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Consider a generalised position vector in $\mathbb{R}^3$, e.g. $\mathbf{r} = \langle x,y,z \rangle$. When I write "$\mathbf{g} = \mathbf{r}$", do I define a vector field $\mathbf{g}$, or do I simply define another position vector $\mathbf{g}$?

To my taste, "$\mathbf{g}(\mathbf{r}) = \mathbf{r}$" would rather indicate definition of the vector field, whereas "$\mathbf{g} = \mathbf{r}$" would rather define another identical position vector.

If it is really ambiguous, maybe the definition sign, i.e. "$:=$", could help to keep those two possibilities apart?

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If we write $a=b$, then we are saying that $a$ and $b$ are exactly the same object. If $\renewcommand{vec}[1]{\mathbf{#1}}\vec{r} = \langle x, y, z \rangle$ is a general position vector, and we write $\vec{g} = \vec{r}$, then we are not saying that $\vec{g}$ is a vector field, and we are not saying that $\vec{g}$ is an identical copy of $\vec{r}$. We are saying that $\vec{g}$ is precisely, exactly, identically the same mathematical object as $\vec{r}$. There should be no ambiguity here.

If you want $\vec{g}$ to denote a vector field, then it would be appropriate to either say "$\vec{g}$ is a vector field" or to write $$\vec{g} : \mathbb{R}^n \to \mathbb{R}^n,$$ then specify the action of $\vec{g}$ on a general vector $\vec{r}$, for example $\vec{g}(\vec{r}) = \vec{r}$.

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