16 7 Surface Integrals Copyright Cengage Learning All rights reserved 1 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 2 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 3 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 cos 0 2 y sin sin z cos 0 That is r sin cos i sin sin j cos k We can compute that r r sin 4 Example 1 Solution cont d Therefore by Formula 2 5 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 x y z then the total mass of the sheet is and the center of mass is where 6 Graphs 7 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 and 8 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 then 9 Example 2 Evaluate S y dS where S is the surface z x y2 0 x 1 0 y 2 See Figure 2 Figure 2 10 Example 2 Solution Since and Formula 4 gives 11 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 by 12 Oriented Surfaces 13 Oriented Surfaces To define surface integrals of vector fields we need to rule out nonorientable surfaces such as the M bius strip shown in Figure 4 It is named after the German geometer August M bius 1790 1868 A M bius strip Figure 4 14 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 Constructing a M bius strip Figure 5 15 Oriented Surfaces If an ant were to crawl along the M bius 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 M bius strip try drawing a pencil line down the middle Therefore a M bius strip really has only one side From now on we consider only orientable two sided surfaces 16 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 6 17 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 The two orientations of an orientable surface Figure 7 18 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 19 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 a sin cos i a sin sin j a cos k for the sphere x2 y2 z2 a2 20 Oriented Surfaces We know that r r a2 sin2 cos i a2 sin2 sin j a2 sin cos k and r r a2 sin So the orientation induced by r 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 8 21 Oriented Surfaces The opposite inward orientation would have been obtained see Figure 9 if we had reversed the order of the parameters because r r r r Negative orientation Figure 9 Positive orientation Figure 8 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 22 Surface Integrals of Vector Fields 23 Surface Integrals of Vector Fields Suppose that S is an oriented surface with unit normal vector n and imagine a fluid with density 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 v 24 Surface Integrals of Vector Fields If we divide S into small patches Sij as in Figure 10 compare with Figure 1 Figure 10 Figure 1 25 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 v n A Sij where v and n are evaluated at some point on Sij Recall that the component of the vector v in the direction of the unit vector n is v n 26 Surface Integrals of Vector Fields By summing these quantities and taking the limit we get according to Definition 1 the surface integral of the function v n over S and this is interpreted physically as the rate of flow through S If we write F v then F is also a vector field on the integral in Equation 7 becomes and 27 Surface Integrals of Vector …
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