Jacobian determinant

The Jacobian or Jacobian is a mathematical quantity, plays a role in the multi-dimensional integral calculus, ie the calculation of surface and volume integrals. In particular, they found in the area formula and the products resulting from this transform set using.

Local behavior of a function

The Jacobian is at a given point important information about the behavior of the function near that point. For example, if the Jacobian of a continuously differentiable function at a point is non-zero, the function is invertible in a neighborhood of. The principle that with positive determinant in the function retains its orientation and with negative Jacobian reverses the orientation. The absolute value of the determinant in point is the value that the function expands the near or shrinks.

Definition

For differentiable functions, the Jacobian is defined as

Thus, as the determinant of the Jacobian matrix.

For the transformation of volume elements, an important application in physics, this definition is sufficient. The area formula of measure and integration theory describes the other hand also, as integrals over functions that map verschiedendimensionale spaces into each other to transform. In this use case is not a square matrix more, so the term is not defined. One manages by the following definition:

The generalized Jacobian of a function is defined as

This refers to the Jacobian matrix and its transpose. The term is called the Gram determinant of.

As long as the picture is not considered self-image, it is customary to omit the prefix generalized. In self-images, though it can lead to misunderstandings, since both definitions generally have different values ​​. It is indeed

In the context of surface or transformation formula, however, the amount will in any case always used.

Examples

In the integration on geometric objects, it is often impractical to integrate on Cartesian coordinates. Means that even in physics, the integral over a radially symmetric potential field whose value depends only on a radius much easier to calculate in spherical coordinates.

To do this, one applies a coordinate transformation. After the transformation rate then applicable in this example:

The following bills are listed at the two coordinate systems:

Spherical coordinates

The conversion formulas of spherical coordinates ( ) are denominated in Cartesian coordinates:

The Jacobian is:

Consequently, the result for the volume element:

Cylindrical coordinates

The conversion formulas of cylindrical coordinates (, , ) are denominated in Cartesian coordinates:

The Jacobian is:

Consequently, the result for the volume element:

Just as well you could have chosen a different sequence of cylindrical coordinates. The Jacobian is then, for example:

In the transformation law, however, only the value of the determinant is one, so the result is independent of the chosen order of variables, is derived according to which.