In
vector calculusVector calculus is a branch of mathematics concerned with differentiation and integration of vector fields. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple...
, the
Jacobian matrix is the
matrixIn mathematics, a matrix is a rectangular array of numbers, such asEntries of a matrix are often denoted by a variable with two subscripts, as shown on the right. Matrices of the same size can be added and subtracted entrywise and matrices of compatible size can be multiplied...
of all first-order
partial derivativeIn mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant...
s of a
vectorA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
-valued
functionIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
. Suppose
F :
Rn →
Rm is a function from
Euclidean n-spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions higher dimensions...
to Euclidean
m-space. Such a function is given by
m real-valued component functions,
y1(
x1,...,
xn), ...,
ym(
x1,...,
xn). The partial derivatives of all these functions (if they exist) can be organized in an
m-by-
n matrix, the Jacobian matrix
J of
F, as follows:
This matrix is also denoted by and . The
i th row (
i = 1, ...,
m) of this matrix is the
gradientIn vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
of the
ith component function
yi: .
The
Jacobian determinant (often simply called the
Jacobian) is the determinant of the Jacobian matrix.
These concepts are named after the
mathematicianA mathematician is a person whose primary area of study and/or research is the field of mathematics. Mathematicians are concerned with particular problems related to logic, space, transformations, numbers and more general ideas which encompass these concepts...
Carl Gustav Jacob Jacobi. The term "Jacobian" is normally , but sometimes also .
Jacobian matrix
The Jacobian of a function describes the orientation of a tangent plane to the function at a given point. In this way, the Jacobian generalizes the
gradientIn vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
of a scalar valued function of multiple variables which itself generalizes the derivative of a scalar-valued function of a scalar. Likewise, the Jacobian can also be thought of as describing the amount of "stretching" that a transformation imposes. For example, if is used to transform an image, the Jacobian of
f, describes how much the image in the neighborhood of is stretched in the
x,
y, and
xy directions.
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 derivativeIn mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant...
s are required to exist.
The importance of the Jacobian lies in the fact that it represents the best linear approximation to a differentiable function near a given point. In this sense, the Jacobian is the derivative of a multivariate function. For a function of
n variables,
n > 1, the derivative of a numerical function must be matrix-valued, or a partial derivative.
If
p is a point in
Rn and
F is
differentiableIn calculus the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point; for example, the derivative of the position of a vehicle with respect to time is the instantaneous velocity...
at
p, then its derivative is given by
JF(
p). In this case, the linear map described by
JF(
p) is the best
linear approximationIn mathematics, a linear approximation is an approximation of a general function using a linear function . They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations.-Definition:Given a twice continuously...
of
F near the point
p, in the sense that
for
x close to
p and where
o is the little o-notation (for , not ) and is the
distanceIn mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space becomes a metric space...
between
x and
p.
In a sense, both gradient and Jacobian are "first derivatives", the former of a scalar function of several variables and the latter of a vector function of several variables. Jacobian of the gradient has a special name: the
Hessian matrixIn mathematics, the Hessian matrix is the square matrix of second-order partial derivatives of a function; that is, it describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named...
which in a sense is the "
second derivativeIn calculus, the second derivative of a function ƒ is the derivative of the derivative of ƒ. Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, the second derivative of the position of a vehicle with respect to time is...
" of the scalar function of several variables in question. (More generally, gradient is a special version of Jacobian; it is the Jacobian of a scalar function of several variables.)
Inverse
According to the
inverse function theoremIn mathematics, specifically differential calculus, the inverse function theorem gives sufficient conditions for a function to be invertible in a neighborhood of a point in its domain...
, the
matrix inverseIn linear algebra, an n-by-n matrix A is called invertible or nonsingular or nondegenerate if there exists an n-by-n matrix B such that...
of the Jacobian matrix of a function is the Jacobian matrix of the
inverse function. That is, for some function
F :
Rn →
Rn and a point
p in
Rn,
.
It follows that the (scalar) inverse of the Jacobian determinant of a transformation is the Jacobian determinant of the inverse transformation.
Examples
Example 1. The transformation from
spherical coordinatesIn mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance from a fixed origin, the elevation angle of that point from a fixed plane, and the azimuth angle of its orthogonal...
(
r,
θ,
φ) to
Cartesian coordinatesA Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length....
(
x1,
x2,
x3) is given by the function
F :
R+ × [0,π) × [0,2π) →
R3 with components:
The Jacobian matrix for this coordinate change is
Example 2. The Jacobian matrix of the function
F :
R3 →
R4 with components
is
This example shows that the Jacobian need not be a square matrix.
In dynamical systems
Consider a
dynamical systemThe dynamical system concept is a mathematical formalization for any fixed "rule" which describes the time dependence of a point's position in its ambient space...
of the form
x' =
F(
x), where
x' is the (component-wise) time derivative of
x, and
F :
Rn →
Rn is continuous and differentiable. If
F(
x0) = 0, then
x0 is a stationary point (also called a fixed point). The behavior of the system near a stationary point is related to the eigenvalues of
JF(
x0), the Jacobian of
F at the stationary point. Specifically, if the eigenvalues all have magnitude less than one, then the point is an attractor, but if any eigenvalue has magnitude greater than one, then the point is unstable.
Jacobian determinant
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
determinantIn algebra, the determinant is a special number associated to any square matrix, that is to say, a rectangular array of numbers where the number of rows and columns are equal. The fundamental geometric meaning of a determinant is a scale factor for measure when the matrix is regarded as a linear...
, known as the
Jacobian determinant. The Jacobian determinant is also called the "Jacobian" in some sources.
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 ∈
Rn if the Jacobian determinant at
p is non-zero. This is the
inverse function theoremIn mathematics, specifically differential calculus, the inverse function theorem gives sufficient conditions for a function to be invertible in a neighborhood of a point in its domain...
. Furthermore, if the Jacobian determinant at
p is
positiveBeing negative or non-negative is a property of a number which is real, or a member of a subset of real numbers such as rational and integer numbers. A negative number is one that is less than zero, such as −, -1.414, -1. A positive number is one that is greater than zero, such as , 1.414, 1...
, then
F preserves orientation near
p; if it is negative,
F reverses orientation. The
absolute valueIn mathematics, the absolute value of a real number is its numerical value without regard to its sign. So, for example, 3 is the absolute value of both 3 and −3.The absolute value of a number is denoted by ....
of the Jacobian determinant at
p gives us the factor by which the function
F expands or shrinks
volumeThe volume of any solid, liquid, gas, plasma, theoretical object, or vacuum is how much three-dimensional space it occupies, often quantified numerically. One-dimensional figures and two-dimensional shapes are assigned zero volume in the three-dimensional space...
s near
p; this is why it occurs in the general substitution rule.
Example
The Jacobian determinant of the function
F :
R3 →
R3 with components
is
From this we see that
F reverses orientation near those points where
x1 and
x2 have the same sign; the function is locally invertible everywhere except near points where
x1 = 0 or
x2 = 0. Intuitively, if you start with a tiny object around the point (1,1,1) and apply
F to that object, you will get an object set with approximately 40 times the volume of the original one.
Uses
The Jacobian determinant is used when making a change of variables when
integratingIntegration is an important concept in mathematics which, together with differentiation, forms one of the main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integralis defined informally...
a function over its domain. To accommodate for the change of coordinates the Jacobian determinant arises as a multiplicative factor within the integral. Normally it is required that the change of coordinates is done in a manner which maintains an injectivity between the coordinates that determine the domain. The Jacobian determinant, as a result, is usually well defined.
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