Lesson 21: Curl and Divergence

Transcription

1 Lesson 21: Curl and Divergence July 24th, 2015

2 In this lesson we examine two differential operators comparable to the gradient. Both are applied to vector fields, but one produces a vector field while the other produces a function. These operators have many applications to problems involving fluid flow and electromagnetism.

3 Let F = P i + Q j + R k be a vector field on R 3, and suppose the partial derivatives of P, Q, and R all exist. The curl of F is the vector field on R 3 defined by curl F = ( R y Q ) ( P i + z z R ) ( Q j + x x P ) k y This formula may be difficult to remember. To make it easier, let s define the differential operator ( del ) as = x i + y j + z k We can think of as a vector field with components x, y, and z.

4 With this set-up, we can compute the curl of F as the cross product i j k F = x y z = P Q R ( R y Q ) ( P i + z z R ) ( Q j + x x P ) k y = curl F Thus we sometimes write curl F = F

5 Example If F(x, y, z) = xz i + xyz j y 2 k, find curl F.

6 Let s calculate curl ( f ): i j k curl ( f ) = f = x y z f x f y f z = ( 2 ) ( f y z 2 f 2 ) ( f i + z y z x 2 f 2 ) f j + x z x y 2 f k y x = 0 i + 0 j + 0 k = 0 If F is conservative, then F = f for some f. Notice that this implies the following: If F is conservative, then curl F = 0.

7 This gives us a way of checking if a vector field on R 3 is not conservative. Namely, if curl F 0, then F is not conservative. For example, we saw that F(x, y, z) = xz i + xyz j y 2 k has curl F = y(2 + x) i + x j + yz k 0. Therefore F is not conservative. The following theorem holds for simply connected domains, but we state it how it is given in the book. Theorem If F is a vector field defined on all of R 3 whose component functions have continuous partial derivatives and curl F = 0, then F is a conservative vector field.

8 Example (a) Show that F(x, y, z) = y 2 z 3 i + 2xyz 3 j + 3xy 2 z 2 k is a conservative vector field. (b) Find a function f such that F = f.

9 Example Compute the curl of F = e xy sin z j + y tan 1 (x/z) k.

10 If F = P i + Q j + R k is a vector field on R 3 and P, Q, and x y R exist, then the divergence of F is the function of three z variables defined by div F = P x + Q y + R z Note that div F is a function whereas curl F is a vector field. Recalling = ( / x) i + ( / y) j + ( / z) k, we can write the divergence of F as div F = F

11 Example If F(x, y, z) = xz i + xyz j y 2 k, compute div F.

12 Let s compute div curl F: div curl F = ( F) = x ( R y Q z ) + ( P y z R ) + ( Q x z x P ) y = 2 R x y 2 Q x z + 2 P y z 2 R y x + 2 Q z x 2 P z y = 0 So we have that and curl = 0 div curl = 0

13 Example Compute the divergence of F(x, y, z) = e xy sin z j + y tan 1 (x/z) k

14 The names curl and divergence come from fluid flow. Imagine fluid flowing through a container, and suppose the velocity of the fluid at any point (x, y, z) is given by the velocity vector field F(x, y, z). Particles near (x, y, z) tend to rotate about the axis pointing in the direction of curl F(x, y, z). The length of curl F measures how quickly particles flow around this axis. If curl F = 0, the fluid flows free from rotations around P and F is called irrotational. div F(x, y, z) represents the net rate of change (w.r.t time) of the mass of fluid flowing from the point (x, y, z) per unit volume, i.e. div F measures the tendency of the fluid to diverge from the point (x, y, z). If div F = 0, then F is called incompressible.

15 Example See #10 in.

16 Example Prove that div (f F) = f div F + F f

17 As a final note, with the concepts of curl and divergence, we can rewrite Green s Theorem in a way that will be useful in the future. Suppose we have a region D in the plane with boundary curve C and a vector field F = P i + Q j all satisfying the assumptions of Green s Theorem. We can view F as a vector field on R 3 with z-component 0. Then and so curl F = ( Q ) P x y k, (curl F) k = ( Q ) P x y k k = Q P. x y Since Q as P x y = (curl F) k, we can rewrite Green s Theorem F dr = (curl F) k da C D

18 This last equation says that the line integral of the tangential component of F along C equals the double integral of the vertical component of curl F over D. We have a similar statement for the normal component of F. Let r(t) = x(t) i + y(t) j, a t b. Then the unit tangent vector is and the outward unit normal is T(t) = x (t) r (t) i + y (t) r (t) j n(t) = y (t) r (t) i x (t) r (t) j A derivation then shows that F n ds = div F(x, y) da C D