

Curl, Div and Grad Vectorvalued Functions
Name: Name
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1993
Question:
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.
Replies:
Here is some simple guidelines. Grad is simplest. It operates on
realvalued 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
vectorvalued 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 3D vectorvalued
functions and produces a vectorvalued 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
vectorvalued 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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Update: June 2012

