1 Copyright © Cengage Learning. All rights reserved. 16.7 Surface Integrals2 Parametric Surfaces 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 This should be compared with the formula for a line integral:3 Parametric Surfaces Observe also that 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), and z = z(u, v) in the formula for f (x, y, z).4 Example 1 Compute the surface integral ∫∫S x2 dS, where S is the unit sphere x2 + y2 + z2 = 1. Solution: We use the parametric representation x = sin φ cos θ y = sin φ sin θ z = cos φ 0 ≤ φ ≤ π 0 ≤ θ ≤ 2π That is, r(φ, θ ) = sin φ cos θ i + sin φ sin θ j + cos φ k We can compute that | rφ × rθ | = sin φ5 Example 1 – Solution Therefore, by Formula 2, cont’d6 Parametric Surfaces If a thin sheet (say, of aluminum foil) has the shape of a surface S and the density (mass per unit area) at the point (x, y, z) is ρ (x, y, z), then the total mass of the sheet is and the center of mass is , where7 Graphs8 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) and so we have Thus and9 Graphs Therefore, in this case, Formula 2 becomes Similar formulas apply when it is more convenient to project S onto the yz-plane or xz-plane. For instance, if S is a surface with equation y = h(x, z) and D is its projection onto the xz-plane, then10 Example 2 Evaluate ∫∫S y dS, where S is the surface z = x + y2, 0 ≤ x ≤ 1, 0 ≤ y ≤ 2. (See Figure 2.) Figure 211 Example 2 – Solution Since and Formula 4 gives12 Graphs If S is a piecewise-smooth surface, that is, 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 by13 Oriented Surfaces14 Oriented Surfaces To define surface integrals of vector fields, we need to rule out nonorientable surfaces such as the Möbius strip shown in Figure 4. [It is named after the German geometer August Möbius (1790–1868).] Figure 4 A Möbius strip15 Oriented Surfaces You can construct one for yourself by taking a long rectangular strip of paper, giving it a half-twist, and taping the short edges together as in Figure 5. Figure 5 Constructing a Möbius strip16 Oriented Surfaces 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). Then, if the ant 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.) Therefore a Möbius strip really has only one side. From now on we consider only orientable (two-sided) surfaces.17 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 = –n1 at (x, y, z). (See Figure 6.) Figure 618 Oriented Surfaces If it is possible to choose a unit normal vector n at every such point (x, y, z) so that n varies continuously over S, then S is called an oriented surface and the given choice of n provides S with an orientation. There are two possible orientations for any orientable surface (see Figure 7). Figure 7 The two orientations of an orientable surface19 Oriented Surfaces For a surface z = g(x, y) given as the graph of g, we use Equation 3 to associate with the surface a natural orientation given by the unit normal vector Since the k-component is positive, this gives the upward orientation of the surface.20 Oriented Surfaces If S is a smooth orientable surface given in parametric form by a vector function r(u, v), then it is automatically supplied with the orientation of the unit normal vector and the opposite orientation is given by –n. For instance, the parametric representation is r(φ, θ ) = a sin φ cos θ i + a sin φ sin θ j + a cos φ k for the sphere x2 + y2 + z2 = a2.21 Oriented Surfaces We know that rφ × rθ = a2 sin2 φ cos θ i + a2 sin2 φ sin θ j + a2 sin φ cos φ k and | rφ × rθ | = a2 sin φ So the orientation induced by r(φ, θ ) is defined by the unit normal vector Observe that n points in the same direction as the position vector, that is, outward from the sphere (see Figure 8). Positive orientation Figure 822 The opposite (inward) orientation would have been obtained (see Figure 9) if we had reversed the order of the parameters because rθ × rφ = –rφ × rθ . For a closed surface, that is, a surface that is the boundary of a solid region E, the convention is that the positive orientation is the one for which the normal vectors point outward from E, and inward-pointing normals give the negative orientation (see Figures 8 and 9). Oriented Surfaces Negative orientation Figure 9 Positive orientation Figure 823 Surface Integrals of Vector Fields24 Surface Integrals of Vector Fields Suppose that S is an oriented surface with unit normal vector n, and imagine a fluid with density ρ (x, y, z) and velocity field v(x, y, z) flowing through S. (Think of S as an imaginary surface that doesn’t impede the fluid flow, like a fishing net across a stream.) Then the rate of flow (mass per unit time) per unit area is ρ v.25 Surface Integrals of Vector Fields If we divide S into small patches Sij, as in Figure 10 (compare with Figure 1). Figure 1 Figure 1026 Surface Integrals of Vector Fields Then Sij is nearly planar and so we can approximate the mass of fluid per unit time crossing Sij in the direction of the normal n by the quantity (ρ v ! n)A(Sij) where ρ, v, and n are evaluated at some point on Sij. (Recall that the component of the vector ρ v in the direction of the unit vector n is ρ v ! n.)27 Surface Integrals of Vector Fields By summing these quantities and taking the limit we get, according to Definition 1, the surface integral …
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