16 Vector Calculus Copyright Cengage Learning All rights reserved 16 9 The Divergence Theorem Copyright Cengage Learning All rights reserved The Divergence Theorem Learning objectives convert a surface integral of a vector field over a closed surface to a triple integral of its divergence over the region bounded by the surface and vice versa and then evaluate replace a surface integral over a complicated surface by a surface integral over a simpler surface and then evaluate 3 The Divergence Theorem Recall that a version of Green s Theorem in vector form is where C is the positively oriented boundary curve of the plane region D If we were seeking to extend this theorem to vector fields on we might make the guess that where S is the boundary surface of the solid region E 4 The Divergence Theorem It turns out that Equation 1 is true under appropriate hypotheses and is called the Divergence Theorem also known as Gauss Theorem Notice its similarity to Green s Theorem and Stokes Theorem in that it relates the integral of a derivative of a function div F in this case over a region to the integral of the original function F over the boundary of the region 5 The Divergence Theorem The boundary of E is a closed surface and we use the convention that the positive orientation is outward that is the unit normal vector n is directed outward from E Thus the Divergence Theorem states that under the given conditions the flux of F across the boundary surface of E is equal to the triple integral of the divergence of F over E 6 The Divergence Theorem Let s consider the region E that lies between the closed surfaces S1 and S2 where S1 lies inside S2 Let n1 and n2 be outward normal vectors of S1 and S2 Then the boundary surface of E is S S1 U S2 and its normal n is given by n n1 on S1 and n n2 on S2 See Figure 3 Figure 3 7 The Divergence Theorem Applying the Divergence Theorem to S we get 8 The Divergence Theorem Another application of the Divergence Theorem occurs in fluid flow Let v x y z be the velocity field of a fluid with constant density Then F v is the rate of flow per unit area 9 The Divergence Theorem If P0 x0 y0 z0 is a point in the fluid and Ba is a ball with center P0 and very small radius a then div F P div F P0 for all points in Ba since div F is continuous We approximate the flux over the boundary sphere Sa as follows This approximation becomes better as a 0 and suggests that 10 The Divergence Theorem Equation 8 says that div F P0 is the net rate of outward flux per unit volume at P0 This is the reason for the name divergence If div F P 0 the net flow is outward near P and P is called a source If div F P 0 the net flow is inward near P and P is called a sink 11 The Divergence Theorem For the vector field in Figure 4 it appears that the vectors that end near P1 are shorter than the vectors that start near P1 Figure 4 The vector field F x2 i y2 j 12 The Divergence Theorem Thus the net flow is outward near P1 so div F P1 0 and P1 is a source Near P2 on the other hand the incoming arrows are longer than the outgoing arrows Here the net flow is inward so div F P2 0 and P2 is a sink We can use the formula for F to confirm this impression Since F x2 i y2 j we have div F 2x 2y which is positive when y x So the points above the line y x are sources and those below are sinks 13
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