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Let $(M,h)$ be a smooth Riemannian manifold of dimension $d\geq 1$ with smooth metric. Set $X:=\{A= \mbox{smooth vector field s.t. } div_h A=0 \}$.

Then $X$ is an infinite dimensional vector space. How to prove this?

Comment: I am aware of the question

How to prove the space of divergence-free vector fields on a manifold is infinite dimensional?

but the proof therein does not convince me completely: I cannot see why one should be able to write locally $dV_h= dx^1\cdots dx^d$

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