# 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 $$\mathbf{J}_\mathbf{F} = \mathbf{J}_\mathbf{f}(\mathbf{\mathbf{g}(\mathbf{x})})\mathbf{J}_\mathbf{g}(\mathbf{x})$$.

## Thoughts

Created: 2022-03-13 Sun 21:44

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