Jacobian matrix and determinant

In vector calculus, the Jacobian matrix (, ) 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 (1804-1851).

The Jacobian matrix is the natural generalization of the derivative and the differential of a usual function to vector valued functions of several variables. 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.

Definition

Let

f

:

R

n

→

R

m

{\textstyle \mathbf {f} :\mathbb {R} ^{n}\to \mathbb {R} ^{m}}

be a function such that each of its first-order partial derivatives exists on

R

n

{\textstyle \mathbb {R} ^{n}}

. This function takes a vector ⁠

x

=

(

x

1

,

…

,

x

n

)

∈

R

n

{\displaystyle \mathbf {x} =(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}}

⁠ as input and produces the vector ⁠

f

(

x

)

=

(

f

1

(

x

)

,

…

,

f

m

(

x

)

)

∈

R

m

{\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 Jf, is the ⁠

m

×

n

{\displaystyle m\times n}

⁠ matrix whose (i, j) entry is

∂

f

i

∂

x

j

;

{\textstyle {\frac {\partial f_{i}}{\partial x_{j}}};}

explicitly

J

f

=

[

∂

f

∂

x

1

⋯

∂

f

∂

x

n

]

=

[

∇

T

f

1

⋮

∇

T

f

m

]

=

[

∂

f

1

∂

x

1

⋯

∂

f

1

∂

x

n

⋮

⋱

⋮

∂

f

m

∂

x

1

⋯

∂

f

m

∂

x

n

]

{\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

∇

T

f

i

{\displaystyle \nabla ^{\mathsf {T}}f_{i}}

is the transpose (row vector) of the gradient of the

i

{\displaystyle i}

-th component.