1
3 y 2.5
2 x 1.5
1
0.5 −2 0 −1.5 −1 −0.5 −0.5 −1 0 −2 −1.5 0.5 −1 −0.5 1 0 0.5 1.5 1 1.5 2 2
Figure 1: Paddle wheel dipped into uid owing in the xy plane.
NOTES ON CURL AND ALL THAT JAZZ
These notes are an attempt to give a more geometric/physical meaning to the notions of curl, divergence, and thereby illuminate the theorems of Green, Gauss, and Stokes.
1. An intuitive description of the curl in two dimensions The curl of a vector eld F = (u(x, y, z), v(x, y, z), w(x, y, z)) is de ned to be
curl(F) = (wy vz, uz wx, vx uy). (1) To gain an intuitive understanding of the curl, we shall think of the vector eld as the velocity eld of a uid. The function u is the x component of the velocity, v is the y component, and w is the z component. We begin with a simpler situation, two-dimensional ow, where F = (u(x, y), v(x, y)). Then
curl(F) = (vx uy)k. (2) We imagine dipping a paddle wheel into the uid with the paddle wheel oriented so that the axis of rotation is in the k direction as in Figure 1. Now let us consider a simple ow, where u and v are linear in x and y: 2
y
x
Figure 2: Vector eld F1 = (y/2 + 1, 0) producing clockwise rotation of paddle wheel.
u(x, y) = A + u1x + u2y
v(x, y) = B + v1x + v2y. (3)
First we consider the ow where the uid moves only in the x direction, F1 = (u(x, y), 0) = (A + u1x + u2y, 0). Note that the velocity component (A + u1x, 0) is the same along any vertical line, and thus will not cause any rotation. However, if u2 > 0, the component (u2y, 0) increases along a vertical line, making the velocity at the top of wheel greater than at the bottom. This will cause the wheel to rotate in a clockwise direction. If u2 < 0, the wheel will rotate in the counterclockwise direction. Therefore F causes rotation only when u = 0. See Figure 2. 1 2 6 Next we consider a simple ow F2 = (0, B + v1x + v2y). This ow moves the uid particles in the y direction. Now the component (0, B + v2y) is constant along any horizontal line and will not cause the wheel to rotate. However, the component (0, v2x) increases from left to right if v2 > 0, causing the wheel to rotate in the counterclockwise direction. If v2 < 0, the wheel will rotate in the clockwise direction. Hence F2 will cause rotation if v2 = 0. See Figure 3. The combined ow F = F + F will cause rotation if v u6 = 0. If v u > 0, the 1 2 1 2 6 1 2 rotation will be counterclockwise and if v1 u2 < 0, the rotation will be clockwise. The quantity v u will be a measure of the speed of rotation. In this special | 1 2| case when the vector eld F is given by (3), v u = vx uy. 1 2 3
y
x
Figure 3: Vector eld F2 = (0, x/2 + 1) causing paddle wheel to rotate in coun- terclockwise direction.
Now let F = (u(x, y), v(x, y)) be a general two-dimensional vector eld and let p0 = (x0, y0). We make a linear approximation to each component of F at p0.
u(x, y) u(p ) + ux(p )(x x ) + uy(p )(y y ) (4) 0 0 0 0 0 A + ux(p )x + uy(p )y 0 0 v(x, y) v(p ) + vx(p )(x x ) + vy(p )(y y ) (5) 0 0 0 0 0 B + vx(p )x + vy(p )y. 0 0
For (x, y) close to (x0, y0), the linear approximations (4) and (5) provide a good description of the vector eld. If we apply the previous discussion to these linear approximations, we deduce that the quantity vx(p ) uy(p ) is a measure of the 0 0 ability of the uid to rotate a small paddle wheel centered at p0, with the quantity vx(p ) uy(p ) giving the direction and speed of rotation. In fact, we will see in 0 0 the Example (iii) below that vx(p0) uy(p0) is twice the angular velocity at which the paddle wheel rotates. | | Examples (i) Let F = (1 y2, 0) on the strip < x < , 1 y 1 . F is { ∞ ∞ } the velocity eld of water owing in the channel occupied by the strip. Note the velocity is zero on the top and bottom edges of the strip, and is greatest in the middle of the strip (y = 0). It is easy to see that curl(F) = 2yk. This means that 4 a paddle wheel placed in the strip in y > 0 will rotate in the counterclockwise direction, and in the clockwise direction in y < 0. On the line y = 0, it will not rotate. (ii) Let ( y, x) F(x, y) = , > 0. r