Jacobian

In mathematics, the Jacobian of a coordinate transformation is the determinant of the functional matrix of Jacobi. This matrix consists of partial derivatives. The Jacobian appears as the weight (measure) in multiple integrals over generalized coordinates. The Jacobian is named after the German mathematician Carl Gustav Jacob Jacobi (1804 - 1851).

Definition

Let f be a map of an open subset T of $\mathbb {R} ^{n}$ into $\mathbb {R} ^{n}$ with continuous first partial derivatives,

$\mathbf {f} :\quad T\rightarrow \mathbb {R} ^{n}.$

That is if

$\mathbf {t} =(t_{1},\;t_{2},\;\ldots ,t_{n})\in T\subset \mathbb {R} ^{n},$

then

${\begin{aligned}x_{1}&=f_{1}(t_{1},t_{2},\ldots ,t_{n})\\x_{2}&=f_{2}(t_{1},t_{2},\ldots ,t_{n})\\\cdots &\cdots \\x_{n}&=f_{n}(t_{1},t_{2},\ldots ,t_{n}),\\\end{aligned}}$

with

$\mathbf {x} =(x_{1},\;x_{2},\;\ldots ,x_{n})\in \mathbb {R} ^{n}.$

The n × n functional matrix of Jacobi consists of partial derivatives

${\begin{pmatrix}{\dfrac {\partial f_{1}}{\partial t_{1}}}&{\dfrac {\partial f_{2}}{\partial t_{1}}}&\ldots &{\dfrac {\partial f_{n}}{\partial t_{1}}}\\\\{\dfrac {\partial f_{1}}{\partial t_{2}}}&{\dfrac {\partial f_{2}}{\partial t_{2}}}&\ldots &\dots \\\\&&\ddots \\\\{\dfrac {\partial f_{1}}{\partial t_{n}}}&\dots &\ldots &{\dfrac {\partial f_{n}}{\partial t_{n}}}\\\end{pmatrix}}.$

The determinant of this matrix is usually written as

$\mathbf {J} _{\mathbf {f} }(\mathbf {t} )\quad {\hbox{or}}\quad {\frac {\partial {\big (}f_{1},f_{2},\ldots ,f_{n}{\Big )}}{\partial {\big (}t_{1},t_{2},\ldots ,t_{n}{\Big )}}}$

Example

Let T be the subset {r, θ, φ | r > 0, 0 < θ<π, 0 <φ <2π} in $\scriptstyle \mathbb {R} ^{3}$ and let f be defined by

${\begin{aligned}x_{1}&=f_{1}(r,\theta ,\phi )=r\sin \theta \cos \phi \\x_{2}&=f_{2}(r,\theta ,\phi )=r\sin \theta \sin \phi \\x_{3}&=f_{3}(r,\theta ,\phi )=r\cos \theta \\\end{aligned}}$

The Jacobi matrix is

${\begin{pmatrix}\sin \theta \cos \phi &\sin \theta \sin \phi &\cos \theta \\r\cos \theta \cos \phi &r\cos \theta \sin \phi &-r\sin \theta \\-r\sin \theta \sin \phi &r\sin \theta \cos \phi &0\\\end{pmatrix}}$

Its determinant can be obtained most conveniently by a Laplace expansion along the third column

$\cos \theta {\begin{vmatrix}r\cos \theta \cos \phi &r\cos \theta \sin \phi \\-r\sin \theta \sin \phi &r\sin \theta \cos \phi \end{vmatrix}}+r\sin \theta {\begin{vmatrix}\sin \theta \cos \phi &\sin \theta \sin \phi \\-r\sin \theta \sin \phi &r\sin \theta \cos \phi \end{vmatrix}}=r^{2}(\cos \theta )^{2}\sin \theta +r^{2}(\sin \theta )^{3}=r^{2}\sin \theta$

The quantities {r, θ, φ} are known as spherical polar coordinates and its Jacobian is r²sinθ.

Coordinate transformation

The map $\mathbf {f} :\;T\rightarrow \mathbb {R} ^{n}$ is a coordinate transformation if (i) f has continuous first derivatives on T (ii) f is one-to-one on T and (iii) the Jacobian of f is not equal to zero on T.

Multiple integration

It can be proved that

$\int _{\mathbf {f} (\mathbf {t} )}\phi (\mathbf {x} )\;\mathrm {d} \mathbf {x} =\int _{T}\phi {\big (}\mathbf {f} (\mathbf {t} ){\big )}\;\mathbf {J} _{\mathbf {f} }(\mathbf {t} )\;\mathrm {d} \mathbf {t} .$

As an example we consider the spherical polar coordinates mentioned above. Here x = f(t) ≡ f(r, θ, φ) covers all of $\mathbb {R} ^{3}$, while T is the region {r > 0, 0 < θ<π, 0 <φ <2π}. Hence the theorem states that

$\iiint \limits _{\mathbb {R} ^{3}}\phi (\mathbf {x} )\;\mathrm {d} \mathbf {x} =\int \limits _{0}^{\infty }\int \limits _{0}^{\pi }\int \limits _{0}^{2\pi }\phi {\big (}\mathbf {x} (r,\theta ,\phi ){\big )}\;r^{2}\sin \theta \;\mathrm {d} r\mathrm {d} \theta \mathrm {d} \phi .$