
16-36 Calculus – Differentiation and Integration
Now, Eq. (16.5) becomes
f x y dx dy f r r r dr d
R G
( , ) ( cos , sin ) | |
∫∫
=
∫∫
q q q
=
∫∫
f r r r dr d
G
( cos , sin )q q q (16.6)
For example, Figs. 16.6 and 16.7 show how the equations x = r cos q, y = r sin q transform the
rectangle
: , /0 1 0 2£ £ £ £θ π into the quarter circle R bounded by x
2
+ y
2
= 1 in the first
quadrant of the xy plane.
rθ-plane
θ
G
I
r
π
2
Figure 16.7
xy-plane
θ
1
= Ox
I
R
O
x
y
θ
π
2
=
Figure 16.8
Example 16.36 Prove that dA = r dr dq when transforming to polar coordinates.
Solution: When transforming from Cartesian to polar coordinates, we have
x r y r= =cos , sin .q q
The Jacobian ...