I attempt to understand the identification and applications of differential forms in vector calculus from Loring Tu's An Introduction to Manifolds (2nd edition) (page no. 43). The author states the following preposition.
Proposition C. On $\mathbb{R}^3$, a vector field $\textbf{F}$ is the gradient of some scalar function $f$ if and only if $\text{curl} \, \textbf{F} = 0$.
If I have understood correctly, then this Proposition C describes the following two conditions:
- Condition 1: $\textbf{F}$ is defined on $\mathbb{R}^3$,
- Condition 2: $\text{curl} \, \textbf{F} = 0$,
which imply $\textbf{F} = \text{grad} \, f$, where $f: \mathbb{R}^3 \to \mathbb{R}$.
My Question: Could you please supply an example of a vector field $\textbf{F}: U \to \mathbb{R}^3$ with $\text{curl} \, \textbf{F} = 0$ such that for $U = \mathbb{R}^3$, $\textbf{F}$ can be written as the gradient of some scalar function but for some $U \neq \mathbb{R}^3$, $\textbf{F}$ can not be written as the gradient of some scalar function?
Please note that the example supplied in the book chooses a vector field $\textbf{F}$ which is not defined on $\mathbb{R}^3$. As a result, I don't see clearly how Condition 1 plays its role.
There can be no such example, since on $U$ $\mathbf F$ could be written as the gradient of the same scalar function (restricted to $U$).
Rather, the condition is necessary because there are vector fields with zero curl that are not defined on all of $\mathbb R^3$ and are not the gradient of a scalar function. An example is afforded by
$$ \mathbf F(\mathbf r)=\frac1{x^2+y^2}\pmatrix{-y\\x\\0}\;, $$
defined everywhere except on the $z$ axis, which has zero curl but is not the gradient of a scalar function since its line integral along a circle around the $z$ axis is non-zero.