6.4 CHANGE OF VARIABLES IN MULTIPLE INTEGRALS
is constant along the line elementKL, the latter has components (∂x/∂u)duand
(∂y/∂u)duin the directions of thex-andy-axes respectively. Similarly, sinceu
is constant along the line elementKN, the latter has corresponding components
(∂x/∂v)dvand (∂y/∂v)dv. Using the result for the area of a parallelogram given
in chapter 7, we find that the area of the parallelogramKLMNis given by
dAuv=∣
∣
∣
∣∂x
∂udu∂y
∂vdv−∂x
∂vdv∂y
∂udu∣
∣
∣
∣=∣
∣
∣
∣∂x
∂u∂y
∂v−∂x
∂v∂y
∂u∣
∣
∣
∣du dv.Defining theJacobianofx,ywith respect tou,vas
J=∂(x, y)
∂(u, v)≡∂x
∂u∂y
∂v−∂x
∂v∂y
∂u,we have
dAuv=∣
∣
∣
∣∂(x, y)
∂(u, v)∣
∣
∣
∣du dv.The reader acquainted with determinants will notice that the Jacobian can also
be written as the 2×2 determinant
J=∂(x, y)
∂(u, v)=∣
∣
∣
∣
∣
∣
∣
∣∂x
∂u∂y
∂u
∂x
∂v∂y
∂v∣
∣
∣
∣
∣
∣
∣
∣.Such determinants can be evaluated using the methods of chapter 8.
So, in summary, the relationship between the size of the area element generatedbydx,dyand the size of the corresponding area element generated bydu,dvis
dx dy=∣
∣
∣
∣∂(x, y)
∂(u, v)∣
∣
∣
∣du dv.This equality should be taken as meaning that when transforming from coordi-
natesx, yto coordinatesu, v, the area elementdx dyshould be replaced by the
expression on the RHS of the above equality. Of course, the Jacobian can, and
in general will, vary over the region of integration. We may express the double
integral in either coordinate system as
I=∫∫Rf(x, y)dx dy=∫∫R′g(u, v)∣
∣
∣
∣∂(x, y)
∂(u, v)∣
∣
∣
∣du dv. (6.12)When evaluating the integral in the new coordinate system, it is usually advisable
to sketch the region of integrationR′in theuv-plane.