Jacobian

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This editable Main Article is under development and subject to a disclaimer.

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

f : R n -> R m {\displaystyle \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 × n and consists of m rows of the first-order partial derivatives of f with respect to x1, ...,xn, respectively. 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:

* For m = 1, the Jacobi matrix appears in the second (linear) term of the Taylor series of f. Here the Jacobi matrix is 1 × 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).

## Contents

* 1 Definition
* 1.1 Example
* 2 Coordinate transformation
* 3 Multiple integration
* 4 Geometric interpretation of the Jacobian
* 5 Reference

## Definition[edit]

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

f : T -> R m . {\displaystyle \mathbf {f} :\quad T\rightarrow \mathbb {R} ^{m}.}

That is if

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

then

x 1 = f 1 ( t 1 , t 2 , … , t n ) x 2 = f 2 ( t 1 , t 2 , … , t n ) ⋯ ⋯ x m = f m ( t 1 , t 2 , … , t n ) , {\displaystyle {\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_{m}&=f_{m}(t_{1},t_{2},\ldots ,t_{n}),\\\\\end{aligned}}}

with

x = ( x 1 , x 2 , … , x m ) ∈ R m . {\displaystyle \mathbf {x} =(x_{1},\;x_{2},\;\ldots ,x_{m})\in \mathbb {R} ^{m}.}

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

( ∂ f 1 ∂ t 1 ∂ f 1 ∂ t 2 … ∂ f 1 ∂ t n ∂ f 2 ∂ t 1 ∂ f 2 ∂ t 2 … … ⋱ ∂ f m ∂ t 1 … … ∂ f m ∂ t n ) . {\displaystyle {\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),

J f ( t ) or ∂ ( f 1 , f 2 , … , f n ) ∂ ( t 1 , t 2 , … , t n ) . {\displaystyle \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[edit]

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

x 1 ≡ x = f 1 ( r , θ , ϕ ) = r sin ⁡ θ cos ⁡ ϕ x 2 ≡ y = f 2 ( r , θ , ϕ ) = r sin ⁡ θ sin ⁡ ϕ x 3 ≡ z = f 3 ( r , θ , ϕ ) = r cos ⁡ θ {\displaystyle {\begin{aligned}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{aligned}}}

The Jacobi matrix is

( sin ⁡ θ cos ⁡ ϕ r cos ⁡ θ cos ⁡ ϕ − r sin ⁡ θ sin ⁡ ϕ sin ⁡ θ sin ⁡ ϕ r cos ⁡ θ sin ⁡ ϕ r sin ⁡ θ cos ⁡ ϕ cos ⁡ θ − r sin ⁡ θ 0 ) {\displaystyle {\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

cos ⁡ θ | r cos ⁡ θ cos ⁡ ϕ − r sin ⁡ θ sin ⁡ ϕ r cos ⁡ θ sin ⁡ ϕ r sin ⁡ θ cos ⁡ ϕ | + r sin ⁡ θ | sin ⁡ θ cos ⁡ ϕ − r sin ⁡ θ sin ⁡ ϕ sin ⁡ θ sin ⁡ ϕ r sin ⁡ θ cos ⁡ ϕ | = r 2 ( cos ⁡ θ ) 2 sin ⁡ θ + r 2 ( sin ⁡ θ ) 3 = r 2 sin ⁡ θ {\displaystyle \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 }

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

## Coordinate transformation[edit]

Let  T ⊂ R n {\displaystyle T\subset \mathbb {R} ^{n}} . The map  f : T -> R n , {\displaystyle \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[edit]

It can be proved [1] that

∫ f ( t ) ϕ ( x ) d x = ∫ T ϕ ( f ( t ) ) J f ( t ) d t . {\displaystyle \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  R 3 {\displaystyle \mathbb {R} ^{3}} , while T is the region {r > 0, 0 < θ<π, 0 <φ <2π}. Hence the theorem states that

∭ R 3 ϕ ( x ) d x = ∫ 0 ∞ ∫ 0 π ∫ 0 2 π ϕ ( x ( r , θ , ϕ ) ) r 2 sin ⁡ θ d r d θ d ϕ . {\displaystyle \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[edit]

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,

d g 1 ≡ lim Δ t 1 -> 0 f ( t 1 + Δ t 1 , t 2 , t 3 ) − f ( t 1 , t 2 , t 3 ) Δ t 1 Δ t 1 = ∂ f ∂ t 1 d t 1 {\displaystyle \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

d g 2 ≡ ∂ f ∂ t 2 d t 2 , d g 3 ≡ ∂ f ∂ t 3 d t 3 {\displaystyle \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,

d V = d g 1 ⋅ ( d g 2 × d g 3 ) = ∂ f ∂ t 1 ⋅ ( ∂ f ∂ t 2 × ∂ f ∂ t 3 ) d t 1 d t 2 d t 3 {\displaystyle \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

∂ f ∂ t 1 ≡ ( ∂ x ∂ t 1 , ∂ y ∂ t 1 , ∂ z ∂ t 1 ) ≡ ( ∂ f 1 ∂ t 1 , ∂ f 2 ∂ t 1 , ∂ f 3 ∂ t 1 ) {\displaystyle {\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

∂ f ∂ t 1 ⋅ ( ∂ f ∂ t 2 × ∂ f ∂ t 3 ) = | ∂ f 1 ∂ t 1 ∂ f 2 ∂ t 1 ∂ f 3 ∂ t 1 ∂ f 1 ∂ t 2 ∂ f 2 ∂ t 2 ∂ f 3 ∂ t 2 ∂ f 1 ∂ t 3 ∂ f 2 ∂ t 3 ∂ f 3 ∂ t 3 | ≡ ∂ ( f 1 , f 2 , f 3 ) ∂ ( t 1 , t 2 , t 3 ) ≡ J f ( t ) . {\displaystyle {\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.

d V = ∂ ( f 1 , f 2 , f 3 ) ∂ ( t 1 , t 2 , t 3 ) d t 1 d t 2 d t 3 ≡ J f ( t ) d t . {\displaystyle \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} .}

## Reference[edit]

1. ↑ T. M. Apostol, Mathematical Analysis, Addison-Wesley, 2nd ed. (1974), sec. 15.10