CV. Changing Variables
in Multiple Integrals
1. Changing variables.
Double integrals in x, y coordinates which are taken over circular regions, or have inte-
grands involving the combination x2
+
y2, are often better done in polar coordinates:
This involves introducing the new variables
r and 19, together with the equations relating
them to x, y in both the forward and backward directions:
Changing the integral to polar coordinates then requires three steps:
A. Changing the integrand
f (x, y) to g(r, 8), by using (2);
B. Supplying the area element in the r, I9 system: dA
=
r dr dB
;
C. Using the region R to determine the limits of integration in the r, I9 system.
In the same way, double integrals involving other types of regions or integrands can
sometimes be simplified by changing the coordinate system from x, y to one better adapted
to the region or integrand. Let's call the new coordinates u and v; then there will be
equations introducing the new coordinates, going in both directions:
(often one will only get or use the equations in one of these directions). To change the
integral to u,v-coordinates, we then have to carry out the three steps A, B, C above.
A
first step is to picture the new coordinate system; for this we use the same idea as for polar
coordinates, namely, we consider the grid formed by the level curves of the new coordinate
functions:
I
\
,u=Uo
(4)
u(x, y)
=
UO,
v(x,y)
=
vo
.
Once we have this, algebraic and geometric intuition will usually handle
steps A and C, but for B we will need a formula: it uses a determinant
called the Jacobian, whose notation and definition are
v=v2
Using it, the formula for the area element in the u, v-system is