every point (x, y, z) on S (except at any boundary point). • There are two unit normal vectors n1 and n2 = –n1at (x, y, z).