16.5 Curl and Divergence
P(x,y,z) Many ways to differentiate a vector field F(x, y, z) = Q(x,y,z) . The nine scalar 1st-derivatives form R(x,y,z) the total derivative or Jacobian matrix:
∂P ∂P ∂P ∂x ∂y ∂z = ∂Q ∂Q ∂Q DF ∂x ∂y ∂z ∂R ∂R ∂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 Definition. If F = Pi + Qj + Rk is a three-dimensional vector field then the curl of F is the vector field
∂R ∂Q ∂P ∂R ∂Q ∂P curl F = F = i + j + k ∇ × ∂y − ∂z ∂z − ∂x ∂x − ∂y defined wherever all partial derivatives exist.1
Differential Operators and Notation ‘Nabla’ or ‘Del’ is the differential operator
∂ ∂ ∂ ∂ x = i + j + k = ∂y ∇ ∂x ∂y ∂z ∂z The gradient of f is the action of the operator on f : ∇ ∂ ∂ ∂ ∂ f ∂ f ∂ f grad f = f = i + j + k f = i + j + k ∇ ∂x ∂y ∂z ∂x ∂y ∂z The cross product notation for curl now makes sense: P ∂x P i j k ∂y ∂z ∂x ∂z ∂x ∂y Q = ∂y Q = ∂x ∂y ∂z = i j + k = ∇ × × QR − PR PQ ··· R ∂z R PQR This construction is easier to remember than the formula in the definition and is most simple in column-vector notation.
Example If F = (x2 3y)i + xzj + (x + yz)k then − ∂ 2 ∂x x 3y z x ∂ − − F = xz = 1 ∇ × ∂y × − ∂ x + yz z + 3 ∂z 1Some authors will treat two-dimensional vector fields and state that if F = Pi + Qj is such then its curl is the scalar quantity ∂Q ∂P , familiar from Green’s Theorem. ∂x − ∂y
1 Curl and Conservatism Recall Section 16.3 where a vector field F = Pi + Qj + Rk on a simply- connected region is conservative if and only if ∂Q ∂P ∂R ∂Q ∂P ∂R = , = , = ∂x ∂y ∂y ∂z ∂z ∂x This says precisely that all parts of the curl vanish. Theorem. If F has continuous first-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 (often written f = 0). ∇ ∇ × ∇ Rotating fields: Interpretation of Curl Curl measures the tendency of objects to rotate. y y
2 2
4 224 4 224 − − x − − x 2 2 − −
(xi + yj) = 0 (yi xj) = 2k ∇ × ∇ × − − No rotation Rotating flow Definition. A vector field 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 field 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 fluid with ve- locity field F = Pi + Qj. Assuming no friction and that the paddle rotates freely, the paddle will rotate with angular velocity
1 ∂Q ∂P ω = (a, b) (a, b) k 2 ∂x − ∂y
More generally, place a paddle at position r and with axis n in a fluid flow F. Then the paddle will rotate around n with angular speed
1 ω = (curl F) n rad/s 2 ·
2 The proof is a little tricky and (at least in 3D) requires Stokes’ Theorem. It is much easier to consider a. . .