In vector calculus, the Jacobian matrix (/dʒəˈkoʊbiən/, /dʒɪ-, jɪ-/) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. If this matrix is square, that is, if the number of variables equals the number of components of function values, then its determinant is called the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian. They are named after Carl Gustav Jacob Jacobi.

The Jacobian matrix is the natural generalization to vector valued functions of several variables of the derivative and the differential of a usual function. This generalization includes generalizations of the inverse function theorem and the implicit function theorem, where the non-nullity of the derivative is replaced by the non-nullity of the Jacobian determinant, and the multiplicative inverse of the derivative is replaced by the inverse of the Jacobian matrix.

The Jacobian determinant is fundamentally used for changes of variables in multiple integrals.

Let ${\textstyle \mathbf {f} :\mathbb {R} ^{n}\to \mathbb {R} ^{m}}$ be a function such that each of its first-order partial derivatives exists on ${\textstyle \mathbb {R} ^{n}}$. This function takes a point ⁠${\displaystyle \mathbf {x} =(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}}$⁠ as input and produces the vector ⁠${\displaystyle \mathbf {f} (\mathbf {x} )=(f_{1}(\mathbf {x} ),\ldots ,f_{m}(\mathbf {x} ))\in \mathbb {R} ^{m}}$⁠ as output. Then the Jacobian matrix of f, denoted J_f, is the ⁠${\displaystyle m\times n}$⁠ matrix whose (i, j) entry is ${\textstyle {\frac {\partial f_{i}}{\partial x_{j}}};}$ explicitly $${\displaystyle \mathbf {J_{f}} ={\begin{bmatrix}{\dfrac {\partial \mathbf {f} }{\partial x_{1}}}&\cdots &{\dfrac {\partial \mathbf {f} }{\partial x_{n}}}\end{bmatrix}}={\begin{bmatrix}\nabla ^{\mathsf {T}}f_{1}\\\vdots \\\nabla ^{\mathsf {T}}f_{m}\end{bmatrix}}={\begin{bmatrix}{\dfrac {\partial f_{1}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\dfrac {\partial f_{m}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{m}}{\partial x_{n}}}\end{bmatrix}}}$$ where ${\displaystyle \nabla ^{\mathsf {T}}f_{i}}$ is the transpose (row vector) of the gradient of the ${\displaystyle i}$-th component.

The Jacobian matrix, whose entries are functions of x, is denoted in various ways; other common notations include Df, ${\displaystyle \nabla \mathbf {f} }$, and ${\textstyle {\frac {\partial (f_{1},\ldots ,f_{m})}{\partial (x_{1},\ldots ,x_{n})}}}$. Some authors define the Jacobian as the transpose of the form given above.

The Jacobian matrix represents the differential of f at every point where f is differentiable. In detail, if h is a displacement vector represented by a column matrix, the matrix product J(x) ⋅ h is another displacement vector, that is the best linear approximation of the change of f in a neighborhood of x, if f(x) is differentiable at x. This means that the function that maps y to f(x) + J(x) ⋅ (y – x) is the best linear approximation of f(y) for all points y close to x. The linear map h → J(x) ⋅ h is known as the derivative or the differential of f at x.

When ${\textstyle m=n}$, the Jacobian matrix is square, so its determinant is a well-defined function of x, known as the Jacobian determinant of f. It carries important information about the local behavior of f. In particular, the function f has a differentiable inverse function in a neighborhood of a point x if and only if the Jacobian determinant is nonzero at x (see inverse function theorem for an explanation of this and Jacobian conjecture for a related problem of global invertibility). The Jacobian determinant also appears when changing the variables in multiple integrals (see substitution rule for multiple variables).

When ${\textstyle m=1}$, that is when ${\textstyle f:\mathbb {R} ^{n}\to \mathbb {R} }$ is a scalar-valued function, the Jacobian matrix reduces to the row vector ${\displaystyle \nabla ^{\mathsf {T}}f}$; this row vector of all first-order partial derivatives of ⁠${\displaystyle f}$⁠ is the transpose of the gradient of ⁠${\displaystyle f}$⁠, i.e. ${\displaystyle \mathbf {J} _{f}=\nabla ^{\mathsf {T}}f}$. Specializing further, when ${\textstyle m=n=1}$, that is when ${\textstyle f:\mathbb {R} \to \mathbb {R} }$ is a scalar-valued function of a single variable, the Jacobian matrix has a single entry; this entry is the derivative of the function ⁠${\displaystyle f}$⁠.