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. [definition: Jacobian Row of a Scalar Function] Let $U \subset \mathbb{R}^m$ be open, let $a \in U$, and let $f: U \to \mathbb{R}$ be differentiable at $a$. The Jacobian row of $f$ at $a$ is the $1 \times m$ matrix whose $j$th entry is $\partial_{x_j}f(a)$. [/definition] In applications, one often meets the partial derivatives first: a formula gives entries, and the calculation produces a matrix before differentiability has been separately checked. The danger is that a row of partial derivatives, or a full rectangular table of them, may fail to describe any genuine first-order approximation. The following criterion is the standard safeguard: it gives a practical regularity condition under which the computed table is warranted to be the Jacobian matrix of a differentiable map. [quotetheorem:327] The gradient of a scalar function carries the same partial derivatives as the Jacobian row but is usually identified with a vector in the domain. With Euclidean [inner product](/page/Inner%20Product) notation, \begin{align*} Df_a(h) &= \nabla f(a) \cdot h, \end{align*} so the Jacobian row is the transpose of the gradient column under the standard Euclidean identification. The hypothesis is stronger than the mere existence of the entries of the matrix. A table of partial derivatives at one point may not describe a valid linear approximation. ## Examples The most familiar Jacobian matrices come from nonlinear coordinate functions in low dimensions. They show how each row tracks one output component while each column tracks one input variable. [example: Nonlinear Map from the Plane to the Plane] Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be defined by \begin{align*} f(x_1,x_2) = (x_1^2x_2 + \sin x_2, e^{x_1-x_2}). \end{align*} Its component functions are $f_1(x_1,x_2)=x_1^2x_2+\sin x_2$ and $f_2(x_1,x_2)=e^{x_1-x_2}$. Differentiating each component with respect to each input variable gives \begin{align*} \partial_{x_1}f_1(x_1,x_2)=2x_1x_2. \end{align*} \begin{align*} \partial_{x_2}f_1(x_1,x_2)=x_1^2+\cos x_2. \end{align*} \begin{align*} \partial_{x_1}f_2(x_1,x_2)=e^{x_1-x_2}. \end{align*} \begin{align*} \partial_{x_2}f_2(x_1,x_2)=-e^{x_1-x_2}. \end{align*} Therefore the Jacobian matrix has first row coming from $f_1$ and second row coming from $f_2$: \begin{align*} Jf_{(x_1,x_2)} = ((2x_1x_2, x_1^2+\cos x_2), (e^{x_1-x_2}, -e^{x_1-x_2})). \end{align*} At $a=(1,0)$, the four entries become \begin{align*} \partial_{x_1}f_1(a)=2\cdot 1\cdot 0=0. \end{align*} \begin{align*} \partial_{x_2}f_1(a)=1^2+\cos 0=1+1=2. \end{align*} \begin{align*} \partial_{x_1}f_2(a)=e^{1-0}=e. \end{align*} \begin{align*} \partial_{x_2}f_2(a)=-e^{1-0}=-e. \end{align*} Thus \begin{align*} Jf_a=((0,2),(e,-e)). \end{align*} Also, \begin{align*} f(a)=f(1,0)=(1^2\cdot 0+\sin 0,e^{1-0})=(0,e). \end{align*} For a displacement $h=(h_1,h_2)$, the matrix action is \begin{align*} Jf_a h=(0h_1+2h_2,eh_1+(-e)h_2)=(2h_2,eh_1-eh_2). \end{align*} Hence the first-order approximation at $a$ is \begin{align*} f(a+h)=f(a)+Jf_a h+o(|h|)=(0,e)+(2h_2,eh_1-eh_2)+o(|h|). \end{align*} The first component changes to first order only in the $x_2$ direction at this point, while the second component changes by $e(h_1-h_2)$ to first order. [/example] This example is square, so determinant information is available. Since $\det Jf_a=-2e$, the linear approximation at $a$ is invertible and reverses orientation. Rectangular Jacobian matrices are just as important. They occur whenever a map changes dimension, such as a parametrised surface or a collection of measured quantities depending on fewer variables. [example: Rectangular Jacobian Matrix] Let $F: \mathbb{R}^2 \to \mathbb{R}^3$ be defined by \begin{align*} F(s,t) = (s,t,s^2+t^2). \end{align*} Its component functions are $F_1(s,t)=s$, $F_2(s,t)=t$, and $F_3(s,t)=s^2+t^2$. Differentiating each component with respect to each parameter gives \begin{align*} \partial_s F_1(s,t)=1. \end{align*} \begin{align*} \partial_t F_1(s,t)=0. \end{align*} \begin{align*} \partial_s F_2(s,t)=0. \end{align*} \begin{align*} \partial_t F_2(s,t)=1. \end{align*} \begin{align*} \partial_s F_3(s,t)=2s. \end{align*} \begin{align*} \partial_t F_3(s,t)=2t. \end{align*} Thus the Jacobian matrix has one row for each output component and one column for each input parameter: \begin{align*} JF_{(s,t)}=((1,0),(0,1),(2s,2t)). \end{align*} The first column is $(1,0,2s)$, which is the velocity obtained by varying $s$ while holding $t$ fixed. The second column is $(0,1,2t)$, which is the velocity obtained by varying $t$ while holding $s$ fixed. For a parameter displacement $h=(h_1,h_2)$, multiplication by the Jacobian gives \begin{align*} JF_{(s,t)}h=(1h_1+0h_2,0h_1+1h_2,2s h_1+2t h_2). \end{align*} Therefore \begin{align*} JF_{(s,t)}h=(h_1,h_2,2s h_1+2t h_2). \end{align*} So the rectangular Jacobian does not have a determinant, but it still sends each infinitesimal parameter displacement to the corresponding tangent vector in $\mathbb{R}^3$. [/example] The rectangular case cannot use a determinant of $JF_{(s,t)}$, but it still encodes tangent directions and ranks. This is the beginning of parametrised submanifold geometry. A major geometric use of Jacobian matrices is to measure how coordinate changes scale area or volume. Polar coordinates provide the standard first calculation. [example: Polar Coordinate Map] Let $P: (0,\infty) \times (0,2\pi) \to \mathbb{R}^2$ be the polar coordinate map \begin{align*} P(r,\theta) = (r\cos\theta, r\sin\theta). \end{align*} Its component functions are $P_1(r,\theta)=r\cos\theta$ and $P_2(r,\theta)=r\sin\theta$. Differentiating each component with respect to $r$ and $\theta$ gives \begin{align*} \partial_r P_1(r,\theta)=\cos\theta. \end{align*} \begin{align*} \partial_\theta P_1(r,\theta)=r(-\sin\theta)=-r\sin\theta. \end{align*} \begin{align*} \partial_r P_2(r,\theta)=\sin\theta. \end{align*} \begin{align*} \partial_\theta P_2(r,\theta)=r\cos\theta. \end{align*} Therefore \begin{align*} JP_{(r,\theta)}=((\cos\theta,-r\sin\theta),(\sin\theta,r\cos\theta)). \end{align*} Using $\det((a,b),(c,d))=ad-bc$, its Jacobian determinant is \begin{align*} \det JP_{(r,\theta)}=(\cos\theta)(r\cos\theta)-(-r\sin\theta)(\sin\theta). \end{align*} \begin{align*} \det JP_{(r,\theta)}=r\cos^2\theta+r\sin^2\theta. \end{align*} \begin{align*} \det JP_{(r,\theta)}=r(\cos^2\theta+\sin^2\theta)=r. \end{align*} Thus the infinitesimal area element in polar coordinates gains the factor $r$. On a region where $P$ is injective, such as one obtained by restricting the angular interval to avoid identifying the two sides of the same ray, this is the factor appearing in the change-of-variables formula \begin{align*} \int_E g(x,y)\,d\mathcal{L}^2(x,y)=\int_{P^{-1}(E)} g(r\cos\theta,r\sin\theta)\, r\, d\mathcal{L}^2(r,\theta) \end{align*} when the hypotheses of the [Area Formula (Classical)](/theorems/25) are satisfied. [/example] The factor $r$ is not a decorative correction. Annuli with larger radius contain more area per unit change in angle, and the Jacobian determinant records that local stretching. The next example shows why partial derivatives alone are not enough. A matrix of partial derivatives can exist at a point even when there is no valid first-order linear approximation there. [example: Existing Partial Derivatives without Differentiability] Define $f: \mathbb{R}^2 \to \mathbb{R}$ by $f(0,0)=0$ and, for $(x_1,x_2)\ne(0,0)$, \begin{align*} f(x_1,x_2) = \frac{x_1x_2}{\sqrt{x_1^2+x_2^2}}. \end{align*} We compute the two partial derivatives at the origin from their one-variable difference quotients. For $h\ne 0$, \begin{align*} \frac{f(h,0)-f(0,0)}{h}=\frac{0-0}{h}=0. \end{align*} Hence $\partial_{x_1}f(0,0)=0$. Similarly, for $h\ne 0$, \begin{align*} \frac{f(0,h)-f(0,0)}{h}=\frac{0-0}{h}=0. \end{align*} Hence $\partial_{x_2}f(0,0)=0$, so the only possible Jacobian row at $(0,0)$ is $(0,0)$. If $f$ were differentiable at $(0,0)$, the total derivative would therefore be the zero linear map, and the defining error condition would require \begin{align*} \frac{f(h)-f(0,0)-0}{|h|}\to 0 \end{align*} as $h\to 0$ in $\mathbb{R}^2$. Test this condition along $h=(t,t)$ with $t\ne 0$. First, \begin{align*} f(t,t)=\frac{t\cdot t}{\sqrt{t^2+t^2}}=\frac{t^2}{\sqrt{2t^2}}. \end{align*} Also, \begin{align*} |(t,t)|=\sqrt{t^2+t^2}=\sqrt{2t^2}. \end{align*} Therefore \begin{align*} \frac{f(t,t)-f(0,0)}{|(t,t)|}=\frac{t^2/\sqrt{2t^2}}{\sqrt{2t^2}}=\frac{t^2}{2t^2}=\frac{1}{2}. \end{align*} This quotient is constantly $\frac{1}{2}$ along the line $x_1=x_2$, so it does not tend to $0$. Thus the partial derivatives at the origin exist, but they do not assemble into a genuine first-order linear approximation there. [/example] This failure example explains the role of differentiability in the definition. The Jacobian matrix is not meant to be an arbitrary array of partial derivatives; it is meant to represent a genuine linear approximation. ## Properties The main algebraic property of Jacobian matrices is that they turn composition into matrix multiplication. This is the coordinate form of the chain rule. [quotetheorem:323] The order in this matrix product matters: the Jacobian of the outer map is evaluated at the intermediate point, and it multiplies the Jacobian of the inner map on the left. This is why Jacobians are useful bookkeeping devices rather than just tables of partial derivatives. They record how every infinitesimal direction is first transformed by one map and then transformed by the next. The differentiability hypotheses are essential here; if one only has partial derivatives without a genuine linear approximation, there is no reliable linear map for composition to multiply. In the square case, taking determinants of the chain-rule identity also explains why local volume factors multiply under successive coordinate changes. Local inverse questions ask whether the first-order approximation loses information. A square Jacobian matrix can fail to be invertible by collapsing some nonzero direction to first order, and the inverse function theorem gives the main positive result when no such collapse occurs. [quotetheorem:8859] The condition $\det J_f(a) \neq 0$ says that the derivative at $a$ is a linear isomorphism: to first order, no direction is flattened and no dimension is lost. The theorem upgrades that infinitesimal information to a local statement about the nonlinear map itself, giving nearby coordinates in which $f$ can be reversed. If the determinant vanishes, this conclusion can fail in simple ways, such as $x \mapsto x^2$ at $0$, where distinct nearby points have the same image. The result is local rather than global: a map may have invertible Jacobian everywhere on a region and still fail to be one-to-one on the whole region. Its main use here is to justify treating a nonvanishing Jacobian determinant as the signal that a change of coordinates is locally legitimate. For integration, the absolute value of the Jacobian determinant is the local volume factor. The sign tracks orientation; the measure formula needs nonnegative scaling. In the following statement, $d\mathcal{L}^n$ means integration with respect to $n$-dimensional [Lebesgue measure](/page/Lebesgue%20Measure). A set is measurable if its size is defined for this measure, and a function is integrable if its integral with respect to that measure is defined and finite in the usual Lebesgue sense. [quotetheorem:8860] The hypotheses prevent several common pathologies. Injectivity ensures that the same target point is not counted multiple times by the parametrisation, while the differentiability and nonvanishing determinant conditions ensure that the map behaves locally like an invertible linear change of scale. The absolute value appears because volume is unsigned: orientation reversal changes the sign of a determinant but not the size of a region. When the Jacobian determinant vanishes, the formula no longer describes an ordinary full-dimensional change of variables, because volume may be collapsed in some direction. Rectangular Jacobians need an invariant that still makes sense without determinants. Rank measures the dimension of the infinitesimal image and is the quantity behind implicit functions, parametrised surfaces, and regular value theory. [definition: Rank of the 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 $a$. The rank of the Jacobian matrix of $f$ at $a$ is \begin{align*} \operatorname{rank} Jf_a. \end{align*} [/definition] Rank is central in the [implicit function theorem](/theorems/52), the constant rank theorem, and the study of regular values. Its geometric meaning comes from viewing $Jf_a$ as the coordinate matrix of the linear map $Df_a$: once that identification is made, the rank is the dimension of the image of this linear approximation. The next issue is therefore not to prove a new rank formula, but to identify exactly which matrix represents $Df_a$ in the standard coordinates. [quotetheorem:7904] This result supplies that coordinate identification: the entries of $Jf_a$ are the partial derivatives of the component functions, and this matrix represents the derivative $Df_a$. With that bridge in place, ordinary linear algebra gives the geometric interpretation of rank as the dimension of the infinitesimal image. ## Relationship to Other Concepts The Jacobian matrix is a coordinate representation of the [total derivative](/page/Derivative). When $n=1$, it is closely related to the gradient. When $m=1$, it records the velocity vector of a parametrised curve. When $m=n$, its determinant controls local orientation and volume distortion. In linear algebra, the Jacobian matrix is just a matrix representing a linear map. The special feature is that the linear map arises from local approximation to a nonlinear function. This is why matrix operations such as determinant, rank, kernel, image, and eigenvalues acquire analytic meaning. In differential equations, if $f: U \to \mathbb{R}^n$ is a vector field and $x^* \in U$ satisfies $f(x^*)=0$, then $Jf_{x^*}$ is the linearisation matrix at the equilibrium. The eigenvalues of $Jf_{x^*}$ often govern the qualitative behaviour of nearby solutions. In differential geometry, the derivative of a smooth map $F:M\to N$ is written $dF_p:T_pM\to T_{F(p)}N$. In local coordinate charts, the matrix of $dF_p$ has the same entry pattern as a Jacobian matrix. Thus the Euclidean Jacobian is the model for coordinate computations on manifolds. The Jacobian determinant should not be confused with the derivative itself. The derivative is linear map data, the Jacobian matrix is coordinate matrix data, and the determinant is a scalar extracted only in the square case. Keeping these distinctions separate prevents many errors in multivariable analysis. ## References [Derivative](/page/Derivative). [Linear Map](/page/Linear%20Map). [Matrix](/page/Matrix). [Chain Rule](/theorems/323). [Inverse Function Theorem](/theorems/51). [Area Formula (Classical)](/theorems/25). Spivak, *Calculus on Manifolds* (1965). Rudin, *Principles of Mathematical Analysis* (1976). Munkres, *Analysis on Manifolds* (1991).