Vector Fields which are Gradients—Special 3D case—The Curl, and Del

Learning Goals: we specialize the previous material to three dimensions and introduce the curl.

Now when deciding whether a vector field is a gradient, checking the derivatives is more work than just antidifferentiating, except in two and three dimensions. In two dimension, there is just one pair of derivatives to check, while in three dimensions there are three pairs. That kind of reflexivity in three dimensions makes for an interesting investigation.

If we have a vector field G = (g1, g2, g3) then it is a gradient of some function if and only if ∂g ∂g ∂g ∂g (remember, we still actually haven’t proven the “if” part!) 1 = 2 , 2 = 3 , and ∂y ∂x ∂z ∂y ∂g ∂g 3 = 1 . Since there are three possible obstructions, let’s make them a three dimensional ∂x ∂z ⎛ ∂g ∂g ∂g ∂g ∂g ∂g ⎞ vector, and look at 3 − 2 , 1 − 3 , 2 − 1 . Notice that we have arranged these so ⎝⎜ ∂y ∂z ∂z ∂x ∂x ∂y ⎠⎟ that the pieces involving y and z are in the x-slot, x and z in the y-slot, and x and y in the z-slot. Does that combination remind you of anything else we have studied? Hint: it should!

∂ Yep, it reminds us of cross products. In fact, if we write g and so forth for all the partial ∂x 2 ⎛ ∂ ∂ ∂ ⎞ derivatives, this looks exactly like the cross product , , × (g ,g ,g ) . ⎝⎜ ∂x ∂y ∂z⎠⎟ 1 2 3

Now that first thing is pretty weird looking, though we have come across stuff like it before. ∂ Remember when we looked at just some bare derivative signs, like and we called it an ∂x operator—a derivative waiting for a function to differentiate? Well, this is just a bunch of them grouped into a vector, all waiting for stuff to operate on. We call this particular operator ∇, pronounced “del” or “nabla”. It is basically a gradient sign without a function to take the gradient of—like it is sitting there waiting to take a gradient.

But instead, it is used as a vector and we take a cross product with it, to get the thing above: ∇ × G. Notice that it is calculated the exact same way we take a regular cross product: ˆi ˆj kˆ ∂ ∂ ∂ . This is also called curl G because it measures how much G is rotating (as we ∂x ∂y ∂z

g1 g2 g3 will see in a bit). Recall that vector fields that seem to rotate can’t be gradients due to the always-increasing problem as you go around a loop in the flow. But derivatives don’t measure things “in the large.” They measure what is happening locally. So the curl measures rotation on a microscopic scale. Let’s see what’s going on.

Let’s look at an example vector field: (4 – y, 0, 0). You should quickly calculate that the curl of this is (0, 0, 1). Here is a picture of part of that vector field, into which we have dropped a small paddle wheel:

Now the vectors push harder on this paddle wheel near the bottom than near the top, so it will tend to rotate counter-clockwise. Now ∂g1 / ∂y = –1, because the x-component of the vector gets larger as you decrease y. That is what is causing the wheel to spin. But note that part of the z- component of the curl is –∂g1 / ∂y which gives the 1 above. Note also that if we compute the torque on the wheel it points straight up, since it is trying to cause the wheel to rotate counterclockwise.

Now for a more general vector field, there is also some component of the torque coming from the vector field pushing harder upward or downward on the right side versus the left. In fact, if the y-component of the vector field increases as we increase x (∂g2 / ∂x > 0) we get a ∂g ∂g counterclockwise (positive) torque. So that’s why the z-component of the curl is 2 − 1 . We ∂x ∂y can get a similar analysis in the other directions—we can even make it more rigorous if we throw in distances and moments of inertia, but that seems to be unnecessarily complicated. The idea is that the curl is the vector field trying to twist around a point in space, and measures exactly how strong and in which direction the field is trying to rotate things.

Things are actually even more interesting than this. The curl combines two rotational effects into one measurement. One is the local twisting—if you held the paddle wheel in place while letting it spin, how would it spin? The other is rotation due to the wheel being pushed along a possible curved path. Think about what you would see if you placed an X on a merry-go-round and then watched it from above. Relative to the merry-go-round, the X doesn’t move. But as the merry- go-round goes ‘round, from above the X will appear to rotate as it moves.

⎛ −y x ⎞ Example: compute the curl of vortex flow, V(x, y,z) = , ,0 . It is a messy ⎝⎜ x2 + y2 x2 + y2 ⎠⎟ calculation with the quotient rule to show that ∇ × V = 0! How could this flow have no rotation? Well, if you dropped a little paddle wheel into the vortex, it would circulate around the center, so would appear to rotate counterclockwise as in the merry-go-round explanation just given. But the inside edge of the paddle wheel is pushed harder than the outside edge, so that if you actually just held the paddle wheel in place the vortex would cause it to rotate clockwise. These two effects exactly cancel each other out, and a paddle wheel dropped into a vortex does not rotate at all!

⎛ ∂ ∂ ∂ ⎞ Let’s get back to the del operator ∇ = , , . Since we’re treating it like a vector, let’s ⎝⎜ ∂x ∂y ∂z⎠⎟ try another operation that we do to vectors: scalar multiplication. Let’s multiply it by a function f, but let’s put the function on the right side of it. Here’s what we get: ⎛ ∂ ∂ ∂ ⎞ ⎛ ∂ ∂ ∂ ⎞ ⎛ ∂ f ∂ f ∂ f ⎞ ∇f = , , f = f , f , f = , , . Hmm, that looks suspiciously like ⎝⎜ ∂x ∂y ∂z⎠⎟ ⎝⎜ ∂x ∂y ∂z ⎠⎟ ⎝⎜ ∂x ∂y ∂z ⎠⎟ the gradient. Well, isn’t that what we wrote at the left end of the sequence of equations? So del gives us the gradient as well as the curl.

Four things come to mind. One: we had to multiply by putting del on the left of the function. ⎛ ∂ ∂ ∂ ⎞ ⎛ ∂ ∂ ∂ ⎞ What if we did f∇? Well this would be f , , = f , f , f which is just a ⎝⎜ ∂x ∂y ∂z⎠⎟ ⎝⎜ ∂x ∂y ∂z⎠⎟ bunch more derivatives waiting for something to differentiate. In other words, we have created a new operator. So ∇ is not really a vector that we can use like other vectors—operators don’t necessarily obey vector rules like commutativity with scalar multiplication. Order and grouping make a big difference! So be careful with the order of your symbols and where you put your parentheses. Similarly, we can take G × ∇, but it doesn’t give us the curl, it gives us a messy new operator waiting for stuff to work on.

The second thing is that the gradient is a kind of derivative. Well, the curl uses the same tools, so is it some kind of derivative as well? If it were, what would you guess that ∇ × (fG) should be? Looks like a product rule, so we might guess something like (f)(∇ × G) + ∇f × G. Or maybe it should be (f)(∇ × G) + G × ∇f? These are different because cross products change sign when you reverse the order of the terms. One of these happens to be right, but I’ll let you play around to figure out which one.

Third, what about other dimensions? In four dimensions could we have an operator like ⎛ ∂ ∂ ∂ ∂ ⎞ ∇ = , , , ? Well why not? If we multiply it by a function on the right, it gives us ⎝⎜ ∂x ∂y ∂z ∂w⎠⎟ the good old gradient! We lose curl, because cross products only work in three dimensions, but everything else seems to work the same.

And fourth, could we take a dot product, like ∇⋅G? This could work in any number of dimensions. Would it give us something useful?