Ask A Scientist Mathematics Archive

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.
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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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