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 \sub \mathbb{R}^n, $$ then

\begin{align} 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{align} $$ with

\mathbf{x} = (x_1,\; x_2,\; \ldots, x_n)\in \mathbb{R}^n. $$ The n &times; 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, &theta;, &phi; | r > 0, 0 < &theta;<&pi;, 0 <&phi; <2&pi;}  in $$\scriptstyle \mathbb{R}^3$$ and let f be defined by

\begin{align} 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{align} $$ 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, &theta;, &phi;} are known as spherical polar coordinates and its Jacobian is r2sin&theta;.

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}. $$ As an example we consider the spherical polar coordinates mentioned above. Here x = f(t) &equiv; f(r, &theta;, &phi;) covers all of $$\mathbb{R}^3$$, while T is the region {r > 0, 0 < &theta;<&pi;, 0 <&phi; <2&pi;}. 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. $$