- Μία Μαθηματική Μήτρα.
Στον Διανυσματικό Λογισμό, the Jacobian is shorthand for either the Jacobian matrix or its [[ορίζουσα] (determinant), the Jacobian determinant.
Jacobian matrix Edit
Its importance lies in the fact that it represents the best γραμμική προσέγγιση to a differentiable function near a given point.
In this sense, the Jacobian is akin to a derivative of a multivariate function.
Suppose F : Rn → Rm is a function from Euclidean n-space 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 of F, as follows:
This matrix is denoted by
- or by
The ith row of this matrix is given by the gradient of the function yi for i=1,...,m.
If p is a point in Rn and F is differentiable at p, then its derivative is given by JF(p) (and this is the easiest way to compute the derivative). In this case, the linear map described by JF(p) is the best linear approximation of F near the point p, in the sense that
for x close to p.
Consider Σφαιρικές Συντεταγμένες (spherical polar coordinates). The following function constitutes a change of variables back to cartesian coordinates.
F : R x [0,2π] x [0,π] → R3 with components:
The Jacobian matrix for this coordinate change is
The Jacobian matrix of the function F : R3 → R4 with components:
This example shows that the Jacobian need not be a square matrix.
Ιακωβιανή Δυναμικών ΣυστημάτωνEdit
Consider a dynamical system of the form x' = F(x), with F : Rn → Rn. If F(x0) = 0, then x0 is a stationary point. The behaviour of the system near a stationary point can often be determined by the eigenvalues of JF(x0), the Jacobian of F at the stationary point.
Jacobian determinant Edit
If m = n, then F is a function from n-space to n-space and the Jacobi matrix is a square matrix. We can then form its determinant, 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 p 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 of the function F : R3 → R3 with components
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. 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 about 40 times the volume of the original one.
The Jacobian determinant is used when making a change of variables when integrating 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.
- ↑ D.K. Arrowsmith and C.M. Place, Dynamical Systems, Section 3.3, Chapman & Hall, London, 1992. ISBN 0-412-39080-9.
- Ian Craw's Undergraduate Teaching Page An easy to understand explanation of Jacobians
- Mathworld A more technical explanation of Jacobians
Αν και θα βρείτε εξακριβωμένες πληροφορίες
"Οι πληροφορίες αυτές μπορεί πρόσφατα
Πρέπει να λάβετε υπ' όψη ότι
- Μην κάνετε χρήση του περιεχομένου της παρούσας εγκυκλοπαίδειας
αν διαφωνείτε με όσα αναγράφονται σε αυτήν
- Όχι, στις διαφημίσεις που περιέχουν απαράδεκτο περιεχόμενο (άσεμνες εικόνες, ροζ αγγελίες κλπ.)