Math Divergence and Curl

Transcription

1 Math 23 - Divergence and Curl Peter A. Perry University of Kentucky November 3, 28

2 Homework Work on Stewart problems for 6.5: - (odd), 2, 3-7 (odd), 2, 23, 25 Finish Homework D2 due tonight Begin Homework D3 due Wednesday, December 5

3 Unit IV: Vector Calculus Lecture 35 Lecture 36 Lecture 37 Lecture 38 Lecture 39 Lecture 4 Vector Fields Line Integrals I Line Integrals II Fundamental Theorem Green s Theorem Curl and Divergence

4 Goals of the Day This lecture is about two very important derivatives of a vector field. You ll learn: How to compute the curl of a vector field and what it measures How to compute the divergence of a vector field and what it measures (Sneak preview) The theorems that give the meaning of divergence and curl

5 Curl If F = Pi + Qj + Rk is a vector field on R 3, and the partial derivatives of P, Q, and R all exist, then the curl of F is a new vector field: curl F = ( R y Q ) ( P i + z z R ) ( Q j + x x P ) k y This new vector field measures the rotation of the vector field F at a given point (x, y, z): Its direction is the axis of rotation, dictated by the right-hand rule Its magnitude is the angular speed of rotation

6 Curl curl F = ( R y Q ) ( P i + z z R ) ( Q j + x x P ) k y The new vector field curl F is sometimes written F because of an easier-to-remember formula: i j k curl F = F = x y z P Q R

7 F(x, y, z) = xi + yj + zk Curl.5.5 If F = xi + yj + zk then i j k F = = x y z x y z F(x, y, z) = yi xj + k If F = yi xj + k then i j k F = = 2k x y z y x.5.5

8 Curl A gradient vector field has zero curl: i j k ( f ) = x y z f f f x y z ( = 2 ) ( f x y 2 f i + 2 ) ( f y x y z 2 f j + 2 ) f z y z x 2 f k x z = so the curl detects conservative vector fields.

9 Divergence The divergence of a vector field F = Pi + Qj + Rk is a scalar function: div F = P x + Q y + R z Sometimes div F is written F: F = x, y, P, Q, R = P z x + Q y + R z The divergence computes the outflow per unit volume of the vector field (thought of as a velocity field)

10 F(x, y, z) = xi + yj + zk Divergence If F = xi + yj + zk then.5.5 F = x x + y y + z z = 3 (same outflow at each point of space) F(x, y, z) = yi xj + k If F = yi xj + k then F = y x x y + = (no outflow anywhere in space).5.5

11 Divergence Remember that ( f ) =? There is an analogous result for the divergence: div curl F = You can see this using the definitions of divergence and curl: div curl F = ( R x y Q ) + ( P z y z R ) + ( Q x z x P ) y The second partial derivatives cancel in pairs by Clairaut s theorem. It turns out that any vector field F can be written as F = f + A for a scalar potential f and a vector potential A.

12 Divergence and Curl If f is a scalar function and F is a vector function, which of these expressions make sense? Do they define a scalar or a vector? Remember that curl F is a vector div F is a scalar (a) curl f (b) grad f (c) div F (d) curl(grad f ) (e) grad F (f) grad(div F) (g) div(grad f ) (h) grad(div f ) (i) curl(curl F) (j) div(div f )

13 Conservative Vector Fields Again Determine whether the vector field F(x, y, z) = y 2 z 3 i + 2xyz 3 j + 3xy 2 z 2 k is conservative and, if so, find a function f so that f = F.

14 Vector Identities Show that div(f F) = f F + f div F

15 Divergence Theorem, Stokes Theorem Divergence Theorem Suppose E is a simple solid region and S is its boundary. Let N be the outward normal to S. Then F N ds = div F dv S E Stokes Theorem Suppose S is an oriented piecewise-smooth surface with outward normal N, bounded by a simple closed curve C with piecewise smooth boundary. Then F dr = curl F N ds C S