This isn't a question about the actual mathematics per se, but more the notation of it.
In Manton and Sutcliff's book 'Topological Solitons', the use the notation
$\nabla \phi^{(\mu)} (\textbf{x}) = \nabla (\phi(\mu\textbf{x})) = \mu \nabla \phi(\mu\textbf{x})$
where we have applied the chain rule for the second equality. My question is, is this the correct way to notate the chain rule when you just have an arbitrary function $\phi$? Something about it doesn't sit right with me, for lack of a better term, as the $\nabla \phi(\mu(\textbf{x}))$ would suggest to me that we still need to apply the chain rule to compute the gradient.
Any clarification would be greatly appreciated!
Presumably they use $\nabla \phi$ to denote just the gradient of $\phi$, i.e., “the gradient of $\phi(\mathbf{x})$”, so that $\nabla \phi(\mu \mathbf{x})$ is what you get when you first compute that gradient and then substitute $\mu \mathbf{x}$ in $\mathbf{x}$'s place: $(\nabla \phi)(\mu \mathbf{x})$.
Whereas the notation $\nabla (\phi(\mu \mathbf{x}))$ means the gradient of the function $\mathbf{x} \mapsto \phi(\mu \mathbf{x})$.