Revision as of 09:15, 13 January 2009 by imported>Paul Wormer
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
- ↑ T. M. Apostol, Mathematical Analysis, Addison-Wesley, 2nd ed. (1974), sec. 15.10