Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Is there a good measure theoretic definition of curl?

To give an idea of the sort of equation that I'm looking for, here's now I define grad and div. For the gradient, say we are given a Fréchet differentiable function $f:X\rightarrow\mathbb{R}$, then we can define $\nabla f(x)$ to be the element in $X$ such that $$ \langle \nabla f(x),\eta \rangle = f^\prime(x)\eta. $$ Hence, $\nabla f(x)$ is the Riesz representative of the Fréchet derivative (Note, we've assumed a Hilbert space.) For the divergence, we have $f:X\rightarrow X$ and can define $$ \nabla \cdot f(x) = \lim_{\Omega\rightarrow \{x\}}\frac{1}{\mu(\Omega)} \int_{\Omega} \nabla\cdot f(x) d\mu =\lim_{\Omega\rightarrow \{x\}}\frac{1}{\mu(\Omega)} \int_{\partial \Omega} f(x)\cdot n d\mu. $$ Here, $\lim_{\Omega\rightarrow \{x\}} 1/\mu(\Omega)$ is shorthand for $\lim_{k\rightarrow \infty} 1/\mu(\Omega_k)$ where $\Omega_{k+1}\subseteq\Omega_k$ and $\lim_{k\rightarrow\infty}\Omega_k = \{x\}$. In addition, $\mu$ denotes some measure. In any case, the first equality follows from the Lebesgue differentiation theorem and the second follows from integration by parts. In this way, we require that $X$ be finite dimensional.

Now, I'd like to get something along these lines for curl. More specifically, I'm interested in a definition of curl that does not use differential forms or the exterior calculus.

I haven't checked this closely, but if it helps, I believe that for $f:X\rightarrow X$, we have that $$ (\nabla \times f(x))\times \delta x = f^\prime(x)\delta x-f^\prime(x)^*\delta x $$ where $f^\prime(x)^*$ denotes the adjoint of the Fréchet derivative of $f$ at $x$. In other words, the cross product between the curl of $f$ and $\delta x$ is the antisymmetric part of the Fréchet derivative.

share|cite|improve this question
2  
What about the definition on the Wikipedia page? en.wikipedia.org/wiki/Curl_%28mathematics%29 – Thomas Nov 26 '12 at 16:06
1  
Not sure if this might be useful (probably not), but you should get the curl if you replace $\cdot$ with $\times$ in the definition of divergence. This holds in classical vector calculus (cfr. Schey, Div grad curl and all of that, Ex. III-29), and should extend to general measure-theoretic setting. – Giuseppe Negro Jan 7 '13 at 2:16
1  
Maybe this answer helps: mathoverflow.net/questions/52829/… – Christian Blatter Sep 10 '13 at 15:44

Let $v=(v_i)$ be a vector field. If $c=(c_i)$ is the curl of $v$, as you did with the divergence, you can write:

$$ c_i(x) = \lim_{A_i\to \{x\}}\frac{1}{\mu(A_i)} \int_{A_i}(\nabla\times v)_i\,d\mu = \lim_{A_i\to \{x\}}\frac{1}{\mu(A_i)} \int_{\partial A_i} (v\times dx)_i, $$

where: $$ (v\times dx)_i = \sum_{j,k}\epsilon_{ijk} v_j\,dx_k $$

are the components of the usual cross product, and $A_i$ are flat surfaces with smooth boundary, orthogonal to $dx_i$. For example, $A_1$ is parallel to the $yz$ plane, and oriented in that way.

Note: integration by parts (componentwise) is exactly equivalent to exterior differentiation and Stokes theorem.

share|cite|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.