Sometimes, it is often advantageous to evaluate
in a coordinate system other than the xy
This may be as a consequence either of the shape of the region, or of the complexity of the integrand.
Calculating the double integral in the new coordinate system can be much simpler.
The formula for change of variables is given by
where the expression
is the so-called Jacobian
of the transformation
, and S
is the pullback
of the region of integration R
which can be computed by substituting
into the definition of R
. Notice, that
in the formula above
means the absolute value of the appropriate determinant.
Supposing that the transformation
is a 1-1 mapping from R
to a region
, the inverse relation is described by the Jacobian
Thus, use of change of variables in a double integral requires the following 3 steps:
- Find the pulback S in the new coordinate system for the initial region of integration R;
- Calculate the Jacobian of the transformation and write down the differential through the new variables: ;
- Replace x and y in the integrand by substituting
and , respectively.