In a classical electrodynamics textbook (Griffiths), it is mentioned that even though the electric field function, $E:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$, is a (3D) vector valued function, the amount of information needed to fully describe it is equivalent to the amount needed to describe a scalar valued function, and this is because it is the gradient of an electric potential function, $U:\mathbb{R}^{3}\rightarrow \mathbb{R}$ and so is completely determined by it. The author goes on to explains that this is so because $E$ has the property that its curl is zero evevrywhere, and this is what restricts the freedom in determining $E$ (and what enables it to be the gradient of a scalar function in the first place).
This is all fine (and fun), but I find myself unable to answer this question: why does the imposition of zero curl provide an amount of information exactly equivalent to two scalar-valued functions (no more, no less)?
An example of the sort of reasoning that leads me to other conclusions: since zero curl is equivalent to equating three pairs of partial derivatives ($\partial E_{x}/\partial y = \partial E_{y}/\partial x$ and so on), leaving 6 out of 9 partial derivatives "free", it seems that "one third of the amount of information" required to describe $E$ has been taken up, as opposed "two thirds"…
I am of course assuming here that the first order partial derivatives determine the behavior of the function (perhaps up to a constant as in the 1D case?), and that these partial derivatives are independent of each other and thus provide equal amounts of information. Is either of these assumptions wrong? Is my whole reasoning off?
Any insight into this question and a possible answer would be appreciated.
*The question is, of course, a general one about vector-valued functions, with $E$ just being a particular case.
**I have not mentioned all the obvious assumption of smoothness necessary for everything to be defined..
***I realise that I'm playing fast-and-loose with the word "information", and that my whole question is very un-formal. Any reference to an area of mathematics which may perhaps put such intuitions on rigorous footing are more then welcome.
Edit: I am not asking why (or when) a vector field having zero curl is equivalent to it being a gradient of some scalar field. I am asking about the amount of information we get about a vector field when we determine its curl.
Best Answer
If the curl were an arbitrary vector field, it would “contain as much information as the field itself”. (I'm using scare quotes because this is all on the same hand-waving level of rigorosity that your arguments address.)
However, the curl satisfies $\nabla\cdot(\nabla\times E)=0$. Thus, we have one scalar constraint on the curl, which reduces the number of free scalar functions in the curl from $3$ to $2$, so specifying the curl reduces the number of free scalar functions in $E$ from $3$ by $2$ to $1$.
Your considerations about the partial derivatives don't work out because the partial derivatives aren't independent. If they were, that would replace one scalar function by three scalar functions. The partial derivatives are subject to $\frac{\partial^2}{\partial x\partial y}=\frac{\partial^2}{\partial y\partial x}$, and likewise for the other two pairs of coordinates.