




In any coordinate system, in computing an integral over a volume, you break
the volume up into little pieces, and sum the value of the integrand at a point
in each piece, times the volume of the piece.
In order to do this using a coordinate system w_{1 },_{ }w_{2
}, w_{3 }, instead of x, y and z, you must know how much volume
is associated with a little box whose boundaries are 6 surfaces, two of which
have each w_{i} fixed, the two separated, in value of this variable,
by dw_{i }.
With the coordinates x,y and z, the region defined in this way is a small rectangular
box with volume dxdydz. If the coordinates, w_{1 },_{ }w_{2
}, w_{3 }, are orthogonal, and ds^{2 }= u_{1}^{2}dw_{1}^{2
}+ u_{2}^{2}dw_{2}^{2} + u_{3}^{2}dw_{3}^{2},
then the volume is u_{1 }u_{2 }u_{3 }dw_{1 }dw_{2
}dw_{3}.
In doing integrals it is sometimes convenient to change variables to arbitrary
coordinates which are not orthogonal.
What then is the volume?
Making a change of dwi in the variable wi produces a change in each of x, y
and z and we can compute these changes. Thus we can compute the vector in x,
y, z space that corresponds to this change:
since ,
with similar formulae for dy and dz, the change dwi produces a vector change
in r of _{.
}The volume we want is the volume in the small parallelopiped determined
by these vectors. But we know the volume in a parallelopiped determined by three
vectors: it is the magnitude of the determinant having them as rows.
Explicitly we find that
dV = dx dy dz = Jdw_{1}dw_{2}dw_{3} where J is called the Jacobian of the transformation from variables x, y, z, to w_{1 },_{ }w_{2 }, w_{3 }, and is given by
The Jacobian tells you how to express the volume element dxdydz in the new coordinates. If you were to go backward, starting from coordinates w_{1 },_{ }w_{2 }, w_{3 }, and changing to x,y and z, the roles of the two sets would be reversed, and the w's would be differentiated with respect to x y and z in the backward Jacobian J_{b}, which obeys J_{b }dx dy dz = dw_{1}dw_{2}dw_{3}. This implies that J_{b}is the reciprocal of J: J_{b }= J^{1}.
This last fact can be very useful. Sometimes you know how to express the w's in terms of the x's and not vice versa; you can then compute J_{b} and determine J to be its reciprocal.
The Jacobian tells us how, in changing variables from any given set of variables
of integration to any other to express the volume element in for the old variables
in terms of the volume element for the new set.
The same argument works in any dimension. Thus for two variables you get dxdy
= J dw_{1}dw_{2 }, with J, the Jacobian being the magnitude
of the two by two determinant,given by the top left corner of the three by three
Jacobian. In four or more dimensions, everything works similalry except the
determinant has more and longer rows.
Reversing the Order of Integration:
When you are doing several integrals sequentially it may happen that the limits
of integration of one integral involve the variable integrated over subsequently.
Sometimes, further, you may be able to perform the integrations if you can reverse
the order of doing them.
You must be very careful when you change the order of integration under these
circumstances: You should draw a picture of the region of integration, and make
sure that the limits you deduce for the reverse order correspons to the same
region of that picture
Here is a simple example.