Jacobian
Given a vector field \(\mathbf{f} : \mathbb{R}^m \rightarrow \mathbb{R}^n\), its Jacobian \(\mathbf{J}\), sometimes denoted \(\mathbf{J}_{\mathbf{f}}\) or \(\frac{\partial(f_1,\cdots f_m)}{\partial (x_1,\cdots x_n)}\) is the \(m \times n\) matrix such that \(\mathbf{J}_{ij} = \frac{\partial f_i}{\partial x_j}\).
If we have a vector field \(\mathbf{F} : \mathbb{R}^n \rightarrow \mathbb{R}^m\) that is a composition of functions such that \(\mathbf{F}(\mathbf{x}) = \mathbf{f}(\mathbf{g}(\mathbf{x}))\), then we have that \(\mathbf{J}_\mathbf{F} = \mathbf{J}_\mathbf{f}(\mathbf{\mathbf{g}(\mathbf{x})})\mathbf{J}_\mathbf{g}(\mathbf{x})\).