Proposition

If $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$, define a vector field $\mathbf{f}$ by $\mathbf{f}(p) = f(p)_p \in \mathbb{R}^n_p$.

Show that every vector field $F$ on $\mathbb{R}^n$ is of the form $\mathbf{f}$ for some $f$.
Show that $\div \mathbf{f} = \trace f’$.

Solution

1

Define $f_i:\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $f_i(p) = F^i(p)$ for each $i$. Let $f(p) = (f_1(p), \cdots, f_n(p))$. Then $F = \mathbf{f}$.

2

\[\begin{align*} \div \mathbf{f} &= D_1F^1 + \cdots + D_nF^n \\ &= D_1f_1 + \cdots + D_nf_n \\ &= f'_{1, 1} + \cdots + f'_{n, n} \\ &= \trace f'. \end{align*}\]