Many ways to differentiate a vector eld F x y z The nine scalar 1st derivatives form 16 5 Curl and Divergence the total derivative or Jacobian matrix P x Q x R x P y Q y R y P z Q z R z DF cid 18 P x y z cid 19 Q x y z R x y z Entries of DF can combined in various ways curl and divergence are two such combinations that have useful physical interpretations Curl De nition If F Pi Qj Rk is a three dimensional vector eld then the curl of F is the vector eld cid 18 R cid 19 cid 18 P cid 19 cid 18 Q curl F F y Q z i z R x j x P y cid 19 k Differential Operators and Notation Nabla or Del is the differential operator de ned wherever all partial derivatives exist 1 i x j y k z x y z cid 19 The gradient of f is the action of the operator on f i f i x j y k z grad f f f x f y j f z k The cross product notation for curl now makes sense P Q R P Q R cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 i j y k x z P Q R cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 y z Q R cid 18 x y z cid 12 cid 12 cid 12 cid 12 i cid 12 cid 12 cid 12 cid 12 x z P R cid 12 cid 12 cid 12 cid 12 j cid 12 cid 12 cid 12 cid 12 x y P Q cid 12 cid 12 cid 12 cid 12 k This construction is easier to remember than the formula in the de nition and is most simple in column vector notation Example If F x2 3y i xzj x yz k then x y z x2 3y xz x yz z x 1 z 3 F quantity Q x P y familiar from Green s Theorem 1 1Some authors will treat two dimensional vector elds and state that if F Pi Qj is such then its curl is the scalar Curl and Conservatism Recall Section 16 3 where a vector eld F Pi Qj Rk on a simply connected region is conservative if and only if R x Q x Q z R y P y P z This says precisely that all parts of the curl vanish Theorem If F has continuous rst derivatives on a simply connected region of R3 then curl F 0 F is conservative One direction of this theorem can be written as follows Corollary If f has continuous second derivatives then curl f 0 Rotating elds Interpretation of Curl Curl measures the tendency of objects to rotate often written f 0 xi yj 0 No rotation yi xj 2k Rotating ow De nition A vector eld F is said to be irrotational if curl F 0 everywhere Local rotation The picture is too simplistic curl measures local rotation The fact that an object in the vector eld F yi xj will travel in circles round the origin is irrelevant to curl It is the fact that the object will also rotate so that it changes the direction it is facing Theorem Suppose that a small paddle with vertical axis is positioned at the point a b in a uid with ve locity eld F Pi Qj Assuming no friction and that the paddle rotates freely the paddle will rotate with angular velocity cid 19 a b x P y a b k cid 18 Q 1 2 1 2 More generally place a paddle at position r and with axis n in a uid ow F Then the paddle will rotate around n with angular speed curl F n rad s 2 22y 4 224x 22y 4 224x The proof is a little tricky and at least in 3D requires Stokes Theorem It is much easier to consider a Duck Race Suppose that the water in a river of width 4 has velocity v cid 0 1 1 4y2 cid 1 i The curl of the ow is x y z 1 1 4y2 0 0 0 0 1 2y v Release several ducks and assume that they move with the water There are two aspects to the motion of the ducks 1 Linear The ducks follow the direction of ow to the right The faster ducks are closer to midstream 2 Rotational The side of a duck closer to midstream will move faster than the side nearer the edge of the river This will cause most ducks to rotate According to the Theorem and interpreting the right hand rule a duck with y 1 will rotate with angular velocity 1 4k i e counter clockwise at 0 25 radians per second 2 v 1 Combine the two and you get the motion in the animation 2 If you ve ever been white water rafting this explains why the guide always wants to stay in the center of the river you travel fastest and you don t spin 2Click it to make it move after opening the pdf in Acrobat on your desktop 3 Global Rotation cid 54 Local Rotation It is important to distinguish between local rotation changing the direction an object faces and global rotation vector elds making objects travel in loops Com pare the following three duck races In all cases the ducks will travel counter clockwise around the origin global rotation Their local rotation changes each time Ducks have local rotation counter clockwise Because of the global rotation of the vector eld there are more arrows on the side of the duck further from the origin Since all arrows have the same length the duck rotates counter clockwise Ducks have zero local rotation The fact that the speed of ow is faster nearer the origin exactly balances the the fact that more arrows are on the outside of the duck y x 0 1 F1 x2 y2 1 2 F1 x2 y2 1 2 k 1 y x 0 F2 1 x2 y2 F2 0 y x 0 1 F3 x2 y2 3 2 F3 x2 y2 3 2 k 1 Ducks have local rotation clockwise Now the speed of ow is so much greater nearer the origin that the duck spins the opposite way 4 Circulation Green s Theorem Revisited cid 73 Suppose that F Pi Qj is a vector eld in two dimensions Then F k Q x P y forms part of Green s Theorem cid 90 cid 90 which we can now rewrite cid 90 cid 90 C F dr cid 72 C F T ds where T is the unit tangent vector of C This is often called the circulation of the vector eld F around C how vigorously F pushes a particle round C Q x F k dA Recall that cid 72 F dr P y dA D D C …