Jacobian Matrix Determinant

A Jacobian Matrix Determinant is a determinant that is applied to a jacobian matrix.

Context:
- It can be defined as $J=\left|\dfrac{\partial\left(y_{1}, \ldots, y_{n}\right)}{\partial\left(x_{1}, \ldots, x_{n}\right)}\right|$
- …
Example(s):
- [math]\displaystyle{ \mathbf f\left(\begin{bmatrix} x\\y\end{bmatrix}\right) = \begin{bmatrix} f_1(x,y)\\f_2(x,y)\end{bmatrix} = \begin{bmatrix} x^2 y \\5 x + \sin y \end{bmatrix}. }[/math] The Jacobian matrix of $\mathbf{f}$ is [math]\displaystyle{ \mathbf J_{\mathbf f}(x, y) = \begin{bmatrix} \dfrac{\partial f_1}{\partial x} & \dfrac{\partial f_1}{\partial y}\\[1em] \dfrac{\partial f_2}{\partial x} & \dfrac{\partial f_2}{\partial y} \end{bmatrix} = \begin{bmatrix} 2 x y & x^2 \\ 5 & \cos y \end{bmatrix} }[/math] and the Jacobian determinant is [math]\displaystyle{ \det(\mathbf J_{\mathbf f}(x, y)) = 2 x y \cos y - 5 x^2 . }[/math]
- …
Counter-Example(s):
See: Equilibrium Point, Square Matrix, Determinant, Continuously Differentiable Function, Invertible, Inverse Function Theorem, Positive Number, Negative Number, Absolute Value, Volume, Substitution Rule.

References

2021a

(Wolfram Mathworld, 2021) ⇒ Eric W. Weisstein (1999-2021). "Jacobian". From MathWorld--A Wolfram Web Resource. Retrieved:2021-1-24.
- QUOTE: Given a set $\mathbf{y=f(x)}$ of $n$ equations in $n$ variables $x_1, \cdots , x_n$, written explicitly as

$\mathbf{y} \equiv\left[\begin{array}{c} f_{1}(\mathbf{x}) \\ f_{2}(\mathbf{x}) \\ \vdots \\ f_{n}(\mathbf{x}) \end{array}\right]$

(1)

or more explicitly as

$\left\{\begin{array}{l} y_{1}=f_{1}\left(x_{1}, \ldots, x_{n}\right) \\ \vdots \\ y_{n}=f_{n}\left(x_{1}, \ldots, x_{n}\right) \end{array}\right.$

(2)

the Jacobian matrix, sometimes simply called “the Jacobian” (Simon and Blume 1994) is defined by

$\mathbf{J}\left(x_{1}, \ldots, x_{n}\right)=\left[\begin{array}{ccc} \frac{\partial y_{1}}{\partial x_{1}} & \cdots & \frac{\partial y_{1}}{\partial x_{n}} \\ \vdots & \ddots & \vdots \\ \frac{\partial y_{n}}{\partial x_{1}} & \cdots & \frac{\partial y_{1}}{\partial x_{n}} \end{array}\right] .$

(3)

The determinant of $\mathbf{J}$ is the Jacobian determinant (confusingly, often called “the Jacobian” as well) and is denoted

$J=\left|\frac{\partial\left(y_{1}, \ldots, y_{n}\right)}{\partial\left(x_{1}, \ldots, x_{n}\right)}\right|$

(4)

2021b

(Wikipedia, 2021) ⇒ https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant Retrieved:2021-1-24.
- In vector calculus, the Jacobian matrix ( ) of a vector-valued function in several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian in literature.
  Suppose $f : ℝ n \to ℝ m$ is a function such that each of its first-order partial derivatives exist on $ℝ n$ . This function takes a point $x \in ℝ n$ as input and produces the vector $f (x) \in ℝ m$ as output. Then the Jacobian matrix of $f$ is defined to be an $m \times n$ matrix, denoted by $J$ , whose $(i, j)$ th entry is [math]\displaystyle{ \mathbf J_{ij} = \frac{\partial f_i}{\partial x_j} }[/math] , or explicitly : [math]\displaystyle{ \mathbf J = \begin{bmatrix} \dfrac{\partial \mathbf{f}}{\partial x_1} & \cdots & \dfrac{\partial \mathbf{f}}{\partial x_n} \end{bmatrix} = \begin{bmatrix} \nabla^T f_1 \\ \vdots \\ \nabla^T f_m \end{bmatrix} = \begin{bmatrix} \dfrac{\partial f_1}{\partial x_1} & \cdots & \dfrac{\partial f_1}{\partial x_n}\\ \vdots & \ddots & \vdots\\ \dfrac{\partial f_m}{\partial x_1} & \cdots & \dfrac{\partial f_m}{\partial x_n} \end{bmatrix} }[/math] where [math]\displaystyle{ \nabla^T f_i }[/math] is the transpose (row vector) of the gradient of the [math]\displaystyle{ i }[/math] component.
  This matrix, whose entries are functions of $x$ , is denoted in various ways; common notations include $D f$ , $J f$ , [math]\displaystyle{ \nabla \mathbf{f} }[/math] , and [math]\displaystyle{ \frac{\partial(f_1,..,f_m)}{\partial(x_1, ..,x_n)} }[/math] . Some authors define the Jacobian as the transpose of the form given above.
  The Jacobian matrix represents the differential of $f$ at every point where $f$ is differentiable. In detail, if $h$ is a displacement vector represented by a column matrix, the matrix product $J (x) \cdot h$ is another displacement vector, that is the best linear approximation of the change of $f$ in a neighborhood of $x$ , if $f (x)$ is differentiable at $x$ .Template:Efn This means that the function that maps $y$ to $f (x) + J (x) \cdot (y - x)$ is the best linear approximation of $f (y)$ for all points $y$ close to $x$ . This linear function is known as the derivative or the differential of $f$ at $x$ .
  When $m = n$ , the Jacobian matrix is square, so its determinant is a well-defined function of $x$ , known as the Jacobian determinant of $f$ . It carries important information about the local behavior of $f$ . In particular, the function $f$ has locally in the neighborhood of a point $x$ an inverse function that is differentiable if and only if the Jacobian determinant is nonzero at $x$ (see Jacobian conjecture). The Jacobian determinant also appears when changing the variables in multiple integrals (see substitution rule for multiple variables).
  When $m = 1$ , that is when $f : ℝ n \to ℝ$ is a scalar-valued function, the Jacobian matrix reduces to a row vector. This row vector of all first-order partial derivatives of $f$ is the gradient of $f$ , i.e. [math]\displaystyle{ \mathbf{J}_{f} = \nabla f }[/math] . Specialising further, when $m = n = 1$ , that is when $f : ℝ \to ℝ$ is a scalar-valued function of a single variable, the Jacobian matrix has a single entry. This entry is the derivative of the function $f$ .
  These concepts are named after the mathematician Carl Gustav Jacob Jacobi (1804–1851).

2021c

(Wikipedia, 2021) ⇒ https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant#Jacobian_determinant Retrieved:2021-1-24.
- If $m = n$ , then $f$ is a function from $ℝ n$ to itself and the Jacobian matrix is a square matrix. We can then form its determinant, known as the Jacobian determinant. The Jacobian determinant is sometimes simply referred to as "the Jacobian".
  The Jacobian determinant at a given point gives important information about the behavior of $f$ near that point. For instance, the continuously differentiable function $f$ is invertible near a point $p \in ℝ n$ if the Jacobian determinant at $p$ is non-zero. This is the inverse function theorem. Furthermore, if the Jacobian determinant at $p$ is positive, then $f$ preserves orientation near $p$ ; if it is negative, $f$ reverses orientation. The absolute value of the Jacobian determinant at $p$ gives us the factor by which the function $f$ expands or shrinks volumes near $p$ ; this is why it occurs in the general substitution rule.
  The Jacobian determinant is used when making a change of variables when evaluating a multiple integral of a function over a region within its domain. To accommodate for the change of coordinates the magnitude of the Jacobian determinant arises as a multiplicative factor within the integral. This is because the $n$ -dimensional $dV$ element is in general a parallelepiped in the new coordinate system, and the $n$ -volume of a parallelepiped is the determinant of its edge vectors.
  The Jacobian can also be used to solve systems of differential equations at an equilibrium point or approximate solutions near an equilibrium point. Its applications include determining the stability of the disease-free equilibrium in disease modelling.

1994

(Simon & Blume, 1994) ⇒ C. P. Simon, and L. E. Blume (1994). "Mathematics for Economists". In: New York: W. W. Norton.