Jacobian

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In mathematics, the Jacobian of a coordinate transformation is the determinant of the functional matrix of Jacobi. This matrix consists of partial derivatives. The Jacobian appears as the weight (measure) in multiple integrals over generalized coordinates. The Jacobian is named after the German mathematician Carl Gustav Jacob Jacobi (1804 - 1851).

Definition

Let f be a map of an open subset T of into with continuous first partial derivatives,

That is if

then

with

The n × n functional matrix of Jacobi consists of partial derivatives

The determinant of this matrix is usually written as

Example

Let T be the subset {r, θ, φ | r > 0, 0 < θ<π, 0 <φ <2π} in and let f be defined by

The Jacobi matrix is

Its determinant can be obtained most conveniently by a Laplace expansion along the third column

The quantities {r, θ, φ} are known as spherical polar coordinates and its Jacobian is r2sinθ.

Coordinate transformation

The map is a coordinate transformation if (i) f has continuous first derivatives on T (ii) f is one-to-one on T and (iii) the Jacobian of f is not equal to zero on T.

Multiple integration

It can be proved [1] that

As an example we consider the spherical polar coordinates mentioned above. Here x = f(t) ≡ f(r, θ, φ) covers all of , while T is the region {r > 0, 0 < θ<π, 0 <φ <2π}. Hence the theorem states that

Geometric interpretation of the Jacobian

The Jacobian has a geometric interpretation which we expound for the example of n = 3.

The following is a vector of infinitesimal length in the direction of increase in t1,

Similarly, we define

The scalar triple product of these three vectors gives the volume of an infinitesimally small parallelepiped,

The components of the first vector are given by

and similar expressions hold for the components of the other two derivatives. It has been shown in the article on the scalar triple product that

where we used that a determinant is invariant under transposition (interchange of rows and columns). Finally.

Reference

  1. T. M. Apostol, Mathematical Analysis, Addison-Wesley, 2nd ed. (1974), sec. 15.10