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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).  
In [[mathematics]], the '''Jacobi matrix''' is the matrix of first-order [[partial derivative]]s of the (vector-valued) function:
:<math>\mathbf{f}:\quad\mathbb{R}^n \rightarrow \mathbb{R}^m</math>
(often '''f''' maps only from and to appropriate subsets of these spaces). The Jacobi matrix is ''m'' &times; ''n''  and consists of ''m'' rows of the first-order [[partial derivatives]] of '''f''' with respect to ''x''<sub>1</sub>, ...,''x''<sub>n</sub>, 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 &times; n (the [[gradient]] of ''f'', a row vector).
 
*The Jacobian appears as the [[weight]] ([[measure (mathematics)|measure]]) in multi-dimensional [[integral]]s 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 '''''x'''''<sub>0</sub>  if and only if the Jacobian at '''''x'''''<sub>0</sub> is non-zero.
 
The Jacobi matrix and its determinant are named after the German mathematician [[Carl Gustav Jacob Jacobi]] (1804 - 1851).
 
==Definition==
==Definition==
Let '''f''' be a map of an open subset ''T'' of <math>\mathbb{R}^n</math> into <math>\mathbb{R}^n</math> with continuous first partial derivatives,
Let '''f''' be a map of an open subset ''T'' of <math>\mathbb{R}^n</math> into <math>\mathbb{R}^m</math> with continuous first partial derivatives,
:<math>
:<math>
\mathbf{f}:\quad T \rightarrow \mathbb{R}^n.
\mathbf{f}:\quad T \rightarrow \mathbb{R}^m.
</math>
</math>
That is if
That is if
Line 16: Line 27:
x_2 &= f_2(t_1, t_2,\ldots, t_n) \\
x_2 &= f_2(t_1, t_2,\ldots, t_n) \\
\cdots & \cdots\\
\cdots & \cdots\\
x_n &= f_n(t_1, t_2,\ldots, t_n), \\
x_m &= f_m(t_1, t_2,\ldots, t_n), \\
\end{align}
\end{align}
</math>
</math>
with
with
:<math>
:<math>
\mathbf{x} = (x_1,\; x_2,\; \ldots, x_n)\in \mathbb{R}^n .
\mathbf{x} = (x_1,\; x_2,\; \ldots, x_m)\in \mathbb{R}^m .
</math>
</math>
The ''n'' &times; ''n'' functional matrix of Jacobi consists of partial derivatives
The ''m'' &times; ''n'' functional matrix of Jacobi consists of partial derivatives
:<math>
:<math>
\begin{pmatrix}
\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_1} & \dfrac{\partial f_1}{\partial t_2} & \ldots &\dfrac{\partial f_1}{\partial t_n}  \\
\\
\\
\dfrac{\partial f_1}{\partial t_2} & \dfrac{\partial f_2}{\partial t_2} & \ldots &\dots\\ \\
\dfrac{\partial f_2}{\partial t_1} & \dfrac{\partial f_2}{\partial t_2} & \ldots &\dots\\ \\
  & &\ddots\\  \\
  & &\ddots\\  \\
\dfrac{\partial f_1}{\partial t_n} & \dots & \ldots &\dfrac{\partial f_n}{\partial t_n}\\
\dfrac{\partial f_m}{\partial t_1} & \dots & \ldots &\dfrac{\partial f_m}{\partial t_n}\\
\end{pmatrix} .
\end{pmatrix} .
</math>
</math>
The [[determinant]] of this matrix is usually written as
The [[determinant]] (which is only defined for square matrices) of this matrix is usually written as (take ''m'' = ''n''),
:<math>
:<math>
\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)}  
\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)} .
</math>
</math>
===Example===
===Example===
Let ''T'' be the  subset {''r'', &theta;, &phi; | ''r'' > 0, 0 < &theta;<&pi;, 0 <&phi; <2&pi;}  in <math>\scriptstyle \mathbb{R}^3</math> and let '''f''' be defined by
Let ''T'' be the  subset {''r'', &theta;, &phi; | ''r'' > 0, 0 < &theta;<&pi;, 0 <&phi; <2&pi;}  in <math>\scriptstyle \mathbb{R}^3</math> and let '''f''' be defined by
:<math>
:<math>
\begin{align}
\begin{align}
x_1 &= f_1(r,\theta, \phi) = r\sin\theta\cos\phi \\
x_1 \equiv x &= f_1(r,\theta, \phi) = r\sin\theta\cos\phi \\
x_2 &= f_2(r,\theta, \phi) = r\sin\theta\sin\phi \\
x_2 \equiv y &= f_2(r,\theta, \phi) = r\sin\theta\sin\phi \\
x_3 &= f_3(r,\theta, \phi) = r\cos\theta \\  
x_3 \equiv z &= f_3(r,\theta, \phi) = r\cos\theta \\  
\end{align}
\end{align}
</math>
</math>
Line 49: Line 61:
:<math>
:<math>
\begin{pmatrix}  
\begin{pmatrix}  
\sin\theta\cos\phi & \sin\theta\sin\phi & \cos\theta \\
\sin\theta\cos\phi   & r\cos\theta\cos\phi & -r\sin\theta\sin\phi \\
r\cos\theta\cos\phi & r\cos\theta\sin\phi & -r\sin\theta \\
\sin\theta\sin\phi   & r\cos\theta\sin\phi & r\sin\theta\cos\phi  \\  
-r\sin\theta\sin\phi &r\sin\theta\cos\phi & 0 \\
\cos\theta           & -r\sin\theta       & 0 \\    
\end{pmatrix}
\end{pmatrix}
</math>
</math>
Its determinant can be obtained most conveniently by a [[Laplace expansion]] along the third column
Its determinant can be obtained most conveniently by a [[Laplace expansion]] along the third row
:<math>
:<math>
\cos\theta  
\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}
\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
+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} =
\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
r^2(\cos\theta)^2 \sin\theta + r^2 (\sin\theta)^3 = r^2\sin\theta
</math>
</math>
The quantities {''r'', &theta;, &phi;} are known as [[spherical polar coordinates]] and its Jacobian is r<sup>2</sup>sin&theta;.
The quantities {''r'', &theta;, &phi;} are known as [[spherical polar coordinates]] and its Jacobian is ''r''<sup>2</sup>sin&theta;.


==Coordinate transformation==
==Coordinate transformation==
The map <math>  \mathbf{f}:\; T \rightarrow \mathbb{R}^n  </math> 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''.
Let <math>T \sub \mathbb{R}^n</math>. The map <math>  \mathbf{f}:\; T \rightarrow \mathbb{R}^n, </math> 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==
==Multiple integration==
Line 95: Line 107:
:<math>
:<math>
\mathrm{d}V =  \mathrm{d}\mathbf{g}_1 \cdot ( \mathrm{d}\mathbf{g}_2\times \mathrm{d}\mathbf{g}_3 ) =
\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
  \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
</math>
</math>
The components of the first vector are given by
The components of the first vector are given by
:<math>
:<math>
\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)
\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)
</math>
</math>
and similar expressions hold for the components of the  other two derivatives.
and similar expressions hold for the components of the  other two derivatives.
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  \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) =
  \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}
\begin{vmatrix}
\dfrac{\partial x}{\partial t_1} & \dfrac{\partial y}{\partial t_1} & \dfrac{\partial z}{\partial t_1} \\
\dfrac{\partial f_1}{\partial t_1} & \dfrac{\partial f_2}{\partial t_1} & \dfrac{\partial f_3}{\partial t_1} \\
\dfrac{\partial x}{\partial t_2} & \dfrac{\partial y}{\partial t_2} & \dfrac{\partial z}{\partial t_2} \\
\dfrac{\partial f_1}{\partial t_2} & \dfrac{\partial f_2}{\partial t_2} & \dfrac{\partial f_3}{\partial t_2} \\
\dfrac{\partial x}{\partial t_3} & \dfrac{\partial y}{\partial t_3} & \dfrac{\partial z}{\partial t_3} \\
\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( x, y, z)}{\partial( t_1, t_2, t_3)} \equiv \mathbf{J}_{\mathbf{f}}(\mathbf{t})
\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}).
</math>
</math>
where we used that a determinant is invariant under transposition (interchange of rows and columns). Finally.
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.
:<math>
:<math>
\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} .
\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} .
</math>
</math>


==Reference==  
==Reference==  
<references />
<references />[[Category:Suggestion Bot Tag]]

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

(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 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 into with continuous first partial derivatives,

That is if

then

with

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

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

Example

Let T be the subset {r, θ, φ | r > 0, 0 < θ<π, 0 <φ <2π} in and let f be defined by

The Jacobi matrix is

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

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

Coordinate transformation

Let . The map 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 [1] that

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

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,

Similarly, we define

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

The components of the first vector are given by

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

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.

Reference

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