Section 17.5: Curl and Divergence 1 Objectives 1. Compute the curl of a vector field. 2. Determine whether a vector field is conservative. 3. Compute the divergence of a vector field. 4. Associate divergence with incompressibility. 2 Assignments 1. Read Section 17.5 2. Problems: 1,5,9,10,11,15,17,21 3. Challenge: 25,26,28,35 4. Read Section 17.8 3 Maple Commands 4 Lecture In this section, we define the curl and divergence of a vector field, and we look at theorems that help us to associate these operators with physical properties. We will also define the Laplace operator, which is actually the divergence of a gradient of a function. One of the most important things to keep in mind about curl and divergence is that the curl of a vector field in IR3 is a vector, while the divergence of a vector field is a scalar function of three variables. 4.1 Curl Let F = P (x, y, z)i + Q(x, y, z)j + R(x, y, z)k. If the partial derivatives of P , Q, and R all exist, the curl of F is the vector field on IR3 defined by ∂R ∂Q ∂P ∂R ∂Q ∂P curl F = − i + − j + − k. ∂y ∂z ∂z ∂x ∂x ∂y We introduce the del notation, or ∇. We define ∇ as ∂ ∂ ∂ del = ∇ = i + j + k , ∂x ∂y ∂z 1 so that ∇f, if f = f(x, y, z), is the same gradient function that we studied in Chapter 15. The we can define the curl of F as i j k ∂ ∂ ∂ curl F = del × F = ∂x ∂y ∂z PQR ∂R ∂Q ∂R ∂P ∂Q ∂P = i − − j − + k − , ∂y ∂z ∂x ∂z ∂x ∂y which is exactly what we have above. 4.1.1 Example 1: Problem 17.5.2 We want to compute the curl of F = (x − 2z)i + (x + y + z)j + (x − 2y)k. Thus ∇ × F = −3i − 3j + k. 4.2 Theorem Theorem: If f is a function of 3 variables with continuous second order partial derivatives, then curl (∇f) = 0. You should look at the proof of this in your text. The proof involves the use of Clairaut’s theorem, which we covered earlier. This theorem tells us that if F is a conservative vector field, then curl F = 0. This is actually an if and only if statement, under certain assumptions. That is, provided we have certain assumptions on F, we can determine whether or not F is conservative by computing its curl. Theorem: If F is a vector field defined on all of IR3 whose component functions have continuous partial derivatives and curl F = 0, then F is a conservative vector field. 4.2.1 Example 2: Problem 17.5.16 We want to determine whether or not F = xy2z3i+2x2yz3j+3x2y2z2k is conservative. First, note that F is defined of all of IR3, that is, its domain is {(x, y, z): x, y, z ∈ IR}. Therefore, we can use the theorem and compute the curl of F to find out if it is conservative. If we look at the j component of ∇ × F, we get −3xy2z2j which is not equal to 0, so F is not conservative. Class Exercises: 1. What are the i and k components of ∇ × F above? 2. Problem 15 gives a conservative vector field; I encourage you to do this problem, as you also have to find a function f such that ∇f = F, which is a little more tedious when F ∈ IR3 than it is when F ∈ IR2. In particular, now when you integrate fx with respect to x, you have to account for a function of y and z, say g(y, z), instead of just g(y). Example 3 in your text shows you how to do this. 2 4.3 Physical interpretations of curl Let F represent a velocity field for a fluid flow. Particles near the point (x, y, z) rotate about the axis that points in the direction of mathbfcurl F. The magnitude of curl F is the measure of how quickly the particles move around the axis. If the curl F = 0 at a point P (x, y, z), then the fluid is free from rotations at P and F is said to be irrotational at P . That means that there isn’t any whirlpool or eddy at P , because the fluid is not moving at that point. In order for a whirlpool to exist, the fluid must be rotating about a point in space. Your text has a nice explanation of this phenomenon using a water wheel example. I encourage you to read it and think about what’s going on. 4.4 Divergence The divergence of a vector field F = P (x, y, z)i + Q(x, y, z)j + R(x, y, z)k is defined as ∂P ∂Q ∂R div F = ∇ · F = + + , ∂x ∂y ∂z which gives a function of three variables. Theorem: If F = P i+Qj+Rk on IR3, and P , Q, and R have continuous second-order partial derivatives, then ∇ · (curl F) = 0. 4.5 Physical interpretation Again, let F(x, y, z) represent a velocity field for a fluid. The divergence of F measures the tendency of the fluid to diverge from the point (x, y, z), so if ∇ · F = 0, the fluid is incompressible. 4.6 The Laplace operator Let F = ∇f. Then ∇ · F = ∇ · (∇f) = ∇2f = 4f. This operator, ∇2 or 4, is known as the Laplace operator, and the Laplace equation is given as ∇2f = 0. 4.7 Vector forms of Green’s Theorem Consider the vector field F = P i + Qj. Then Z ZZ F · dr = (∇ × F) · k dA, C D 3 which gives the vector form of Green’s Theorem. However, perhaps the more useful vector form of Green’s Theorem is given by Z ZZ F · n ds = ∇ · F dA. C D This extends to three dimensions and becomes the divergence theorem in Section 17.9. However, note that this version of Green’s Theorem simply states that the line integral of the normal component of F (what you get from F · n) along C is the double integral of the divergence of F over the region D that is enclosed by C. If we think of C as a boundary region, then the normal components somehow tell us about the action of F on the boundary. Integrating these normal components tells us how perhaps a fluid is moving within an enclosed region. 4.