Jacobian

In mathematics, the Jacobi matrix is the matrix of first-order partial derivatives of the (vector-valued) function:

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

The Jacobi matrix is m × n and consists of m rows of n first-order partial derivatives of f with respect to x₁, ...,x_n. This 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:

• The Jacobi matrix appears in the second (linear) term of the Taylor series of f.

• 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 $\mathbb {R} ^{n}$ 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

${\frac {\partial \mathbf {f} }{\partial t_{1}}}\equiv \left({\frac {\partial x}{\partial t_{1}}},{\frac {\partial y}{\partial t_{1}}},{\frac {\partial z}{\partial t_{1}}}\right)\equiv \left({\frac {\partial f_{1}}{\partial t_{1}}},{\frac {\partial f_{2}}{\partial t_{1}}},{\frac {\partial f_{3}}{\partial t_{1}}}\right)$

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

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

Finally.

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