1 1 Copyright © Cengage Learning. All rights reserved. 16 Vector CalculusCopyright © Cengage Learning. All rights reserved. 16.7 Surface Integrals3 3 Surface Integrals Learning objectives: § compute the surface integral of a function over a parametric surface (by converting it into a double integral over the parameter domain D) (two formulas) § find a unit normal vector to an orientable surface § determine the upward pointing unit normal vector to the graph of a surface z = f (x,y) § find the unit normal vector determined by a parametrization of a surface § determine the positively oriented (outward pointing) unit normal vector to a closed surface § compute the surface integral of a vector field over a parametric surface (two formulas)4 4 Surface Integrals The relationship between surface integrals and surface area is much the same as the relationship between line integrals and arc length. Suppose f is a function of three variables whose domain includes a surface S. We will define the surface integral of f over S in such a way that, in the case where f (x, y, z) = 1, the value of the surface integral is equal to the surface area of S. We start with parametric surfaces and then deal with the special case where S is the graph of a function of two variables.5 5 Parametric Surfaces6 6 Parametric Surfaces Suppose that 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 We first assume that the parameter domain D is a rectangle and we divide it into subrectangles Rij with dimensions Δu and Δv. Then the surface S is divided into corresponding patches Sij as in Figure 1. Figure 17 7 Parametric Surfaces We evaluate f at a point in each patch, multiply by the area ΔSij of the patch, and form the Riemann sum Then we take the limit as the number of patches increases and define the surface integral of f over the surface S as Notice the analogy with the definition of a line integral and also the analogy with the definition of a double integral.8 8 Parametric Surfaces To evaluate the surface integral in Equation 1 we approximate the patch area ΔSij by the area of an approximating parallelogram in the tangent plane. In our discussion of surface area we made the approximation ΔSij ≈ | ru × rv | Δu Δv where are the tangent vectors at a corner of Sij.9 9 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:10 10 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).11 11 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 , where12 12 Graphs13 13 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 and14 14 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, then15 15 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 by16 16 Oriented Surfaces17 17 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 strip18 18 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 strip19 19 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.20 20 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 621 21 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 surface22 22 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.23 23 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. Positive orientation Figure 824 24 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
View Full Document
Unlocking...