Jacobian matrix and determinant

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In vector calculus, the Jacobian matrix (Template:IPAc-en,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Template:IPAc-en) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. If this matrix is square, that is, if the number of variables equals the number of components of function values, then its determinant is called the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> They are named after Carl Gustav Jacob Jacobi.

The Jacobian matrix is the natural generalization to vector valued functions of several variables of the derivative and the differential of a usual function. This generalization includes generalizations of the inverse function theorem and the implicit function theorem, where the non-nullity of the derivative is replaced by the non-nullity of the Jacobian determinant, and the multiplicative inverse of the derivative is replaced by the inverse of the Jacobian matrix.

The Jacobian determinant is fundamentally used for changes of variables in multiple integrals.

Definition

Let <math display="inline">\mathbf{f}: \mathbb{R}^n \to \mathbb{R}^m</math> be a function such that each of its first-order partial derivatives exists on <math display="inline>\mathbb{R}^n</math>. This function takes a point Template:Tmath as input and produces the vector Template:Tmath as output. Then the Jacobian matrix of Template:Math, denoted Template:Math, is the Template:Tmath matrix whose Template:Math entry is <math display="inline">\frac{\partial f_i}{\partial x_j};</math> explicitly <math display="block">\mathbf{J_f} = \begin{bmatrix}

 \dfrac{\partial \mathbf{f}}{\partial x_1} & \cdots & \dfrac{\partial \mathbf{f}}{\partial x_n}

\end{bmatrix} = \begin{bmatrix}

 \nabla^{\mathsf{T}} f_1 \\  
 \vdots \\
 \nabla^{\mathsf{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>\nabla^{\mathsf{T}} f_i</math> is the transpose (row vector) of the gradient of the <math>i</math>-th component.

The Jacobian matrix, whose entries are functions of Template:Math, is denoted in various ways; other common notations include Template:Math, <math>\nabla \mathbf{f}</math>, and <math display="inline">\frac{\partial(f_1,\ldots,f_m)}{\partial(x_1,\ldots,x_n)}</math>.<ref>Template:Cite book</ref><ref>Template:Cite book</ref> Some authors define the Jacobian as the transpose of the form given above.

The Jacobian matrix represents the differential of Template:Math at every point where Template:Math is differentiable. In detail, if Template:Math is a displacement vector represented by a column matrix, the matrix product Template:Math is another displacement vector, that is the best linear approximation of the change of Template:Math in a neighborhood of Template:Math, if Template:Math is differentiable at Template:Math.Template:Efn This means that the function that maps Template:Math to Template:Math is the best linear approximation of Template:Math for all points Template:Math close to Template:Math. The linear map Template:Math is known as the derivative or the differential of Template:Math at Template:Math.

When <math display="inline">m=n</math>, the Jacobian matrix is square, so its determinant is a well-defined function of Template:Math, known as the Jacobian determinant of Template:Math. It carries important information about the local behavior of Template:Math. In particular, the function Template:Math has a differentiable inverse function in a neighborhood of a point Template:Math if and only if the Jacobian determinant is nonzero at Template:Math (see inverse function theorem for an explanation of this and Jacobian conjecture for a related problem of global invertibility). The Jacobian determinant also appears when changing the variables in multiple integrals (see substitution rule for multiple variables).

When <math display="inline">m=1</math>, that is when <math display="inline"> f: \mathbb{R}^n \to \mathbb{R}</math> is a scalar-valued function, the Jacobian matrix reduces to the row vector <math>\nabla^{\mathsf{T}} f</math>; this row vector of all first-order partial derivatives of Template:Tmath is the transpose of the gradient of Template:Tmath, i.e. <math>\mathbf{J}_{f} = \nabla^{\mathsf{T}} f</math>. Specializing further, when <math display="inline">m=n=1</math>, that is when <math display="inline">f: \mathbb{R} \to \mathbb{R}</math> is a scalar-valued function of a single variable, the Jacobian matrix has a single entry; this entry is the derivative of the function Template:Tmath.

These concepts are named after the mathematician Carl Gustav Jacob Jacobi (1804–1851).

Jacobian matrix

The Jacobian of a vector-valued function in several variables generalizes the gradient of a scalar-valued function in several variables, which in turn generalizes the derivative of a scalar-valued function of a single variable. In other words, the Jacobian matrix of a scalar-valued function of several variables is (the transpose of) its gradient and the gradient of a scalar-valued function of a single variable is its derivative.

At each point where a function is differentiable, its Jacobian matrix can also be thought of as describing the amount of "stretching", "rotating" or "transforming" that the function imposes locally near that point. For example, if Template:Math is used to smoothly transform an image, the Jacobian matrix Template:Math, describes how the image in the neighborhood of Template:Math is transformed.

If a function is differentiable at a point, its differential is given in coordinates by the Jacobian matrix. However, a function does not need to be differentiable for its Jacobian matrix to be defined, since only its first-order partial derivatives are required to exist.

If Template:Math is differentiable at a point Template:Math in Template:Math, then its differential is represented by Template:Math. In this case, the linear transformation represented by Template:Math is the best linear approximation of Template:Math near the point Template:Math, in the sense that

<math display="block">\mathbf f(\mathbf x) - \mathbf f(\mathbf p) = \mathbf J_{\mathbf f}(\mathbf p)(\mathbf x - \mathbf p) + o(\|\mathbf x - \mathbf p\|) \quad (\text{as } \mathbf{x} \to \mathbf{p}),</math>

where Template:Math is a quantity that approaches zero much faster than the distance between Template:Math and Template:Math does as Template:Math approaches Template:Math. This approximation specializes to the approximation of a scalar function of a single variable by its Taylor polynomial of degree one, namely

<math display="block">f(x) - f(p) = f'(p) (x - p) + o(x - p) \quad (\text{as } x \to p).</math>

In this sense, the Jacobian may be regarded as a kind of "first-order derivative" of a vector-valued function of several variables. In particular, this means that the gradient of a scalar-valued function of several variables may too be regarded as its "first-order derivative".

Composable differentiable functions Template:Math and Template:Math satisfy the chain rule, namely <math>\mathbf{J}_{\mathbf{g} \circ \mathbf{f}}(\mathbf{x}) = \mathbf{J}_{\mathbf{g}}(\mathbf{f}(\mathbf{x})) \mathbf{J}_{\mathbf{f}}(\mathbf{x})</math> for Template:Math in Template:Math.

The Jacobian of the gradient of a scalar function of several variables has a special name: the Hessian matrix, which in a sense is the "second derivative" of the function in question.

Jacobian determinant

File:Jacobian determinant and distortion.svg

A nonlinear map <math>f \colon \mathbb{R}^{2} \to \mathbb{R}^{2}</math> sends a small square (left, in red) to a distorted parallelogram (right, in red). The Jacobian at a point gives the best linear approximation of the distorted parallelogram near that point (right, in translucent white), and the Jacobian determinant gives the ratio of the area of the approximating parallelogram to that of the original square.

If Template:Math, then Template:Math is a function from Template:Math 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 Template:Math near that point. For instance, the continuously differentiable function Template:Math is invertible near a point Template:Math if the Jacobian determinant at Template:Math is non-zero. This is the inverse function theorem. Furthermore, if the Jacobian determinant at Template:Math is positive, then Template:Math preserves orientation near Template:Math; if it is negative, Template:Math reverses orientation. The absolute value of the Jacobian determinant at Template:Math gives us the factor by which the function Template:Math expands or shrinks volumes near Template:Math; 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 Template:Math-dimensional Template:Math element is in general a parallelepiped in the new coordinate system, and the Template:Math-volume of a parallelepiped is the determinant of its edge vectors.

The Jacobian can also be used to determine the stability of equilibria for systems of differential equations by approximating behavior near an equilibrium point.

Inverse

According to the inverse function theorem, the matrix inverse of the Jacobian matrix of an invertible function Template:Math is the Jacobian matrix of the inverse function. That is, the Jacobian matrix of the inverse function at a point Template:Math is

<math display="block">\mathbf J_{\mathbf{f}^{-1}}(\mathbf{p}) = {\mathbf J^{-1}_{\mathbf{f}}(\mathbf{f}^{-1}(\mathbf{p}))},</math>

and the Jacobian determinant is

<math display="block">\det(\mathbf{J}_{\mathbf{f}^{-1}}(\mathbf{p})) = \frac{1}{\det(\mathbf{J}_{\mathbf{f}}(\mathbf{f}^{-1}(\mathbf{p})))}.</math>

If the Jacobian is continuous and nonsingular at the point Template:Math in Template:Math, then Template:Math is invertible when restricted to some neighbourhood of Template:Math. In other words, if the Jacobian determinant is not zero at a point, then the function is locally invertible near this point.

The (unproved) Jacobian conjecture is related to global invertibility in the case of a polynomial function, that is a function defined by n polynomials in n variables. It asserts that, if the Jacobian determinant is a non-zero constant (or, equivalently, that it does not have any complex zero), then the function is invertible and its inverse is a polynomial function.

Critical points

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If Template:Math is a differentiable function, a critical point of Template:Math is a point where the rank of the Jacobian matrix is not maximal. This means that the rank at the critical point is lower than the rank at some neighbour point. In other words, let Template:Math be the maximal dimension of the open balls contained in the image of Template:Math; then a point is critical if all minors of rank Template:Math of Template:Math are zero.

In the case where Template:Math, a point is critical if the Jacobian determinant is zero.

Examples

Example 1

Consider a function Template:Math with Template:Math given by

<math display="block">\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>

Then we have

<math display="block">f_1(x, y) = x^2 y</math>

and

<math display="block">f_2(x, y) = 5 x + \sin y.</math>

The Jacobian matrix of Template:Math is

<math display="block">\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 display="block">\det(\mathbf J_{\mathbf f}(x, y)) = 2 x y \cos y - 5 x^2.</math>

Example 2: polar-Cartesian transformation

The transformation from polar coordinates Template:Math to Cartesian coordinates (x, y), is given by the function Template:Math with components

<math display="block">\begin{align} x &= r \cos \varphi ; \\ y &= r \sin \varphi . \end{align}</math>

<math display="block">\mathbf J_{\mathbf F}(r, \varphi) = \begin{bmatrix}

   \frac{\partial x}{\partial r} & \frac{\partial x}{\partial\varphi}\\[0.5ex]
   \frac{\partial y}{\partial r} & \frac{\partial y}{\partial\varphi}
 \end{bmatrix}

= \begin{bmatrix}

   \cos\varphi & - r\sin \varphi \\
   \sin\varphi &   r\cos \varphi
 \end{bmatrix}</math>

The Jacobian determinant is equal to Template:Math. This can be used to transform integrals between the two coordinate systems:

<math display="block">\iint_{\mathbf F(A)} f(x, y) \,dx \,dy = \iint_A f(r \cos \varphi, r \sin \varphi) \, r \, dr \, d\varphi .</math>

Example 3: spherical-Cartesian transformation

The transformation from spherical coordinates Template:Math<ref>Joel Hass, Christopher Heil, and Maurice Weir. Thomas' Calculus Early Transcendentals, 14e. Pearson, 2018, p. 959.</ref> to Cartesian coordinates (x, y, z), is given by the function Template:Math with components

<math display="block">\begin{align} x &= \rho \sin \varphi \cos \theta ; \\ y &= \rho \sin \varphi \sin \theta ; \\ z &= \rho \cos \varphi . \end{align}</math>

The Jacobian matrix for this coordinate change is

<math display="block">\mathbf J_{\mathbf F}(\rho, \varphi, \theta) = \begin{bmatrix}

 \dfrac{\partial x}{\partial \rho} & \dfrac{\partial x}{\partial \varphi} & \dfrac{\partial x}{\partial \theta} \\[1em]
 \dfrac{\partial y}{\partial \rho} & \dfrac{\partial y}{\partial \varphi} & \dfrac{\partial y}{\partial \theta} \\[1em]
 \dfrac{\partial z}{\partial \rho} & \dfrac{\partial z}{\partial \varphi} & \dfrac{\partial z}{\partial \theta}
 \end{bmatrix}

= \begin{bmatrix}

     \sin \varphi \cos \theta & \rho \cos \varphi \cos \theta & -\rho \sin \varphi \sin \theta \\
     \sin \varphi \sin \theta & \rho \cos \varphi \sin \theta & \rho \sin \varphi \cos \theta \\
     \cos \varphi & - \rho \sin \varphi & 0
 \end{bmatrix}.</math>

The determinant is Template:Math. Since Template:Math is the volume for a rectangular differential volume element (because the volume of a rectangular prism is the product of its sides), we can interpret Template:Math as the volume of the spherical differential volume element. Unlike rectangular differential volume element's volume, this differential volume element's volume is not a constant, and varies with coordinates (Template:Math and Template:Math). It can be used to transform integrals between the two coordinate systems:

<math display="block">\iiint_{\mathbf F(U)} f(x, y, z) \,dx \,dy \,dz = \iiint_U f(\rho \sin \varphi \cos \theta, \rho \sin \varphi\sin \theta, \rho \cos \varphi) \, \rho^2 \sin \varphi \, d\rho \, d\varphi \, d\theta .</math>

Example 4

The Jacobian matrix of the function Template:Math with components

<math display="block">\begin{align} y_1 &= x_1 \\ y_2 &= 5 x_3 \\ y_3 &= 4 x_2^2 - 2 x_3 \\ y_4 &= x_3 \sin x_1 \end{align}</math>

is

<math display="block">\mathbf J_{\mathbf F}(x_1, x_2, x_3) = \begin{bmatrix}

 \dfrac{\partial y_1}{\partial x_1} & \dfrac{\partial y_1}{\partial x_2} & \dfrac{\partial y_1}{\partial x_3} \\[1em]
 \dfrac{\partial y_2}{\partial x_1} & \dfrac{\partial y_2}{\partial x_2} & \dfrac{\partial y_2}{\partial x_3} \\[1em]
 \dfrac{\partial y_3}{\partial x_1} & \dfrac{\partial y_3}{\partial x_2} & \dfrac{\partial y_3}{\partial x_3} \\[1em]
 \dfrac{\partial y_4}{\partial x_1} & \dfrac{\partial y_4}{\partial x_2} & \dfrac{\partial y_4}{\partial x_3} \end{bmatrix}

= \begin{bmatrix}

 1 & 0 & 0 \\
 0 & 0 & 5 \\
 0 & 8 x_2 & -2 \\
 x_3\cos x_1 & 0 & \sin x_1 \end{bmatrix}.</math>

This example shows that the Jacobian matrix need not be a square matrix.

Example 5

The Jacobian determinant of the function Template:Math with components

<math display="block">\begin{align}

 y_1 &= 5x_2 \\
 y_2 &= 4x_1^2 - 2 \sin (x_2 x_3) \\
 y_3 &= x_2 x_3

\end{align}</math>

is

<math display="block">\begin{vmatrix}

 0 & 5 & 0 \\
 8 x_1 & -2 x_3 \cos(x_2 x_3) & -2 x_2 \cos (x_2 x_3) \\
 0 & x_3 & x_2

\end{vmatrix} = -8 x_1 \begin{vmatrix}

 5 & 0 \\
 x_3 & x_2

\end{vmatrix} = -40 x_1 x_2.</math>

From this we see that Template:Math reverses orientation near those points where Template:Math and Template:Math have the same sign; the function is locally invertible everywhere except near points where Template:Math or Template:Math. Intuitively, if one starts with a tiny object around the point Template:Math and apply Template:Math to that object, one will get a resulting object with approximately Template:Math times the volume of the original one, with orientation reversed.

Other uses

Dynamical systems

Consider a dynamical system of the form <math>\dot{\mathbf{x}} = F(\mathbf{x})</math>, where <math>\dot{\mathbf{x}}</math> is the (component-wise) derivative of <math>\mathbf{x}</math> with respect to the evolution parameter <math>t</math> (time), and <math>F \colon \mathbb{R}^{n} \to \mathbb{R}^{n}</math> is differentiable. If <math>F(\mathbf{x}_{0}) = 0</math>, then <math>\mathbf{x}_{0}</math> is a stationary point (also called a steady state). By the Hartman–Grobman theorem, the behavior of the system near a stationary point is related to the eigenvalues of <math>\mathbf{J}_{F} \left( \mathbf{x}_{0} \right)</math>, the Jacobian of <math>F</math> at the stationary point.<ref>Template:Cite book </ref> Specifically, if the eigenvalues all have real parts that are negative, then the system is stable near the stationary point. If any eigenvalue has a real part that is positive, then the point is unstable. If the largest real part of the eigenvalues is zero, the Jacobian matrix does not allow for an evaluation of the stability.<ref>Template:Cite book</ref>

Newton's method

A square system of coupled nonlinear equations can be solved iteratively by Newton's method. This method uses the Jacobian matrix of the system of equations.

Regression and least squares fitting

The Jacobian serves as a linearized design matrix in statistical regression and curve fitting; see non-linear least squares. The Jacobian is also used in random matrices, moments, local sensitivity and statistical diagnostics.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>

See also

• Center manifold
• Hessian matrix
• Pushforward (differential)

Further reading

• Template:Cite book
• Template:Cite book

External links

• Template:Springer
• Mathworld A more technical explanation of Jacobians