Jacobian

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

${\displaystyle \mathbf{𝐟}:{\mathbb{ℝ}}^n\to {\mathbb{ℝ}}^m}$

(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, ...,x_n, 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).

Definition[edit]

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

${\displaystyle \mathbf{𝐟}:T\to {\mathbb{ℝ}}^m.}$

That is if

${\displaystyle \mathbf{𝐭}=(t_1,t_2,\dots ,t_n)\in T\subset {\mathbb{ℝ}}^n,}$

then

${\displaystyle \begin{array}{rl}{x}_1& =f_1(t_1,t_2,\dots ,t_n)\\ x_2& =f_2(t_1,t_2,\dots ,t_n)\\ \cdots & \cdots \\ x_m& =f_m(t_1,t_2,\dots ,t_n),\\ \end{array}}$

with

${\displaystyle \mathbf{𝐱}=(x_1,x_2,\dots ,x_m)\in {\mathbb{ℝ}}^m.}$

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

${\displaystyle \left(\begin{array}{cccc}{\displaystyle \frac{\partial f_1}{\partial t_1}}& {\displaystyle \frac{\partial f_1}{\partial t_2}}& \dots & {\displaystyle \frac{\partial f_1}{\partial t_n}}\\ \\ {\displaystyle \frac{\partial f_2}{\partial t_1}}& {\displaystyle \frac{\partial f_2}{\partial t_2}}& \dots & \dots \\ \\ & & \ddots \\ \\ {\displaystyle \frac{\partial f_m}{\partial t_1}}& \dots & \dots & {\displaystyle \frac{\partial f_m}{\partial t_n}}\\ \end{array}\right).}$

The determinant (which is only defined for square matrices) of this matrix is usually written as (take m = n),

${\displaystyle {\mathbf{𝐉}}_{\mathbf{𝐟}}(\mathbf{𝐭})\text{or}\frac{\partial (f_1,f_2,\dots ,f_n)}{\partial (t_1,t_2,\dots ,t_n)}.}$

Example[edit]

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

${\displaystyle \begin{array}{rl}{x}_1\equiv x& =f_1(r,\theta ,\varphi )=r\sin \theta \cos \varphi \\ x_2\equiv y& =f_2(r,\theta ,\varphi )=r\sin \theta \sin \varphi \\ x_3\equiv z& =f_3(r,\theta ,\varphi )=r\cos \theta \\ \end{array}}$

The Jacobi matrix is

${\displaystyle \left(\begin{array}{ccc}\sin \theta \cos \varphi & r\cos \theta \cos \varphi & -r\sin \theta \sin \varphi \\ \sin \theta \sin \varphi & r\cos \theta \sin \varphi & r\sin \theta \cos \varphi \\ \cos \theta & -r\sin \theta & 0\\ \end{array}\right)}$

Its determinant can be obtained most conveniently by a Laplace expansion along the third row

${\displaystyle \cos \theta \left|\begin{array}{cc}r\cos \theta \cos \varphi & -r\sin \theta \sin \varphi \\ r\cos \theta \sin \varphi & r\sin \theta \cos \varphi \end{array}\right|+r\sin \theta \left|\begin{array}{cc}\sin \theta \cos \varphi & -r\sin \theta \sin \varphi \\ \sin \theta \sin \varphi & r\sin \theta \cos \varphi \end{array}\right|=r^2(\cos \theta {)}^2\sin \theta +r^2(\sin \theta {)}^3=r^2\sin \theta }$

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

Coordinate transformation[edit]

Let ${\displaystyle T\subset {\mathbb{ℝ}}^n}$. The map ${\displaystyle \mathbf{𝐟}:T\to {\mathbb{ℝ}}^n,}$ 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

${\displaystyle \underset{\mathbf{𝐟}(\mathbf{𝐭})}{\int }\varphi (\mathbf{𝐱})\mathrm{d}\mathbf{𝐱}=\underset{T}{\int }\varphi (\mathbf{𝐟}(\mathbf{𝐭})){\mathbf{𝐉}}_{\mathbf{𝐟}}(\mathbf{𝐭})\mathrm{d}\mathbf{𝐭}.}$

As an example we consider the spherical polar coordinates mentioned above. Here x = f(t) ≡ f(r, θ, φ) covers all of ${\displaystyle {\mathbb{ℝ}}^3}$, while T is the region {r > 0, 0 < θ<π, 0 <φ <2π}. Hence the theorem states that

${\displaystyle \underset{{\mathbb{ℝ}}^3}{\iiint }\varphi (\mathbf{𝐱})\mathrm{d}\mathbf{𝐱}=\underset{0}{\overset{\mathrm{\infty }}{\int }}\underset{0}{\overset{\pi }{\int }}\underset{0}{\overset{2\pi }{\int }}\varphi (\mathbf{𝐱}(r,\theta ,\varphi ))r^2\sin \theta \mathrm{d}r\mathrm{d}\theta \mathrm{d}\varphi .}$

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,

${\displaystyle \mathrm{d}{\mathbf{𝐠}}_1\equiv \underset{\mathrm{\Delta }{t}_1\to 0}{\lim }\frac{\mathbf{𝐟}(t_1+\mathrm{\Delta }{t}_1,t_2,t_3)-\mathbf{𝐟}(t_1,t_2,t_3)}{\mathrm{\Delta }{t}_1}\mathrm{\Delta }{t}_1=\frac{\partial \mathbf{𝐟}}{\partial t_1}\mathrm{d}{t}_1}$

Similarly, we define

${\displaystyle \mathrm{d}{\mathbf{𝐠}}_2\equiv \frac{\partial \mathbf{𝐟}}{\partial t_2}\mathrm{d}{t}_2,\mathrm{d}{\mathbf{𝐠}}_3\equiv \frac{\partial \mathbf{𝐟}}{\partial t_3}\mathrm{d}{t}_3}$

The scalar triple product of these three vectors gives the volume of an infinitesimally small parallelepiped,

${\displaystyle \mathrm{d}V=\mathrm{d}{\mathbf{𝐠}}_1\cdot (\mathrm{d}{\mathbf{𝐠}}_2\times \mathrm{d}{\mathbf{𝐠}}_3)=\frac{\partial \mathbf{𝐟}}{\partial t_1}\cdot (\frac{\partial \mathbf{𝐟}}{\partial t_2}\times \frac{\partial \mathbf{𝐟}}{\partial t_3})\mathrm{d}{t}_1\mathrm{d}{t}_2\mathrm{d}{t}_3}$

The components of the first vector are given by

${\displaystyle \frac{\partial \mathbf{𝐟}}{\partial t_1}\equiv (\frac{\partial x}{\partial t_1},\frac{\partial y}{\partial t_1},\frac{\partial z}{\partial t_1})\equiv (\frac{\partial f_1}{\partial t_1},\frac{\partial f_2}{\partial t_1},\frac{\partial f_3}{\partial t_1})}$

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

${\displaystyle \frac{\partial \mathbf{𝐟}}{\partial t_1}\cdot (\frac{\partial \mathbf{𝐟}}{\partial t_2}\times \frac{\partial \mathbf{𝐟}}{\partial t_3})=\left|\begin{array}{ccc}{\displaystyle \frac{\partial f_1}{\partial t_1}}& {\displaystyle \frac{\partial f_2}{\partial t_1}}& {\displaystyle \frac{\partial f_3}{\partial t_1}}\\ {\displaystyle \frac{\partial f_1}{\partial t_2}}& {\displaystyle \frac{\partial f_2}{\partial t_2}}& {\displaystyle \frac{\partial f_3}{\partial t_2}}\\ {\displaystyle \frac{\partial f_1}{\partial t_3}}& {\displaystyle \frac{\partial f_2}{\partial t_3}}& {\displaystyle \frac{\partial f_3}{\partial t_3}}\\ \end{array}\right|\equiv \frac{\partial (f_1,f_2,f_3)}{\partial (t_1,t_2,t_3)}\equiv {\mathbf{𝐉}}_{\mathbf{𝐟}}(\mathbf{𝐭}).}$

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.

${\displaystyle \mathrm{d}V=\frac{\partial (f_1,f_2,f_3)}{\partial (t_1,t_2,t_3)}\mathrm{d}{t}_1\mathrm{d}{t}_2\mathrm{d}{t}_3\equiv {\mathbf{𝐉}}_{\mathbf{𝐟}}(\mathbf{𝐭})\mathrm{d}\mathbf{𝐭}.}$

Reference[edit]