Jacobian

In mathematics, the Jacobi matrix is the matrix of first-order partial derivatives of the (vector-valued) function:
 * $$\mathbf{f}:\quad\mathbb{R}^n \rightarrow \mathbb{R}^m$$

(often f maps only from and to appropriate subsets of these spaces). The Jacobi matrix is m &times; n, i.e., consists of m rows and n columns. Row k contains the first-order partial derivatives of fk with respect to x1, ...,xn, respectively. The Jacobi matrix is also known as the functional matrix of Jacobi. The determinant of the Jacobi matrix for n = m is known as the Jacobian. The Jacobi matrix and its determinant have several uses in mathematics:


 * For m = 1 (f is a scalar valued function of n variables), the Jacobi matrix appears in the second (linear) term of the Taylor series of f.  Here the Jacobi matrix is 1 &times; n (the gradient of f, a row vector).


 * The Jacobian appears as the weight (measure) in multi-dimensional integrals over generalized coordinates, i.e, over non-Cartesian coordinates.


 * The inverse function theorem states that if m = n and f is continuously differentiable, then f is invertible in the neighborhood of  a point x0  if and only if the Jacobian at x0 is non-zero.

The Jacobi matrix and its determinant are 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 $$\mathbb{R}^m$$ with continuous first partial derivatives,

\mathbf{f}:\quad T \rightarrow \mathbb{R}^m. $$ 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_m &= f_m(t_1, t_2,\ldots, t_n), \\ \end{align} $$ with

\mathbf{x} = (x_1,\; x_2,\; \ldots, x_m)\in \mathbb{R}^m. $$ The m &times; n functional matrix of Jacobi consists of partial derivatives

\begin{pmatrix} \dfrac{\partial f_1}{\partial t_1} & \dfrac{\partial f_1}{\partial t_2} & \ldots &\dfrac{\partial f_1}{\partial t_n}  \\ \\ \dfrac{\partial f_2}{\partial t_1} & \dfrac{\partial f_2}{\partial t_2} & \ldots &\dots\\ \\ & &\ddots\\ \\ \dfrac{\partial f_m}{\partial t_1} & \dots & \ldots &\dfrac{\partial f_m}{\partial t_n}\\ \end{pmatrix}. $$ The determinant (which is only defined for square matrices) of this matrix is usually written as (take m = n),

\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 ℝ3 and let f be defined by

\begin{align} x_1 \equiv x &= f_1(r,\theta, \phi) = r\sin\theta\cos\phi \\ x_2 \equiv y &= f_2(r,\theta, \phi) = r\sin\theta\sin\phi \\ x_3 \equiv z &= f_3(r,\theta, \phi) = r\cos\theta \\ \end{align} $$ The Jacobi matrix is

\begin{pmatrix} \sin\theta\cos\phi  & r\cos\theta\cos\phi & -r\sin\theta\sin\phi \\ \sin\theta\sin\phi  & r\cos\theta\sin\phi &  r\sin\theta\cos\phi  \\ \cos\theta          &  -r\sin\theta       & 0 \\ \end{pmatrix}. $$ Its determinant can be obtained most conveniently by a Laplace expansion along the third row

\begin{align} \frac{\partial(x_1, x_2, x_2)}{\partial(r, \theta, \phi)} &= \cos\theta \begin{vmatrix} r\cos\theta\cos\phi & -r\sin\theta\sin\phi \\ r\cos\theta\sin\phi &r\sin\theta\cos\phi \end{vmatrix} +r\sin\theta \begin{vmatrix} \sin\theta\cos\phi & -r\sin\theta\sin\phi \\ \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. \end{align} $$ The quantities {r, &theta;, &phi;} are known as spherical polar coordinates and its Jacobian is r2sin&theta;.

Coordinate transformation
Let $$T \sub \mathbb{R}^n$$. 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. $$

Geometric interpretation of the Jacobian
The Jacobian has a geometric interpretation which is illustrated for the example n = 3.

The following is a vector of infinitesimal length in the direction of increase in t1,

\mathrm{d}\mathbf{g}_1 \equiv \lim_{\Delta t_1 \rightarrow 0} \frac{\mathbf{f}(t_1+\Delta t_1, t_2, t_3) - \mathbf{f}(t_1, t_2, t_3)}{\Delta t_1}\Delta t_1 = \frac{\partial \mathbf{f}}{\partial t_1} \mathrm{d}t_1 $$ Similarly, we define

\mathrm{d}\mathbf{g}_2 \equiv \frac{\partial \mathbf{f}}{\partial t_2} \mathrm{d}t_2,\quad \mathrm{d}\mathbf{g}_3 \equiv \frac{\partial \mathbf{f}}{\partial t_3} \mathrm{d}t_3 $$ The scalar triple product of these three vectors gives the volume of an infinitesimally small parallelepiped,

\mathrm{d}V = \mathrm{d}\mathbf{g}_1 \cdot ( \mathrm{d}\mathbf{g}_2\times \mathrm{d}\mathbf{g}_3 ) = \frac{\partial \mathbf{f}}{\partial t_1} \cdot \left(\frac{\partial \mathbf{f}}{\partial t_2} \times \frac{\partial \mathbf{f}}{\partial t_3}\right) \; \mathrm{d}t_1\mathrm{d}t_2\mathrm{d}t_3 $$ The components of the first vector are given by

\frac{\partial \mathbf{f}}{\partial t_1} \equiv \left( \frac{\partial x}{\partial t_1}, \frac{\partial y}{\partial t_1}, \frac{\partial z}{\partial t_1} \right) \equiv \left( \frac{\partial f_1}{\partial t_1}, \frac{\partial f_2}{\partial t_1}, \frac{\partial f_3}{\partial t_1} \right) $$ and similar expressions hold for the components of the other two derivatives. It has been shown in the article on the scalar triple product that

\frac{\partial \mathbf{f}}{\partial t_1} \cdot \left(\frac{\partial \mathbf{f}}{\partial t_2} \times \frac{\partial \mathbf{f}}{\partial t_3}\right) = \begin{vmatrix} \dfrac{\partial f_1}{\partial t_1} & \dfrac{\partial f_2}{\partial t_1} & \dfrac{\partial f_3}{\partial t_1} \\ \dfrac{\partial f_1}{\partial t_2} & \dfrac{\partial f_2}{\partial t_2} & \dfrac{\partial f_3}{\partial t_2} \\ \dfrac{\partial f_1}{\partial t_3} & \dfrac{\partial f_2}{\partial t_3} & \dfrac{\partial f_3}{\partial t_3} \\ \end{vmatrix} \equiv \frac{\partial( f_1, f_2, f_3)}{\partial( t_1, t_2, t_3)} \equiv \mathbf{J}_{\mathbf{f}}(\mathbf{t}). $$ Note that a determinant is invariant under transposition (interchange of rows and columns), so that the transposed determinant being given is of no concern. Finally.

\mathrm{d}V = \frac{\partial( f_1, f_2, f_3)}{\partial( t_1, t_2, t_3)}\; \mathrm{d}t_1\mathrm{d}t_2\mathrm{d}t_3 \equiv \mathbf{J}_{\mathbf{f}}(\mathbf{t})\; \mathrm{d}\mathbf{t}. $$