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

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

Geometric interpretation of the Jacobian

The Jacobian has a geometric interpretation which we expound for the example of 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,

$\mathrm {d} V=\mathrm {d} \mathbf {g} _{1}\cdot (\mathrm {d} \mathbf {g} _{2}\times \mathrm {d} \mathbf {g} _{3})={\frac {\partial \mathbf {f} }{\partial t_{1}}}\cdot \left({\frac {\partial \mathbf {f} }{\partial t_{2}}}\times {\frac {\partial \mathbf {f} }{\partial t_{3}}}\right)\;\mathrm {d} t_{1}\mathrm {d} t_{3}\mathrm {d} t_{3}$

The components of the first vector are given by

${\frac {\partial \mathbf {f} }{\partial t_{1}}}\equiv \left({\frac {\partial x}{\partial t_{1}}},{\frac {\partial y}{\partial t_{1}}},{\frac {\partial z}{\partial t_{1}}}\right)$

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

${\frac {\partial \mathbf {f} }{\partial t_{1}}}\cdot \left({\frac {\partial \mathbf {f} }{\partial t_{2}}}\times {\frac {\partial \mathbf {f} }{\partial t_{3}}}\right)={\begin{vmatrix}{\dfrac {\partial x}{\partial t_{1}}}&{\dfrac {\partial y}{\partial t_{1}}}&{\dfrac {\partial z}{\partial t_{1}}}\\{\dfrac {\partial x}{\partial t_{2}}}&{\dfrac {\partial y}{\partial t_{2}}}&{\dfrac {\partial z}{\partial t_{2}}}\\{\dfrac {\partial x}{\partial t_{3}}}&{\dfrac {\partial y}{\partial t_{3}}}&{\dfrac {\partial z}{\partial t_{3}}}\\\end{vmatrix}}\equiv {\frac {\partial (x,y,z)}{\partial (t_{1},t_{2},t_{3})}}\equiv \mathbf {J} _{\mathbf {f} }(\mathbf {t} )$

where we used that a determinant is invariant under transposition (interchange of rows and columns). Finally.

$\mathrm {d} V={\frac {\partial (x,y,z)}{\partial (t_{1},t_{2},t_{3})}}\;\mathrm {d} t_{1}\mathrm {d} t_{3}\mathrm {d} t_{3}\equiv \mathbf {J} _{\mathbf {f} }(\mathbf {t} )\;\mathrm {d} \mathbf {t} .$