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Chapter 16 – Vector Calculus 16.5 Curl and Divergence Objectives: Understand the operations of curl and divergence Use curl and divergence to obtain vector forms of Green’s Theorem 16.5 Curl and Divergence 1 Vector Calculus Here, we define two operations that: ◦ Can be performed on vector fields. ◦ Play a basic role in the applications of vector calculus to fluid flow, electricity, and magnetism. Each operation resembles differentiation. However, one produces a vector field whereas the other produces a scalar field. 16.5 Curl and Divergence 2 Definition - Curl Suppose F = P i + Q j + R k is a vector field on 3 and the partial derivatives of P, Q, and R all exist, then the curl of F is the vector field defined by: R Q P R Q P curl F i k j y z z x x y 16.5 Curl and Divergence 3 Curl As a memory aid, let’s rewrite Equation 1 using operator notation. ◦ We introduce the vector differential operator (“del”) as: i j k x y z 16.5 Curl and Divergence 4 Curl i F x P j k y Q z R R Q P R Q P i k j y z z x x y curl F 16.5 Curl and Divergence 5 Curl Thus, the easiest way to remember Definition 1 is by means of the symbolic expression curl F F 16.5 Curl and Divergence 6 Theorem 3 If f is a function of three variables that has continuous second-order partial derivatives, then curl f 0 16.5 Curl and Divergence 7 Conservative Vector Field A conservative vector field is one for which F f So, Theorem 3 can be rephrased as: If F is conservative, then curl F = 0. ◦ This gives us a way of verifying that a vector field is not conservative. 16.5 Curl and Divergence 8 Theorem 4 If F is a vector field defined on all of 3 whose component functions have continuous partial derivatives and curl F = 0, then F is a conservative vector field. NOTE: Theorem 4 is the 3-D version of Theorem 6 in Section 16.3 NOTE: This theorem says that it is true if the domain is simply-connected—that is, “has no hole.” 16.5 Curl and Divergence 9 Curl The reason for the name curl is that the curl vector is associated with rotations. ◦ One connection is explained in Exercise 37. ◦ Another occurs when F represents the velocity field in fluid flow (Example 3 in Section 16.1). 16.5 Curl and Divergence 10 Curl Particles near (x, y, z) in the fluid tend to rotate about the axis that points in the direction of curl F(x, y, z). ◦ The length of this curl vector is a measure of how quickly the particles move around the axis. 16.5 Curl and Divergence 11 Irrotational Curl If curl F = 0 at a point P, the fluid is free from rotations at P. F is called irrotational at P. ◦ That is, there is no whirlpool or eddy at P. 16.5 Curl and Divergence 12 If curl F = 0, a tiny paddle wheel moves with the fluid but doesn’t rotate about its axis. If curl F ≠ 0, the paddle wheel rotates about its axis. ◦ We give a more detailed explanation in Section 16.8 as a consequence of Stokes’ Theorem. 16.5 Curl and Divergence 13 Example 1 – pg. 1068 # 21 Show that any vector field of the form F( x, y, z) f ( x) i g ( y) j h( z) k where f, g, and h are differentiable functions, is irrotational. 16.5 Curl and Divergence 14 Definition - Divergence If F = P i + Q j + R k is a vector field on 3 and ∂P/∂x, ∂Q/∂y, and ∂R/∂z exist, the divergence of F is the function of three variables defined by: P Q R div F x y z 16.5 Curl and Divergence 15 Divergence In terms of the gradient operator i j k x y z the divergence of F can be written symbolically as the dot product of del and F: div F F 16.5 Curl and Divergence 16 Curl versus Divergence Observe that: ◦ Curl F is a vector field. ◦ Div F is a scalar field. 16.5 Curl and Divergence 17 Theorem 11 If F = P i + Q j + R k is a vector field on 3 and P, Q, and R have continuous secondorder partial derivatives, then div curl F = 0 16.5 Curl and Divergence 18 Divergence Again, the reason for the name divergence can be understood in the context of fluid flow. ◦ If F(x, y, z) is the velocity of a fluid (or gas), div F(x, y, z) represents the net rate of change (with respect to time) of the mass of fluid (or gas) flowing from the point (x, y, z) per unit volume. 16.5 Curl and Divergence 19 Incompressible Divergence In other words, div F(x, y, z) measures the tendency of the fluid to diverge from the point (x, y, z). If div F = 0, F is said to be incompressible. 16.5 Curl and Divergence 20 Example 2 – pg. 1068 # 22 Show that any vector of the form F( x, y, z ) f ( y, z ) i g ( x, z ) j h( x, y) k is incompressible. 16.5 Curl and Divergence 21 Example 3 – pg. 1068 Find the following for the given vector field: a) The curl b) The divergence 2. F( x, y, z ) x yzi xy zj xyz k 2 2 8. F( x, y, z ) e , e , e x xy 2 xyz 16.5 Curl and Divergence 22 Gradient Vector Fields Another differential operator occurs when we compute the divergence of a gradient vector field f. ◦ If f is a function of three variables, we have: div f f f f f 2 2 2 x y z 2 2 2 16.5 Curl and Divergence 23 Laplace Operator This expression occurs so often that we abbreviate it as 2f. The operator 2 is called the Laplace operator due to its relation to Laplace’s equation 2 2 2 f f f 2 f 2 2 2 0 x y z 16.5 Curl and Divergence 24 Green’s Theorem – Vector Form Hence, we can now rewrite the equation in Green’s Theorem in the vector form using curl as equation 12: F dr curl F k dA C D 16.5 Curl and Divergence 25 Green’s Theorem – Vector Form Equation 12 expresses the line integral of the tangential component of F along C as the double integral of the vertical component of curl F over the region D enclosed by C. ◦ We now write a similar formula involving the normal component of F and the divergence. (see book for proof) 16.5 Curl and Divergence 26 Green’s Theorem – Vector Form F n ds div F x , y dA C D 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. 16.5 Curl and Divergence 27 Example 4 – pg. 1068 Determine whether or not the vector field is conservative. If it is conservative, find a function f such that F = f. 14. F ( x, y, z ) xyz 2 i x 2 yz 2 j x 2 y 2 zk 16. F ( x, y, z ) e z i j xe z k 16.5 Curl and Divergence 28