Surface Area, Divergence and Curl

Divergence and Curl, Del Notation

Let f be a vector field mapping n space into n space. We usually think of f as a fluid flow, even if the fluid is rather abstract, like an electric field. At a point x in space, f(x) represents the speed and direction of the fluid at that point.

The divergence of f is the sum of the partial of f₁ with respect to x₁, plus the partial of f₂ with respect to x₂, and so on up to the partial of f_n with respect to x_n. This is written ∇.f, and pronounced del dot f.

Remember that the traditional dot product multiplies corresponding elements and takes the sum. So the notation ∇.f is suppose to remind you to apply the del operator, which represents differentiation with respect to x₁ x₂ x₃ … x_n, to the component functions f₁ f₂ f₃ … f_n, and take the sum. The result is the divergence of f.

If f measures the flow of a fluid as it moves through space, the divergence measures the rate at which the fluid is spreading out. A positive divergence indicates the fluid is expanding, and a negative divergence compresses the fluid at that point. If the fluid is incompressible, like water, we can safely set the divergence to 0. All this will be made rigorous later; I'm just providing some intuition here.

Continue to think of f as the motion of a fluid. The curl of f measures the spin of a tiny paddle wheel, placed at any point in the stream. In other words, the fluid moves faster on one side of the wheel than the other, hence turning the wheel. If f has component functions f₁ f₂ and f₃, the curl is defined as follows.

∇×f = [ f₃∂y-f₂∂z, f₁∂z-f₃∂x, f₂∂x-f₁∂y ]

Notice that the curl of f is a vector field. The length of the curl gives the rate of rotation of our tiny wheel, and the direction indicates the axis of rotation. In other words, the curl is a spin vector.

Once again the del cross f notation is suppose to help you remember the formula. If "del" means differentiation with respect to x y and z, then build the following matrix. The second row is the del operator, taking partials with respect to x y and z, and the third row is the component functions of f. The top row becomes the cross product, or the curl of f.

curl	?	?	?
∇	∂x	∂y	∂z
f	f₁	f₂	f₃

Let f rotate all of 3 space counterclockwise about the z axis. The component functions are as follows.

f₁ = -y
f₂ = x
f₃ = z

Compute the curl and get the vector (0,0,2). Place a tiny wheel anywhere in space and it spins about a vector that is parallel to the z axis. And, it spins at the same rate, 2, no matter where it is placed.

Notice that the divergence is well defined in any number of dimensions, but the curl is not. In fact there seems to be no analog for the curl in anything other than 3 space. Yet curl is almost as important as divergence, as we will see when we get to Maxwell's equations. Does this mean there is something special about three dimensions? I don't know. Perhaps you can enlighten me.

Identities

Let g be a scalar field with continuous mixed partials and consider ∇×∇g. Since ∇g produces the partials, ∇×∇g is an expression in mixed partials. Since these mixed partials are continuous, they are equal. Do the algebra, and ∇×∇g = 0.

Let's try to interpret the above. Let g be a potential field, perhaps the elevation of a surface. The gradient gives the direction that a marble would roll, downhill, if left to its own devices. Place two marbles next to each other and shrink them down to an impossibly small size. They will travel downhill together, tracing parallel paths. One marble is never pulled downhill faster than the other. A horizontal wheel placed between them does not spin, since the marbles travel together on either side of the wheel. The curl of the gradient of g is 0.

Next let f be a vector field, such that the component functions have continuous mixed partials, and consider ∇.∇×f. The curl ∇×f includes, in its first component, the partial of f₃ with respect to y. When we apply del dot, we get the mixed partial of f₃ with respect to y, with respect to x. Now the second component includes - the partial of f₃ with respect to x. This is differentiated with respect to y to get the "other" mixed partial. The two mixed partials are equal, and when they are subtracted, the result is 0. This happens across the board, hence ∇.∇×f = 0.

The Laplacian

Let g be a scalar field and consider ∇.∇g, which is also written ∇²g. Remember that ∇g produces a list of the first partials, and the next del operator takes partials again, and adds them up. Therefore ∇²g is the sum of the second partials of g with respect to each of the coordinate variables. This is called the laplacian of g, after Pierre Laplace.

If the laplacian of g is 0 then g is a harmonic function. In other words, ∇²g = 0.