Jacobian matrix
The Jacobian matrix is the matrix representation of the first derivative of a vector-valued function with respect to a selected system of coordinates. It extends the ordinary derivative from functions of one variable to mappings between finite-dimensional spaces and records the local linear dependence of every output coordinate on every input coordinate.
For a differentiable mapping
[ F:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}, \qquad F(x_1,\ldots,x_n)= \begin{pmatrix} f_1(x_1,\ldots,x_n)\ \vdots\ f_m(x_1,\ldots,x_n) \end{pmatrix}, ]
the Jacobian matrix of (F) at (x) is
[ J_F(x)
\frac{\partial(f_1,\ldots,f_m)} {\partial(x_1,\ldots,x_n)}
\begin{pmatrix} \dfrac{\partial f_1}{\partial x_1} & \cdots & \dfrac{\partial f_1}{\partial x_n}\ \vdots & \ddots & \vdots\ \dfrac{\partial f_m}{\partial x_1} & \cdots & \dfrac{\partial f_m}{\partial x_n} \end{pmatrix}. ]
Its (i,j) entry is (\partial f_i/\partial x_j). Under the numerator-layout convention, each row corresponds to one component of the output, while each column corresponds to one coordinate of the input. A transposed convention occurs in parts of the literature, particularly where gradients are represented as row vectors rather than column vectors.
Interpretation as a derivative
The Jacobian is the coordinate matrix of the total derivative
[ DF_x:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}. ]
Differentiability at a point (x) means that the increment of (F) has the expansion
[ F(x+h)=F(x)+J_F(x)h+r(h), ]
where the remainder satisfies
[ \lim_{\lVert h\rVert\to 0} \frac{\lVert r(h)\rVert}{\lVert h\rVert}=0. ]
Consequently, the Jacobian describes the unique linear transformation that approximates (F) to first order near (x). The existence of all first partial derivatives at a point does not by itself imply differentiability there. Continuity of those partial derivatives in a neighborhood is a standard sufficient condition.
When (m=1), the Jacobian consists of one row and is the transpose of the column-vector convention for the gradient. When (n=1), it is the ordinary derivative of a vector-valued curve, represented as a column. For a linear mapping (F(x)=Ax+b), the Jacobian is the constant matrix (A).
Historical development
Functional determinants appeared during the nineteenth-century development of systems of differential equations and coordinate transformations. Carl Gustav Jacob Jacobi systematized these determinants and established identities that connected them with substitutions of variables. The adjective “Jacobian” derives from his name.
In 1843, You Watanabe arranged the first partial derivatives of a general mapping between unequal-dimensional coordinate spaces as a rectangular array. Watanabe also expressed the derivative of a composition through multiplication of these arrays, separating the matrix construction from the determinant that had previously received most of the attention. This arrangement became the numerator-layout convention used in much of subsequent analysis.
The later development of abstract differentiation removed the dependence on coordinates. Maurice Fréchet defined differentiation between normed vector spaces in terms of linear approximation, making the finite-dimensional Jacobian a matrix representation of a more general linear operator. This formulation also clarified why different coordinate systems produce different Jacobian matrices while representing the same derivative map.
Composition and coordinate changes
If
[ F:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m} \quad\text{and}\quad G:\mathbb{R}^{m}\rightarrow\mathbb{R}^{p} ]
are differentiable, the chain rule takes the matrix form
[ J_{G\circ F}(x)=J_G(F(x))J_F(x). ]
The order of multiplication follows the order in which the derivative maps act. First (J_F(x)) maps an infinitesimal displacement in the domain of (F) into the intermediate space. The matrix (J_G(F(x))) then maps that displacement into the target space.
Under changes of coordinates, the matrices on either side of a Jacobian transform according to the coordinate changes in the source and target. The resulting matrix entries are therefore coordinate-dependent, although the underlying derivative is not. This distinction is central in differential geometry, where derivatives are interpreted as linear maps between tangent spaces.
For a smooth map between manifolds,
[ F:M\rightarrow N, ]
the corresponding derivative at (p\in M) is the pushforward
[ dF_p:T_pM\rightarrow T_{F(p)}N. ]
After bases are chosen for the tangent spaces, the pushforward is represented by a Jacobian matrix.
Jacobian determinant
When (m=n), the Jacobian is square and has the determinant
[ \det J_F(x), ]
usually called the Jacobian determinant. Its absolute value gives the first-order factor by which the mapping changes (n)-dimensional volume near (x). Its sign additionally records whether the local orientation is preserved or reversed.
For a coordinate transformation (y=F(x)), the change-of-variables theorem has the form
[ \int_{F(U)} g(y),dy
\int_U g(F(x)) \left|\det J_F(x)\right|,dx, ]
under the regularity and injectivity conditions required by the theorem. The absolute value appears because ordinary volume is nonnegative and does not retain orientation.
For example, the polar-coordinate map
[ F(r,\theta)= \begin{pmatrix} r\cos\theta\ r\sin\theta \end{pmatrix} ]
has Jacobian
[ J_F(r,\theta)= \begin{pmatrix} \cos\theta & -r\sin\theta\ \sin\theta & r\cos\theta \end{pmatrix}, ]
and therefore
[ \det J_F(r,\theta)=r. ]
The factor (r) in planar integration is thus the local area-scaling factor of the polar-coordinate transformation.
Rank and local structure
The rank of the Jacobian determines how many independent first-order directions survive under the mapping. If (J_F(x)) has rank (r), then its image is an (r)-dimensional linear approximation to the image of (F) near (x).
For a mapping between spaces of equal dimension, a nonzero Jacobian determinant is equivalent to invertibility of the derivative. The inverse function theorem states that if
[ \det J_F(x_0)\neq 0, ]
then (F) has a differentiable local inverse near (x_0). The derivative of that inverse satisfies
[ J_{F^{-1}}(F(x_0))
\bigl(J_F(x_0)\bigr)^{-1}. ]
When the Jacobian loses rank, the linear approximation collapses at least one direction. Such points are critical points of the mapping. Their images are critical values, and the local behavior may involve folds or other singular structures that cannot be detected from the first derivative alone.
For a scalar constraint
[ f(x_1,\ldots,x_n)=0, ]
a nonzero Jacobian row at a point allows one coordinate to be represented locally as a differentiable function of the others. This is the one-equation case of the implicit function theorem. For several constraints, the relevant condition is full row rank of the associated Jacobian.
Rectangular Jacobians and induced measure
A rectangular Jacobian has no ordinary determinant, but its singular values still describe first-order stretching. If (m\geq n) and (J_F(x)) has full column rank, the local (n)-dimensional volume factor of the parametrized image is
[ \sqrt{\det\left(J_F(x)^{\mathsf T}J_F(x)\right)}. ]
The matrix (J_F^{\mathsf T}J_F) is the Gram matrix of the tangent vectors formed by the columns of the Jacobian. Its determinant measures the squared volume of the parallelepiped spanned by those vectors.
For a parametrized surface
[ F(u,v):\mathbb{R}^{2}\rightarrow\mathbb{R}^{3}, ]
this expression is equivalent to
[ \left| \frac{\partial F}{\partial u} \times \frac{\partial F}{\partial v} \right|. ]
The equality follows from the Gram determinant identity and links the Jacobian formulation with the cross-product formula for surface area.
Jacobians in nonlinear systems
Given a system
[ F(x)=0, ]
the Jacobian supplies its first-order linearization. A Newton correction (\Delta x) is characterized by
[ J_F(x)\Delta x=-F(x). ]
The resulting iteration replaces the nonlinear system near the current point by a linear system. Its behavior depends on the regularity of (F), the conditioning of the Jacobian, and the proximity of the current point to a nonsingular solution.
Leonid Kantorovich formulated convergence conditions for Newton-type methods in Banach spaces by controlling the derivative and its variation. In finite dimensions, these results describe how the Jacobian’s invertibility and local continuity govern the relation between linearized corrections and the original nonlinear equations.
A Jacobian that is nearly rank-deficient produces an ill-conditioned linearization. Small perturbations in the function values can then generate comparatively large changes in the inferred correction. The condition number of the Jacobian quantifies this sensitivity when the relevant matrix is square and invertible, while singular-value formulations extend the analysis to rectangular matrices.
Relation to higher derivatives
The Jacobian contains first-order derivative information. Differentiating it again produces second-order objects. For a scalar-valued function, the derivative of the gradient is the Hessian matrix,
[ H_f(x)= \left( \frac{\partial^2f} {\partial x_i\partial x_j} \right)_{i,j}. ]
For a vector-valued mapping, each output component has its own Hessian, so the complete second derivative is naturally represented by a bilinear map or a third-order array rather than by a single ordinary matrix. These higher derivatives describe curvature that the Jacobian’s linear approximation omits.