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The Jacobian for Vector-Valued Functions

The derivative of a vector-valued function $\mathbf{f}: \mathbb{R}^n \to \mathbb{R}^m$ at a point $\mathbf{a}$ is the Jacobian matrix $J\mathbf{f}(\mathbf{a}) \in \mathbb{R}^{m \times n}$ . Row $i$ of this matrix is the gradient (transposed) of the $i$ -th component function $f_i$ .

The Jacobian is defined by the best linear approximation property: $\mathbf{f}(\mathbf{a}+\mathbf{h}) \approx \mathbf{f}(\mathbf{a}) + J\mathbf{f}(\mathbf{a})\mathbf{h}$ with error $o(\|\mathbf{h}\|)$ as $\mathbf{h} \to \mathbf{0}$ . The Jacobian is the unique $m\times n$ matrix satisfying this.

Computing the Jacobian: fill row $i$ with the partial derivatives of $f_i$ with respect to $x_1, \ldots, x_n$ . Entry $(i,j)$ is $\partial f_i / \partial x_j$ .

Formal View

Definition 13.2 — Jacobian Matrix of a Vector-Valued Function

The Jacobian matrix of

\mathbf{f} = (f_1,\ldots,f_m)^T: D \subseteq \mathbb{R}^n \to \mathbb{R}^m

\mathbf{a}

is:

J\mathbf{f}(\mathbf{a}) = \begin{bmatrix} \partial f_1/\partial x_1 & \cdots & \partial f_1/\partial x_n \\ \vdots & \ddots & \vdots \\ \partial f_m/\partial x_1 & \cdots & \partial f_m/\partial x_n \end{bmatrix}_{\mathbf{x}=\mathbf{a}} \in \mathbb{R}^{m\times n}

Special cases: $m=1$ gives a $1\times n$ row vector (gradient transposed); $n=m$ gives an $n\times n$ square matrix whose determinant is the Jacobian determinant.

Example 13.2 — Computing a Jacobian

For

\mathbf{f}(x,y) = (x^2+y^2, xy)^T

J\mathbf{f} = \begin{bmatrix} 2x & 2y \\ y & x \end{bmatrix}

(1,2)

J\mathbf{f}(1,2) = \begin{bmatrix}2&4\\2&1\end{bmatrix}

Interactive Visualization

Matrix-Vector Multiplication

Why This Matters

The Jacobian is the multivariable derivative and appears in the chain rule, change-of-variables, and all first-order analysis of nonlinear maps.

Backpropagation in neural networks computes Jacobians of layer-wise functions
Change of variables in integration uses the Jacobian determinant
Inverse function theorem: a map is locally invertible iff its Jacobian is invertible

Learning Resources

The Jacobian Matrix

3Blue1Brown

Visual intuition for the Jacobian as the multivariable derivative.

14 min

Computing Jacobian Matrices

Khan Academy

Step-by-step computation of Jacobian matrices for vector-valued functions.

12 min

Quiz

Question 1

For $\mathbf{f}(x,y,z) = (x^2z, y+z^2)^T$ , the Jacobian $J\mathbf{f}$ is a matrix of size:

Question 2

For $\mathbf{f}(x,y,z) = (x^2z, y+z^2)^T$ at the point $(1,1,1)$ , the $(1,3)$ entry of $J\mathbf{f}$ (row 1, col 3) is:

Common Mistakes

Transposing the Jacobian — rows correspond to output components, columns to input variables.
Computing $\partial f_j/\partial x_i$ instead of $\partial f_i/\partial x_j$ .
Forgetting that the Jacobian is evaluated at a specific point, not a general formula (though the formula is useful).