Divergence 1 Divergence If F P i Q j R k is a vector field on and P x Q y and R z exist then the divergence of F is the function of three variables defined by Observe that curl F is a vector field but div F is a scalar field 2 Divergence In terms of the gradient operator x i y j z k the divergence of F can be written symbolically as the dot product of and F 3 Divergence If F is a vector field on then curl F is also a vector field on As such we can compute its divergence The next theorem shows that the result is 0 4 Vector Forms of Green s Theorem 5 Vector Forms of Green s Theorem If C is given by the vector equation r t x t i y t j a t b then the unit tangent vector is 6 Vector Forms of Green s Theorem The outward unit normal vector to C is given by See Figure 2 Figure 2 7 Vector Forms of Green s Theorem Then from the equation we have 8 Vector Forms of Green s Theorem by Green s Theorem But the integrand in this double integral is just the divergence of F 9 Vector Forms of Green s Theorem So we have a second vector form of Green s Theorem This version says that the line integral of the normal component of F along C is equal to the double integral of the divergence of F over the region D enclosed by C 10 16 9 The Divergence Theorem 11 The Divergence Theorem We write Green s Theorem in a vector version as 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 12 The Divergence Theorem It turns out that Equation 1 is true under appropriate hypotheses and is called the Divergence 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 We state the Divergence Theorem for regions E that are simultaneously of types 1 2 and 3 and we call such regions simple solid regions For instance regions bounded by ellipsoids or rectangular boxes are simple solid regions 13 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 14 Example 1 Find the flux of the vector field F x y z z i y j x k over the unit sphere x2 y2 z2 1 Solution First we compute the divergence of F The unit sphere S is the boundary of the unit ball B given by x2 y2 z2 1 15 Example 1 Solution cont d Thus the Divergence Theorem gives the flux as 16 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 normals 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 17 The Divergence Theorem Applying the Divergence Theorem to S we get The image cannot be displayed Your computer may not have enough memory to open the image or the image may have been corrupted Restart your computer and then open the file again If the red x still appears you may have to delete the image and then insert it again The image cannot be displayed Your computer may not have enough memory to open the image or the image may have been corrupted Restart your computer and then open the file again If the red x still appears you may have to delete the image and then insert it again 18 Example 3 We considered the electric field where the electric charge Q is located at the origin and is a position vector Use the Divergence Theorem to show that the electric flux of E through any closed surface S2 that encloses the origin is 19 Example 3 Solution The difficulty is that we don t have an explicit equation for S2 because it is any closed surface enclosing the origin The simplest such surface would be a sphere so we let S1 be a small sphere with radius a and center the origin You can verify that div E 0 Therefore Equation 7 gives 20 Example 3 Solution cont d The point of this calculation is that we can compute the surface integral over S1 because S1 is a sphere The normal vector at x is x x Therefore since the equation of S1 is x a 21 Example 3 Solution cont d Thus we have This shows that the electric flux of E is 4 Q through any closed surface S2 that contains the origin This is a special case of Gauss s Law for a single charge The relationship between and 0 is 1 4 0 22 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 23 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 24 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 25 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 26 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 …
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