The derivative of a single-variable function records the best linear approximation to the function near a point. For a map between Euclidean spaces, the same idea survives, but the linear approximation is no longer multiplication by a number. It is a [linear map](/page/Linear%20Map) from an input space to an output space, and the Jacobian matrix is the coordinate matrix of that linear map. This is the bridge between [derivatives](/page/Derivative), [matrices](/page/Matrix), and the local geometry of maps.

The Jacobian matrix is where multivariable calculus becomes computable. It lets the chain rule become matrix multiplication, turns local invertibility into a determinant condition in the [inverse function theorem](/theorems/51), and, in the square case, uses its determinant to measure infinitesimal volume distortion in the change of variables theorem. Rectangular Jacobians play an analogous geometric role through ranks, tangent vectors, and area factors for parametrised objects. In differential equations it is the matrix used to linearise a nonlinear vector field near an equilibrium. In geometry it is the coordinate-level shadow of the differential of a smooth map.

## Definition

Computations need entries rather than an abstract linear map. Since the input has $m$ standard coordinate directions and the output has $n$ standard coordinate components, the representing matrix must have $n$ rows and $m$ columns. This is the page's central object: the matrix that turns the first-order linear approximation of a map into something calculable.

[definition: Jacobian Matrix] Let $U \subset \mathbb{R}^m$ be open, let $a \in U$, and let $f: U \to \mathbb{R}^n$ be differentiable at $a$, with component functions $f_1, \ldots, f_n$. The Jacobian matrix of $f$ at $a$ is the matrix $Jf_a \in \mathbb{R}^{n \times m}$ representing $Df_a$ with respect to the standard bases of $\mathbb{R}^m$ and $\mathbb{R}^n$: \begin{align*} (Jf_a)_{ij} &= \partial_{x_j} f_i(a) = \frac{\partial f_i}{\partial x_j}(a), \qquad 1 \le i \le n,\ 1 \le j \le m. \end{align*} [/definition]

The notation in this matrix has two independent indexing jobs to do: $j$ selects an input direction, while $i$ selects the output quantity being measured in that direction. To make that entry formula unambiguous, a vector-valued map must first be read through its scalar coordinate outputs. Component functions are the bookkeeping device that turns the single statement "$f$ has $n$ outputs" into $n$ ordinary real-valued functions whose partial derivatives can be placed into rows.

[definition: Component Functions] Let $U \subset \mathbb{R}^m$ be a set, and let $f: U \to \mathbb{R}^n$ be a function. The component functions of $f$ are the functions $f_i: U \to \mathbb{R}$, for $1 \le i \le n$, such that \begin{align*} f(x) &= (f_1(x), \ldots, f_n(x)) \end{align*} for every $x \in U$. [/definition]

With this convention, the row index records the output component, and the column index records the input direction. Vectors are multiplied on the right:

\begin{align*} Df_a(h) &= Jf_a h, \qquad h \in \mathbb{R}^m. \end{align*}

The matrix also depends on a more intrinsic object: the total derivative. This is the analytic object that the Jacobian matrix represents. It answers the question: which linear map gives the best first-order prediction of the output change caused by a small input displacement?

[definition: Total Derivative] Let $U \subset \mathbb{R}^m$ be open, let $a \in U$, and let $f: U \to \mathbb{R}^n$ be a function. The total derivative of $f$ at $a$ is a linear map $Df_a: \mathbb{R}^m \to \mathbb{R}^n$ such that \begin{align*} f(a+h) &= f(a) + Df_a(h) + o(|h|) \end{align*} as $h \to 0$ in $\mathbb{R}^m$. Here $r(h)=o(|h|)$ means that the error term is negligible compared with the size of the displacement: $|r(h)|/|h| \to 0$ as $h\to 0$. [/definition]

Some questions about a map ask whether nearby points are separated, folded, or orientation-reversed. These questions require a square matrix, so the equal-dimensional case deserves a separate name.

[definition: Square Jacobian Matrix] Let $U \subset \mathbb{R}^n$ be open, let $a \in U$, and let $f: U \to \mathbb{R}^n$ be differentiable at $a$. The Jacobian matrix $Jf_a$ is called a square Jacobian matrix. [/definition]

Local volume change is governed by the determinant of the linear approximation. Naming the determinant separately gives a compact way to state inverse, orientation, and integration results.

[definition: Jacobian Determinant] Let $U \subset \mathbb{R}^n$ be open, let $a \in U$, and let $f: U \to \mathbb{R}^n$ be differentiable at $a$. The Jacobian determinant of $f$ at $a$ is \begin{align*} \det Jf_a. \end{align*} [/definition]

In standard Euclidean coordinates, the determinant is computed from the matrix $Jf_a$. More invariantly, a determinant can be assigned to a finite-dimensional endomorphism once the domain and codomain are identified; the coordinate volume and orientation interpretations depend on the surrounding choice of bases and Euclidean structure. This distinction is often quiet in multivariable calculus, but it becomes important when comparing coordinate systems or working on manifolds.

## Equivalent Characterisations

The entry formula for $Jf_a$ is useful for computation, while the approximation formula is useful for analysis. The point needing justification is that, under differentiability, the rectangular table of first partial derivatives is not merely a formal array: it is the coordinate matrix of the derivative map itself.

This identification is what lets the same symbol $Jf_a$ serve two roles without ambiguity: it is both the table of first partial derivatives and the matrix of the linear map that gives the best first-order approximation near $a$. The differentiability hypothesis is essential here. Without it, a displayed table of partial derivatives may exist without controlling the error term in the approximation formula, so the table need not represent the derivative. Once this identification is fixed for vector-valued maps, the scalar-valued case needs its own convention because the derivative lands naturally in the [dual space](/page/Dual%20Space).

Scalar-valued maps need a separate convention because their derivatives are linear functionals. The same partial derivatives can be viewed either as a row matrix acting on displacements or, after Euclidean identification, as the gradient vector.