Let $\mathbf{v}(\mathbf{x})\in \mathbb{R}^n$ is a vector field. Assume that we perform a change of variables $\tilde{\mathbf{x}} := (\tilde{x}_1, \tilde{x}_2, \dots, \tilde{x}_n) = T(x_1, x_2, \dots, x_n)$ where $T: \mathbb{R}^n \to \mathbb{R}^n$ is a bijection, then the vector field $\mathbf{v}(\mathbf{x})$ changes to $\tilde{\mathbf{v}}(\tilde{\mathbf{x}})$. My question is: What is the relation between $\mathbf{v}(\mathbf{x})$ and $\tilde{\mathbf{v}}(\tilde{\mathbf{x}})$?
To clearly understand, let the transformation $T$ be something simple, like $T(\mathbf{x}) = \mathbf{Ox}$, where $\mathbf{O}$ is an orthogonal matrix. This transformation is just pure rotation around the origin. So the new vector field $\tilde{\mathbf{v}}(\tilde{\mathbf{x}})$ is also just the rotation of $\mathbf{v}(\mathbf{x})$. But we not only rotate the coordinate, we also rotate the vector assigned to each point in $\mathbb{R}^n$. Hence, we have the following relation
$$ \tilde{\mathbf{v}}(\tilde{\mathbf{x}}) = \mathbf{O}\mathbf{v}(\mathbf{O}^{-1}\tilde{\mathbf{x}}) = \mathbf{O}\mathbf{v}(\mathbf{x}) $$
But what about the general case? From the above argument, I think the relation might be $\tilde{\mathbf{v}}(\tilde{\mathbf{x}}) = T(\mathbf{v}(\mathbf{x}))$, but I don't know whether that is true or not. If not, then does it rely on the Jacobian matrix $\mathbf{J}_T$ of $T$?