Jacobian: General Definition

The Jacobian matrix can be defined for functions between arbitrary finite-dimensional vector spaces. For $\mathbf{f}: \mathbb{R}^n \to \mathbb{R}^m$, the Jacobian $J\mathbf{f}(\mathbf{a})$ is the unique linear map from $\mathbb{R}^n$ to $\mathbb{R}^m$ that best approximates $\mathbf{f}$ near $\mathbf{a}$.

The defining property is: $\lim_{\mathbf{h}\to\mathbf{0}} \frac{\|\mathbf{f}(\mathbf{a}+\mathbf{h}) - \mathbf{f}(\mathbf{a}) - J\mathbf{f}(\mathbf{a})\mathbf{h}\|}{\|\mathbf{h}\|} = 0$. In coordinates, the matrix of this linear map has $(i,j)$ entry $\partial f_i / \partial x_j$.

For scalar functions ($m=1$), the Jacobian is a $1 \times n$ row vector — the gradient transposed. The gradient itself is the column vector $\nabla f = (\partial f/\partial x_1, \ldots, \partial f/\partial x_n)^T$. Many authors write $Df$ for the row vector and $\nabla f$ for the column vector; they contain the same information.