Curl of a Vector FieldDate: 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/ |
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