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9.7 Divergence and Curl Vector Fields: R2 R2 F(x,y)= P(x,y) i + Q(x,y) j F(x,y,z) = P(x,y,z) i + Q(x,y,z) j + R(x,y,z) k Example 1: Graph the 2-dim vector field R3 R3 F(x,y)= -y i + x j Draw vectors of the same length Download m-file Divergence P Q R divF x y z Scalar divF F Example 2 2 3 4 5 2 4 6 IF F ( x y z )i (4 x y z ) j ( y z )k Find div F Curl R Q P R Q P curlF ( )i ( ) j( )k y z z x x y vector Example 2 2 3 4 5 2 4 6 IF F ( x y z )i (4 x y z ) j ( y z )k Find curl F Curl i j k curl F F x y z P Q R Remarks: 1) curl ( grad f ) f 0 2) div (curl F ) ( F ) 0 Cross product of the del operator and the vector F WHY Vector Fields The motion of a wind or fluid can be described by a vector field. The concept of a force field plays an important role in mechanics, electricity, and magnetism. Physical Interpretations Curl was introduced by Maxwell James Clerk Maxwell (1831-1879) Scottish Physicist [b. Edinburgh, Scotland, June 13, 1831, d. Cambridge, England, November 5, 1879] He published his first scientific paper at age 14, entered the University of Edinburgh at 16, and graduated from Cambridge University. Physical Interpretations Curl is easily understood in connection with the flow of fluids. If a paddle device, such as shown in fig, is inserted in a flowing fluid, the the curl of the velocity field F is a measure of the tendency of the fluid to turn the device about its vertical axis w. If curl F = 0 then flow of the fluid is said to be irrotational. Which means that it is free of vortices or whirlpools that would cause the paddle to rotate. Note: “irrotational” does not mean that the fluid does not rotate. Physical Interpretations P Q R divF x y z The volume of the fluid flowing through an element of surface area per unit time that is , the flux of the vector field F through the area. The divergence of a velocity field F near a point p(x,y,z) is the flux per unit volume. If div F(p) > 0 then p is said to be a source for F. since there is a net outward flow of fluid near p If div F(p) < 0 then p is said to be a sink for F. since there is a net inward flow of fluid near p If div F(p) = 0 then there are no sources or sinks near p. The divergence of a vector field can also be interpreted as a measure of the rate of change of the density of the fluid at a point. If div F = 0 the fluid is said to be incompressible