Curl of a Vector Field

Date: 05/22/99 at 01:03:01
From: R . Arumainathan
Subject: The curl of a vector field

Can  you explain what does the term curl means as a purely 
mathematical term, and how do we define it in terms of fluid 
mechanics? Does the curl of a vector field need to be only in three-
dimensional space? Can it be two dimensions? Please explain why or why 
not.

Thank you.

Date: 05/22/99 at 10:52:11
From: Doctor Anthony
Subject: Re: The curl of a vector field

The 'del' operator is (d/dx, d/dy, d/dz).

Just as we use D as the 'D' operator in differential equations, and 
apply normal algebraic rules in the way it is manipulated, so del is 
a VECTOR operator standing for the vector with components

     (d/dx, d/dy, d/dz)   these should of course be 'curly' d's 

and this vector can be manipulated like any other vector.

Because it is a vector we have  del(s) = grad s where s is a scalar.

                                del.(a) = div(a) where a is a vector. 

                                del x (a) = curl(a)         "

where . stands for scalar product and  x  stands for vector product.

So for curl(a) we have

    (d/dx, d/dy, d/dz) x (ai, aj, ak) = |i       j         k|
                                        |d/dx   d/dy    d/dx|
                                        |ai      aj       ak| 

You can use the same device for finding curl(a) in other dimensions.
  
- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/   

Date: 05/23/99 at 11:03:41
From: Doctor Mitteldorf
Subject: Re: The curl of a vector field

Dear Mr. Arumainathan,

I'd like to add a few words to Doctor Anthony's answer.  

Derivatives of a vector field are constructed from pairs of component 
directions. For example, the vector field has components (Fx, Fy, Fz) 
and each of these can be differentiated in the directions (x,y,z).  
The components that contribute to the curl are only those that have 
different directions for the two parts, because there's antisymmetry 
in the formula: for example d(Fx)/dy - d(Fy)/dx. So the three 
components of the curl are constructed from the three pairs x-y, y-z, 
and x-z.

In n dimensions there are n(n-1)/2 different combinations of 2 
components. For example, in two dimensions there's x-y and that's all 
n(n-1)/2 = 1. In three dimensions, there's x-y, y-z, and x-z, and n
(n-1)/2 = 3.  In 4 dimensions you have w-x, w-y, w-z, x-y, x-z, y-z, 
and n(n-1)/2 = 6.

This means that the curl in two dimensions is a scalar, a single 
number. The curl in 3 dimensions is a "vector" only by chance; if you 
identify x-y with z, z-x with y and y-z with x you get the familiar 
formula for the curl of a vector field. Technically, it is a "pseudo-
vector" meaning that when your coordinate system rotates, the curl 
rotates with it, but when you invert the coordinate system 
(x -> -x, y -> -y, z -> -z) a vector inverts, (v -> -v) while a 
pseudo-vector stays the same. The curl is composed of expressions like 
d(Fx)/dy, and when the coordinate system inverts, both Fx and dy add a 
minus sign, and the two minus signs cancel out.

In 4 or larger dimensions, the curl is an anti-symmetric tensor, 
consisting of 6 components like d(Fx)/dy - d(Fy)/dx. (In 3 dimensions, 
there is the natural correspondence between antisymmetric tensors and 
pseudovectors.)

This story is actually the tip of a very large iceberg. Einstein 
taught himself to differentiate vector fields in 4 dimensions, and by 
the time he was finished, he knew that there was only one form the 
gravitational field equations could take; hence was born the General 
Theory of Relativity.

You can read more about differentiation in n dimensions in math texts 
on "differential geometry", or in physics books on General 
Relativity. Spivack's "Calculus on Manifolds" is a classic.

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/