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| {{Calculus |Multivariable}}
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| In [[vector calculus]], the '''Jacobian matrix''' ({{IPAc-en|dʒ|i-|ˈ|k|oʊ|b|i|ə|n}}, {{IPAc-en|j|i-|ˈ|k|oʊ|b|i|ə|n}}) is the [[matrix (mathematics)|matrix]] of all first-order [[partial derivative]]s of a [[real coordinate space|vector]]-valued [[function (mathematics)|function]]. Specifically, suppose <math>F \colon \mathbb{R}^n \rightarrow \mathbb{R}^m</math> is a function (which takes as input [[Tuple|real ''n''-tuples]] and produces as output real ''m''-tuples). Such a function is given by ''m'' real-valued component functions,
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| <math>F_1(x_1,\dotsc,x_n),\dotsc,F_m(x_1,\dotsc,x_n)</math>. The partial derivatives of all these functions with respect to the variables <math>x_1,\dotsc,x_n</math> (if they exist) can be organized in an ''m''-by-''n'' matrix, the Jacobian matrix <math>J</math> of <math>F</math>, as follows:
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| :<math>J=\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>
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| This matrix, whose entries are functions of <math>x_1,\dotsc,x_n</math>, is also denoted by <math>J_F(x_1,\dotsc,x_n)</math> and <math>\frac{\partial(F_1,\dotsc,F_m)}{\partial(x_1,\dotsc,x_n)}</math>. (Note that some books define the Jacobian as the [[transpose]] of the matrix given above.)
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| The Jacobian matrix is important because if the function ''F'' is [[differentiable]] at a point <math>p=(x_1,\dotsc,x_n)</math> (this is a slightly stronger condition than merely requiring that all partial derivatives exist there), then the Jacobian matrix defines a [[linear map]] <math>\mathbb{R}^n \rightarrow \mathbb{R}^m</math>, which is the best linear approximation of the function ''F'' near the point ''p''. This linear map is thus the generalization of the usual notion of derivative, and is called the ''derivative'' or the ''differential'' of ''F'' at ''p''.
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| In the case <math>m=n</math> the Jacobian matrix is a square matrix, and its [[determinant]], a function of <math>x_1,\dotsc,x_n</math>, is 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 ''p'' an [[inverse function]] that is differentiable if and only if the Jacobian determinant is nonzero at ''p'' (see [[Jacobian conjecture]]). The Jacobian determinant occurs also when changing the variables in multi-variable integrals (see [[Integration_by_substitution#Substitution_for_multiple_variables|substitution rule for multiple variables]]).
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| If ''m'' = 1, the Jacobian matrix has a single row, and may be identified with a vector, which is the [[gradient]].
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| These concepts are named after the [[mathematician]] [[Carl Gustav Jacob Jacobi]] (1804-1851).
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| ==A simple example==
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| Consider the function <math>F \colon \mathbb{R}^2 \rightarrow \mathbb{R}^2</math> given by
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| :<math>F(x,y)=\begin{bmatrix} x^2 y \\
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| 5x + \sin(y)
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| \end{bmatrix}. </math>
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| Then we have
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| :<math>F_1(x,y)=x^2 y</math>
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| and
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| :<math>F_2(x,y)=5x + \sin(y)</math>
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| and the Jacobian matrix of ''F'' is
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| :<math>J_F(x,y)=\begin{bmatrix} \dfrac{\partial F_1}{\partial x} & \dfrac{\partial F_1}{\partial y}\\
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| \dfrac{\partial F_2}{\partial x} & \dfrac{\partial F_2}{\partial y}
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| \end{bmatrix}=\begin{bmatrix} 2xy & x^2\\
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| 5 & \cos(y)
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| \end{bmatrix} </math>
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| and the Jacobian determinant is
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| :<math>\det(J_F(x,y))=2xy \cos(y) - 5x^2.</math>
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| == Jacobian matrix ==
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| The Jacobian generalizes the [[gradient]] of a [[scalar (mathematics)|scalar]]-valued function of multiple variables, which itself generalizes the derivative of a scalar-valued function of a single variable. In other words, the Jacobian for a scalar-valued multivariable function is the gradient and that of a scalar-valued function of single variable is simply its derivative. The Jacobian can also be thought of as describing the amount of "stretching", "rotating" or "transforming" that a transformation imposes locally. For example, if <math>(x_2,y_2)=f(x_1,y_1)</math> is used to transform an image, the Jacobian of <math>f</math>, <math>J(x_1,y_1)</math> describes how the image in the neighborhood of <math>(x_1,y_1)</math> is transformed.
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| If a function is differentiable at a point, its derivative is given in coordinates by the Jacobian, but a function doesn't need to be differentiable for the Jacobian to be defined, since only the [[partial derivative]]s are required to exist.
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| If '''p''' is a point in '''R'''<sup>''n''</sup> and ''F'' is [[derivative|differentiable]] at '''p''', then its derivative is given by ''J<sub>F</sub>''('''p'''). In this case, the [[linear map]] described by ''J<sub>F</sub>''('''p''') is the best [[linear approximation]] of ''F'' near the point '''p''', in the sense that
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| :<math>F(\mathbf{x}) = F(\mathbf{p}) + J_F(\mathbf{p})(\mathbf{x}-\mathbf{p}) + o(\|\mathbf{x}-\mathbf{p}\|)</math>
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| for '''x''' close to '''p''' and where ''o'' is the [[Big_O_notation#Little-o_notation|little o-notation]] (for <math>x\to p</math>) and <math>\|\mathbf{x}-\mathbf{p}\|</math> is the [[Euclidean distance|distance]] between '''x''' and '''p'''.
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| Compare this to a [[Taylor series]] for a scalar function of a scalar argument, truncated to first order:
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| :<math>f(x) = f(p) + f'(p) ( x - p ) + o(x-p).</math>
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| In a sense, both the [[gradient]] and Jacobian are "[[derivative|first derivatives]]" {{mdash}} the former the first derivative of a ''scalar function'' of several variables, the latter the first derivative of a ''vector function'' of several variables.
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| 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.
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| ===Inverse===
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| According to the [[inverse function theorem]], the [[Invertible matrix|matrix inverse]] of the Jacobian matrix of an [[invertible function]] is the Jacobian matrix of the ''inverse'' function. That is, if the Jacobian of the function ''F'': '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup> is continuous and nonsingular at the point ''p'' in '''R'''<sup>''n''</sup>, then ''F'' is invertible when restricted to some neighborhood of ''p'' and
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| :<math> (J_{F^{-1}})(F(p)) = [ (J_F)(p) ]^{-1}.\ </math>
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| ===Uses===
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| ====Dynamical systems====
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| Consider a [[dynamical system]] of the form ''x''<nowiki>'</nowiki> = ''F''(''x''), where ''x''<nowiki>'</nowiki> is the (component-wise) time derivative of ''x'', and ''F'': '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup> is continuous and differentiable. If ''F''(''x''<sub>0</sub>) = 0, then ''x''<sub>0</sub> is a stationary point (also called a critical point, not to be confused with a fixed point). The behavior of the system near a stationary point is related to the [[eigenvalue]]s of ''J''<sub>''F''</sub>(''x''<sub>0</sub>), the Jacobian of ''F'' at the stationary point.<ref>D.K. Arrowsmith and C.M. Place, ''Dynamical Systems'', Section 3.3, Chapman & Hall, London, 1992. ISBN 0-412-39080-9.</ref> Specifically, if the eigenvalues all have real parts that are less than 0, then the system is stable near the stationary point, if any eigenvalue has a real part that is greater than 0, then the point is unstable. If the largest real part of the eigenvalues is equal to 0, the Jacobian matrix does not allow for an evaluation of the stability.
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| ====Newton's method====
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| A system of coupled nonlinear equations can be solved iteratively by [[Newton%27s_method#Nonlinear_systems_of_equations|Newton's method]]. This method uses the Jacobian matrix of the system of equations.
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| The following is the detail code in MATLAB (although there is a built in 'jacobian' command)
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| <source lang="matlab"> | |
| function s = newton(f, x, tol)
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| % f is a multivariable function handle, x is a starting point, both given as row vectors
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| % s is solution of f(s)=0 found by Newton's method
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| if nargin == 2
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| tol = 10^(-5);
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| end
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| while 1
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| % if x and f(x) are row vectors, we need transpose operations here
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| y = x' - jacob(f, x)\f(x)'; % get the next point
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| if norm(f(y))<tol % check error tolerate
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| s = y';
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| return;
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| end
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| x = y';
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| end
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| function j = jacob(f, x) % approximately calculate Jacobian matrix
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| k = length(x);
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| j = zeros(k, k);
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| x2 = x;
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| dx = 0.001;
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| for m = 1: k
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| x2(m) = x(m)+dx;
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| j(m, :) = (f(x2)-f(x))/dx; % partial derivatives in m-th row
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| x2(m) = x(m);
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| end
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| </source> | |
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| ==Jacobian determinant==
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| If ''m'' = ''n'', then ''F'' is a function from ''n''-space to ''n''-space 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 called "the Jacobian."
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| 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''' ∈ '''R'''<sup>''n''</sup> if the Jacobian determinant at '''p''' is non-zero. This is the [[inverse function theorem]]. Furthermore, if the Jacobian determinant at '''p''' is [[positive number|positive]], then ''F'' preserves orientation near '''p'''; if it is [[negative number|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 [[volume]]s near '''p'''; this is why it occurs in the general [[substitution rule]].
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| The Jacobian determinant is used when making a [[Integration by substitution#Substitution for multiple variables|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.
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| The Jacobian can also be used to solve [[matrix differential equation|systems of differential equations]] at an [[equilibrium point]] or approximate solutions near an equilibrium point.
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| ==Further examples==
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| '''Example 1.''' The transformation from [[Spherical coordinate system|spherical coordinates]] (''r'', ''θ'', ''φ'') to [[Cartesian coordinate system|Cartesian coordinates]] (''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>), is given by the function ''F'': '''R'''<sup>+</sup> × [0,π] × [0,2π) → '''R'''<sup>3</sup> with components:
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| :<math> x_1 = r\, \sin\theta\, \cos\phi \,</math>
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| :<math> x_2 = r\, \sin\theta\, \sin\phi \,</math>
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| :<math> x_3 = r\, \cos\theta. \,</math>
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| The Jacobian matrix for this coordinate change is
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| :<math>J_F(r,\theta,\phi) =\begin{bmatrix}
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| \dfrac{\partial x_1}{\partial r} & \dfrac{\partial x_1}{\partial \theta} & \dfrac{\partial x_1}{\partial \phi} \\[3pt]
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| \dfrac{\partial x_2}{\partial r} & \dfrac{\partial x_2}{\partial \theta} & \dfrac{\partial x_2}{\partial \phi} \\[3pt]
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| \dfrac{\partial x_3}{\partial r} & \dfrac{\partial x_3}{\partial \theta} & \dfrac{\partial x_3}{\partial \phi} \\
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| \end{bmatrix}=\begin{bmatrix}
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| \sin\theta\, \cos\phi & r\, \cos\theta\, \cos\phi & -r\, \sin\theta\, \sin\phi \\
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| \sin\theta\, \sin\phi & r\, \cos\theta\, \sin\phi & r\, \sin\theta\, \cos\phi \\
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| \cos\theta & -r\, \sin\theta & 0
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| \end{bmatrix}. </math>
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| The [[determinant]] is ''r''<sup>2</sup> sin ''θ''. As an example, since ''dV'' = ''dx''<sub>1</sub> ''dx''<sub>2</sub> ''dx''<sub>3</sub> this determinant implies that the [[differential volume element]] ''dV'' = ''r''<sup>2</sup> sin ''θ'' ''dr'' ''dθ'' ''dφ''. Nevertheless this determinant varies with coordinates.
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| '''Example 2.''' The Jacobian matrix of the function ''F'': '''R'''<sup>3</sup> → '''R'''<sup>4</sup> with components
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| :<math> y_1 = x_1 \, </math>
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| :<math> y_2 = 5x_3 \, </math>
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| :<math> y_3 = 4x_2^2 - 2x_3 \, </math>
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| :<math> y_4 = x_3 \sin(x_1) \, </math>
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| is
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| :<math>J_F(x_1,x_2,x_3) =\begin{bmatrix}
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| \dfrac{\partial y_1}{\partial x_1} & \dfrac{\partial y_1}{\partial x_2} & \dfrac{\partial y_1}{\partial x_3} \\[3pt]
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| \dfrac{\partial y_2}{\partial x_1} & \dfrac{\partial y_2}{\partial x_2} & \dfrac{\partial y_2}{\partial x_3} \\[3pt]
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| \dfrac{\partial y_3}{\partial x_1} & \dfrac{\partial y_3}{\partial x_2} & \dfrac{\partial y_3}{\partial x_3} \\[3pt]
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| \dfrac{\partial y_4}{\partial x_1} & \dfrac{\partial y_4}{\partial x_2} & \dfrac{\partial y_4}{\partial x_3} \\
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| \end{bmatrix}=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 5 \\ 0 & 8x_2 & -2 \\ x_3\cos(x_1) & 0 & \sin(x_1) \end{bmatrix}. </math>
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| This example shows that the Jacobian need not be a square matrix.
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| '''Example 3.'''
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| :<math>x=r\cos\phi;</math> | |
| :<math>y=r\sin\phi.</math>
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| <math>J(r,\phi)=\begin{bmatrix} {\partial x\over\partial r} & {\partial x\over \partial\phi} \\ {\partial y\over \partial r} & {\partial y\over \partial\phi} \end{bmatrix}=\begin{bmatrix} {\partial (r\cos\phi)\over \partial r} & {\partial (r\cos\phi)\over \partial \phi} \\ {\partial(r\sin\phi)\over \partial r} & {\partial (r\sin\phi)\over \partial\phi} \end{bmatrix}=\begin{bmatrix} \cos\phi & -r\sin\phi \\ \sin\phi & r\cos\phi \end{bmatrix}</math>
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| The Jacobian determinant is equal to <math>r</math>.
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| This shows how an integral in the [[Cartesian coordinate system]] is transformed into an integral in the [[polar coordinate system]]:
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| :<math>\iint_A dx\, dy= \iint_B r \,dr\, d\phi.</math>
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| '''Example 4.'''
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| The Jacobian determinant of the function ''F'': '''R'''<sup>3</sup> → '''R'''<sup>3</sup> with components
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| :<math>\begin{align}
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| y_1 &= 5x_2 \\
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| y_2 &= 4x_1^2 - 2 \sin (x_2x_3) \\
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| y_3 &= x_2 x_3
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| \end{align}</math>
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| is
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| :<math>\begin{vmatrix}
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| 0 & 5 & 0 \\
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| 8 x_1 & -2 x_3 \cos(x_2 x_3) & -2x_2\cos(x_2 x_3) \\
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| 0 & x_3 & x_2
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| \end{vmatrix} = -8 x_1 \cdot \begin{vmatrix}
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| 5 & 0 \\
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| x_3 & x_2
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| \end{vmatrix} = -40 x_1 x_2.</math>
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| From this we see that ''F'' reverses orientation near those points where ''x''<sub>1</sub> and ''x''<sub>2</sub> have the same sign; the function is [[locally]] invertible everywhere except near points where ''x''<sub>1</sub> = 0 or ''x''<sub>2</sub> = 0. Intuitively, if one starts with a tiny object around the point (1,2,3) and apply ''F'' to that object, one will get a resulting object with approximately 40×1×2 = 80 times the volume of the original one.
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| ==See also==
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| * [[Hessian matrix]]
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| * [[Pushforward (differential)]]
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| ==Notes==
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| {{Reflist}}
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| ==External links==
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| * {{springer|title=Jacobian|id=p/j054080}}
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| * [http://mathworld.wolfram.com/Jacobian.html Mathworld] A more technical explanation of Jacobians
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| [[Category:Multivariable calculus]]
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| [[Category:Differential calculus]]
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| [[Category:Generalizations of the derivative]]
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| [[Category:Determinants]]
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| [[Category:Matrices]]
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