Chapter 16 – Vector Calculus

1. 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

2. 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

3. Definition - Curl • Suppose F = Pi + Qj + Rk 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: 16.5 Curl and Divergence

4. Curl • As a memory aid, let’s rewrite Equation 1 using operator notation. • We introduce the vector differential operator (“del”) as: 16.5 Curl and Divergence

5. Curl 16.5 Curl and Divergence

6. Curl • Thus, the easiest way to remember Definition 1 is by means of the symbolic expression 16.5 Curl and Divergence

7. Theorem 3 • If f is a function of three variables that has continuous second-order partial derivatives, then 16.5 Curl and Divergence

8. Conservative Vector Field • A conservative vector field is one for which • 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. Example 1 – pg. 1068 # 21 • Show that any vector field of the form where f, g, and hare differentiable functions, is irrotational. 16.5 Curl and Divergence

15. Definition - Divergence • If F = Pi + Qj + Rk is a vector field on 3and ∂P/∂x, ∂Q/∂y, and ∂R/∂zexist, the divergence of F is the function of three variables defined by: 16.5 Curl and Divergence

16. Divergence • In terms of the gradient operator the divergence of F can be written symbolically as the dot product of del and F: 16.5 Curl and Divergence

17. Curl versus Divergence • Observe that: • Curl F is a vector field. • Div F is a scalar field. 16.5 Curl and Divergence

18. Theorem 11 • If F = Pi + Qj + Rk is a vector field on 3 and P, Q, and R have continuous second-order partial derivatives, then div curl F = 0 16.5 Curl and Divergence

19. 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

20. 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

21. Example 2 – pg. 1068 # 22 • Show that any vector of the form is incompressible. 16.5 Curl and Divergence

22. Example 3 – pg. 1068 • Find the following for the given vector field: • The curl • The divergence 16.5 Curl and Divergence

23. 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: 16.5 Curl and Divergence

24. Laplace Operator • This expression occurs so often that we abbreviate it as 2f. • The operator is called the Laplace operator due to its relation to Laplace’s equation 16.5 Curl and Divergence

25. 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: 16.5 Curl and Divergence

26. 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

27. Green’s Theorem – Vector Form • 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

28. 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. 16.5 Curl and Divergence