Suppose I have $n$ linearly independent vector fields defined on $\mathbb{R}^n$. Under what circumstances are they $\partial/\partial_i$ for some global coordinate system on $\mathbb{R}^n$?
I'll tag this with algebraic topology, because I suspect that might come into the answer.
Let $\Bbb R^n$ be equipped with a frame $F$ that's the standard one at $0$ (for our mild convenience in a second). We want to know if there is a diffeomorphism $f: \Bbb R^n \to \Bbb R^n$ so that (writing the Jacobian using the God-given Euclidean coordinates) $df_x F_x = I$ (here, I'm writing frames as matrices, so $I$ is the identity matrix, representing the standard frame of the tangent space at $f(x)$.). You then want to know if you can solve the equation $df_x = F_x^{-1}$ at all; converting the set of $n$ vector fields $F^{-1}$ into a set of $n$ 1-forms, we first off need to know that they are exact. There's your first smooth obstruction.
Once you know that $F^{-1}$ is exact, if we specify $f(0) = 0$, $f$ exists and is completely determined by $df = F^{-1}$ (integrate along paths!). So now we actually have a function. Is it a diffeomorphism? Locally, it is ($df$ is pointwise invertible!), but we also have a properness condition - Hadamard's global inverse function theorem says that if $f$ is also proper, it is a diffeomorphism. But I don't really know any reasonable conditions that let us control properness of $f$ in terms of $F$ (but maybe someone else does?)