Calculus 



Integral calculus
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In vector calculus, the Jacobian matrix (//, //) is the matrix of all firstorder partial derivatives of a vectorvalued function. Specifically, suppose $F:\; \backslash mathbb\{R\}^n\; \backslash rightarrow\; \backslash mathbb\{R\}^m$ is a function (which takes as input real ntuples and produces as output real mtuples). Such a function is given by m realvalued component functions,
$F\_1(x\_1,\backslash ldots,x\_n),\backslash ldots,F\_m(x\_1,\backslash ldots,x\_n)$. The partial derivatives of all these functions with respect to the variables $x\_1,\backslash ldots,x\_n$ (if they exist) can be organized in an mbyn matrix, the Jacobian matrix $J$ of $F$, as follows:
 $J=\backslash begin\{bmatrix\}\; \backslash dfrac\{\backslash partial\; F\_1\}\{\backslash partial\; x\_1\}\; \&\; \backslash cdots\; \&\; \backslash dfrac\{\backslash partial\; F\_1\}\{\backslash partial\; x\_n\}\; \backslash \backslash \; \backslash vdots\; \&\; \backslash ddots\; \&\; \backslash vdots\; \backslash \backslash \; \backslash dfrac\{\backslash partial\; F\_m\}\{\backslash partial\; x\_1\}\; \&\; \backslash cdots\; \&\; \backslash dfrac\{\backslash partial\; F\_m\}\{\backslash partial\; x\_n\}\; \backslash end\{bmatrix\}.$
This matrix, whose entries are functions of $x\_1,\backslash ldots,x\_n$, is also denoted by $J\_F(x\_1,\backslash ldots,x\_n)$ and $\backslash frac\{\backslash partial(F\_1,\backslash ldots,F\_m)\}\{\backslash partial(x\_1,\backslash ldots,x\_n)\}$. (Note that some books define the Jacobian as the transpose of the matrix given above.)
The Jacobian matrix is important because if the function F is differentiable at a point $p=(x\_1,\backslash ldots,x\_n)$ (this is a slightly stronger condition than merely requiring that all partial derivatives exist there), then the Jacobian matrix defines a linear map $\backslash mathbb\{R\}^n\; \backslash rightarrow\; \backslash mathbb\{R\}^m$, which is the best linear approximation of the function F near the point p. This linear map is thus the generalization the usual notion of derivative, and is called the derivative or the differential of F at p.
In the case $m=n$ the Jacobian matrix is a square matrix, and its determinant, a function of $x\_1,\backslash ldots,x\_n$, is the Jacobian determinant of F. It carries important information about the local behavior of F. In particular, the function F is locally invertible in the neighborhood of a point p if and only if the Jacobian determinant is nonzero at p (see Jacobian conjecture). The Jacobian determinant occurs also when changing the variables in multivariable integrals (see substitution rule for multiple variables).
If m = 1, the Jacobian matrix has a single row, and may be identified with a vector, which is the gradient.
These concepts are named after the mathematician Carl Gustav Jacob Jacobi (18041851).
A simple example
Consider the function $F:\; \backslash mathbb\{R\}^2\; \backslash rightarrow\; \backslash mathbb\{R\}^2$ given by
 $F(x,y)=\backslash begin\{bmatrix\}\; x^2\; y\; \backslash \backslash $
5x + \sin(y)
\end{bmatrix}.
Then we have
 $F\_1(x,y)=x^2\; y$
and
 $F\_2(x,y)=5x\; +\; \backslash sin(y)$
and the Jacobian matrix of F is
 $J\_F(x,y)=\backslash begin\{bmatrix\}\; \backslash dfrac\{\backslash partial\; F\_1\}\{\backslash partial\; x\}\; \&\; \backslash dfrac\{\backslash partial\; F\_1\}\{\backslash partial\; y\}\backslash \backslash $
\dfrac{\partial F_2}{\partial x} & \dfrac{\partial F_2}{\partial y}
\end{bmatrix}=\begin{bmatrix} 2xy & x^2\\
5 & \cos(y)
\end{bmatrix}
and the Jacobian determinant is
 $\backslash det(J\_F(x,y))=2xy\; \backslash cos(y)\; \; 5x^2.$
Jacobian matrix
The Jacobian generalizes the gradient of a scalarvalued function of multiple variables, which itself generalizes the derivative of a scalarvalued function of a single variable. In other words, the Jacobian for a scalarvalued multivariable function is the gradient and that of a scalarvalued 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 $(x\_2,y\_2)=f(x\_1,y\_1)$ is used to transform an image, the Jacobian of $f$, $J(x\_1,y\_1)$ describes how the image in the neighborhood of $(x\_1,y\_1)$ is transformed.
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 derivatives are required to exist.
If p is a point in R^{n} and F is differentiable at p, then its derivative is given by J_{F}(p). In this case, the linear map described by J_{F}(p) is the best linear approximation of F near the point p, in the sense that
 $F(\backslash mathbf\{x\})\; =\; F(\backslash mathbf\{p\})\; +\; J\_F(\backslash mathbf\{p\})(\backslash mathbf\{x\}\backslash mathbf\{p\})\; +\; o(\backslash \backslash mathbf\{x\}\backslash mathbf\{p\}\backslash )$
for x close to p and where o is the little onotation (for $x\backslash to\; p$) and $\backslash \backslash mathbf\{x\}\backslash mathbf\{p\}\backslash $ is the distance between x and p.
Compare this to a Taylor series for a scalar function of a scalar argument, truncated to first order:
 $f(x)\; =\; f(p)\; +\; f\text{'}(p)\; (\; x\; \; p\; )\; +\; o(xp).$
In a sense, both the gradient and Jacobian are "first derivatives" Template: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.
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.
Inverse
According to the inverse function theorem, the 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^{n} → R^{n} is continuous and nonsingular at the point p in R^{n}, then F is invertible when restricted to some neighborhood of p and
 $(J\_\{F^\{1\}\})(F(p))\; =\; [\; (J\_F)(p)\; ]^\{1\}.\backslash $
Uses
Dynamical systems
Consider a dynamical system of the form x' = F(x), where x' is the (componentwise) time derivative of x, and F : R^{n} → R^{n} is continuous and differentiable. If F(x_{0}) = 0, then x_{0} 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 eigenvalues of J_{F}(x_{0}), the Jacobian of F at the stationary point.^{[1]} 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.
Newton's method
A system of coupled nonlinear equations can be solved iteratively by Newton's method. This method uses the Jacobian matrix of the system of equations.
The following is the detail code in MATLAB (although there is a built in 'jacobian' command)
function s = newton(f, x, tol)
% f is a multivariable function handle, x is a starting point, both given as row vectors
% s is solution of f(s)=0 found by Newton's method
if nargin == 2
tol = 10^(5);
end
while 1
% if x and f(x) are row vectors, we need transpose operations here
y = x'  jacob(f, x)\f(x)'; % get the next point
if norm(f(y))
function j = jacob(f, x) % approximately calculate Jacobian matrix
k = length(x);
j = zeros(k, k);
x2 = x;
dx = 0.001;
for m = 1: k
x2(m) = x(m)+dx;
j(m, :) = (f(x2)f(x))/dx; % partial derivatives in mth row
x2(m) = x(m);
end
Jacobian determinant
If m = n, then F is a function from nspace to nspace 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."
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^{n} if the Jacobian determinant at p is nonzero. 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 rectilinear ndimensional dV element is in general a parallelapiped in the new coordinate system, and the nvolume 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.
Further examples
Example 1. The transformation from spherical coordinates (r, θ, φ) to Cartesian coordinates (x_{1}, x_{2}, x_{3}), is given by the function F : R^{+} × [0,π] × [0,2π) → R^{3} with components:
 $x\_1\; =\; r\backslash ,\; \backslash sin\backslash theta\backslash ,\; \backslash cos\backslash phi\; \backslash ,$
 $x\_2\; =\; r\backslash ,\; \backslash sin\backslash theta\backslash ,\; \backslash sin\backslash phi\; \backslash ,$
 $x\_3\; =\; r\backslash ,\; \backslash cos\backslash theta.\; \backslash ,$
The Jacobian matrix for this coordinate change is
 $J\_F(r,\backslash theta,\backslash phi)\; =\backslash begin\{bmatrix\}$
\dfrac{\partial x_1}{\partial r} & \dfrac{\partial x_1}{\partial \theta} & \dfrac{\partial x_1}{\partial \phi} \\[3pt]
\dfrac{\partial x_2}{\partial r} & \dfrac{\partial x_2}{\partial \theta} & \dfrac{\partial x_2}{\partial \phi} \\[3pt]
\dfrac{\partial x_3}{\partial r} & \dfrac{\partial x_3}{\partial \theta} & \dfrac{\partial x_3}{\partial \phi} \\
\end{bmatrix}=\begin{bmatrix}
\sin\theta\, \cos\phi & r\, \cos\theta\, \cos\phi & r\, \sin\theta\, \sin\phi \\
\sin\theta\, \sin\phi & r\, \cos\theta\, \sin\phi & r\, \sin\theta\, \cos\phi \\
\cos\theta & r\, \sin\theta & 0
\end{bmatrix}.
The determinant is r^{2} sin θ. As an example, since dV = dx_{1} dx_{2} dx_{3} this determinant implies that the differential volume element dV = r^{2} sin θ dr dθ dϕ. Nevertheless this determinant varies with coordinates.
Example 2. The Jacobian matrix of the function F : R^{3} → R^{4} with components
 $y\_1\; =\; x\_1\; \backslash ,$
 $y\_2\; =\; 5x\_3\; \backslash ,$
 $y\_3\; =\; 4x\_2^2\; \; 2x\_3\; \backslash ,$
 $y\_4\; =\; x\_3\; \backslash sin(x\_1)\; \backslash ,$
is
 $J\_F(x\_1,x\_2,x\_3)\; =\backslash begin\{bmatrix\}$
\dfrac{\partial y_1}{\partial x_1} & \dfrac{\partial y_1}{\partial x_2} & \dfrac{\partial y_1}{\partial x_3} \\[3pt]
\dfrac{\partial y_2}{\partial x_1} & \dfrac{\partial y_2}{\partial x_2} & \dfrac{\partial y_2}{\partial x_3} \\[3pt]
\dfrac{\partial y_3}{\partial x_1} & \dfrac{\partial y_3}{\partial x_2} & \dfrac{\partial y_3}{\partial x_3} \\[3pt]
\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 & 8x_2 & 2 \\ x_3\cos(x_1) & 0 & \sin(x_1) \end{bmatrix}.
This example shows that the Jacobian need not be a square matrix.
Example 3.
 $x\backslash ,=r\backslash ,\backslash cos\backslash ,\backslash phi;$
 $y\backslash ,=r\backslash ,\backslash sin\backslash ,\backslash phi.$
$J(r,\backslash phi)=\backslash begin\{bmatrix\}\; \{\backslash partial\; x\backslash over\backslash partial\; r\}\; \&\; \{\backslash partial\; x\backslash over\; \backslash partial\backslash phi\}\; \backslash \backslash \; \{\backslash partial\; y\backslash over\; \backslash partial\; r\}\; \&\; \{\backslash partial\; y\backslash over\; \backslash partial\backslash phi\}\; \backslash end\{bmatrix\}=\backslash begin\{bmatrix\}\; \{\backslash partial\; (r\backslash cos\backslash phi)\backslash over\; \backslash partial\; r\}\; \&\; \{\backslash partial\; (r\backslash cos\backslash phi)\backslash over\; \backslash partial\; \backslash phi\}\; \backslash \backslash \; \{\backslash partial(r\backslash sin\backslash phi)\backslash over\; \backslash partial\; r\}\; \&\; \{\backslash partial\; (r\backslash sin\backslash phi)\backslash over\; \backslash partial\backslash phi\}\; \backslash end\{bmatrix\}=\backslash begin\{bmatrix\}\; \backslash cos\backslash phi\; \&\; r\backslash sin\backslash phi\; \backslash \backslash \; \backslash sin\backslash phi\; \&\; r\backslash cos\backslash phi\; \backslash end\{bmatrix\}$
The Jacobian determinant is equal to $r$.
This shows how an integral in the Cartesian coordinate system is transformed into an integral in the polar coordinate system:
 $\backslash iint\_A\; dx\backslash ,\; dy=\; \backslash iint\_B\; r\; \backslash ,dr\backslash ,\; d\backslash phi.$
Example 4.
The Jacobian determinant of the function F : R^{3} → R^{3} with components
 $\backslash begin\{align\}$
y_1 &= 5x_2 \\
y_2 &= 4x_1^2  2 \sin (x_2x_3) \\
y_3 &= x_2 x_3
\end{align}
is
 $\backslash begin\{vmatrix\}$
0 & 5 & 0 \\
8 x_1 & 2 x_3 \cos(x_2 x_3) & 2x_2\cos(x_2 x_3) \\
0 & x_3 & x_2
\end{vmatrix} = 8 x_1 \cdot \begin{vmatrix}
5 & 0 \\
x_3 & x_2
\end{vmatrix} = 40 x_1 x_2.
From this we see that F reverses orientation near those points where x_{1} and x_{2} have the same sign; the function is locally invertible everywhere except near points where x_{1} = 0 or x_{2} = 0. Intuitively, if you start with a tiny object around the point (1,2,3) and apply F to that object, you will get a resulting object with approximately 40×1×2=80 times the volume of the original one.
See also
Notes
External links
 Template:Springer
 Mathworld A more technical explanation of Jacobians
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