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Date: Around 1993

I am taking a E&M course at UGA and would like help in understanding physically what the grad, div, and curl are. I can compute them, but I am not sure what they mean. Tell me about the vector field that they operate on.

Here is some simple guidelines. Grad is simplest. It operates on real-valued functions of several variables, and simply points in the direction of greatest increase of the function (the gradient) with magnitude proportional to the rate of increase in that direction. Div operates on vector-valued functions of the same number of variables, and produces a real- valued answer. The answer is positive at a particular point if the vector- valued function generally points outward from that point and is negative if the function generally points inward. Curl operates on 3-D vector-valued functions and produces a vector-valued answer. If you point your thumb (I cannot remember if it is right or left, but it does not matter much) along the "curl" then your fingers will curl in the general direction in which the function in question circles around the point where you evaluated the curl. A vector-valued function with zero curl is called "irrotational", and gradients of differentiable functions always have this property. Curl is the hardest to picture, so it is good to know some examples. Vortex motion in a liquid is the best, perhaps. If you look at the velocity as a function of position near a vortex, the curl of the velocity is a constant, equal to the "vorticity", pointing along the axis of the vortex. Also, curls appear in describing electromagnetism all the time - in particular, B = curl(A) defines magnetic fields from vector potentials.

Arthur Smith

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