gradient, divergence, curl

[AriAstronomer]

Hey everyone,
Sorry to throw some technical math on you, but it's been bothering me a while.
So I'm a third year undergrad, and have thoroughly been using gradients, curls, and divergences now. I know that the purpose of the gradient is to measure the rate of maximum change, and on a contour map, it would be the steepest descent down a mountain. But what about the divergence and curl? What is the conceptual significance of them? I asked my prof, and he kind of swept it under the rug, saying divergence has to deal with rate of change, and curl has to do with rotational motion, but if there are better answers out there I would really appreciate it.
Thanks in advance.

[Ken G]

H But what about the divergence and curl? What is the conceptual significance of them?

The significance of divergence is that of a local source (or sink, if negative). A vector field that has nonzero divergence at some point (and note divergence is just a number at each point in the vector field) is like a flow that would require a source at that point, of that strength, to maintain that flow field in a steady state. A classic example is electric (or gravitational) field-- where there is charge, the field "diverges" away (or toward) the charge, and where there is no charge, the field doesn't disappear, it continues on with zero divergence like the water flying away from a sprinkler (all the divergence is at the sprinkler). Mathematically, divergence can be pictured best with "Gauss' law", which says that if you integrate a divergence over a volume, you get the net flux passing through the closed surface of that volume. In the limit as you shrink the volume to zero, and divide by that volume, you get the local divergence.

Curl measures the tendency of a vector field to rotate around a point. It means if you had a cork floating in the vector field, being carried along by it, would the cork spin as it translates along? Mathematically, you form a closed loop in the flow field, and integrate the flow vector along the closed loop. Then make that loop smaller and smaller, and keep dividing by its length, and in the limit as it shrinks to zero, you get the local curl. Curl also is not just a number-- that's only its magnitude. Since there is some freedom as how to orient the loop in a 3D flow field, we get a different curl for each orientation, and what we get is the component, perpendicular to the loop, of the curl vector. The place it shows up is when you have an electric current, it generates a local curl in the B field, and that curl aligns with the current. So just like with the E field, having zero curl doesn't make the field go away, it just makes it continue without any vorticity.

[korjik]

Hey everyone,
Sorry to throw some technical math on you, but it's been bothering me a while.
So I'm a third year undergrad, and have thoroughly been using gradients, curls, and divergences now. I know that the purpose of the gradient is to measure the rate of maximum change, and on a contour map, it would be the steepest descent down a mountain. But what about the divergence and curl? What is the conceptual significance of them? I asked my prof, and he kind of swept it under the rug, saying divergence has to deal with rate of change, and curl has to do with rotational motion, but if there are better answers out there I would really appreciate it.
Thanks in advance.

Take a sink. I mean a real sink with water in it. Pull the plug at the bottom. The water will start swirling around and down the drain. If you take any little bit of the water, you will have two components to the velocity. One will be the swirl around the drain, the other the motion towards the drain. The swirl is the curl and the motion towards the drain is the (negative) divergence.

That is a very bare bones description, but I hope it helps the visualizing.

[AriAstronomer]

Thanks guys. Your explanations helped beautifully. The thing that finally clicked for me was the thing Ken G put in brackets (all the divergence is at the sprinkler). The drain is a really nice way to visualize it.

[Gsquare]

Hey everyone,
Sorry to throw some technical math on you, but it's been bothering me a while.

.. But what about the divergence and curl? What is the conceptual significance of them? I asked my prof, and he kind of swept it under the rug, saying divergence has to deal with rate of change, and curl has to do with rotational motion, but if there are better answers out there I would really appreciate it.

There's a great little book called, "Div, Grad, Curl and All That" by Schey ..
see here to buy..
http://www.amazon.com/Div-Grad-Curl-.../dp/0393969975

Or you can downoad it....
http://www.10xdownloads.com/your-fil...%20Calcul&cr=1

G^2