Graphics Reference
In-Depth Information
y
q
x = p cos q
y = p sin q
0
B
≤
q
≤
p
r = q
X
r = p
x
Y
B'
C'
A'
C
A
x = r cos r
x = r
y = 0
r
p
y = r sin r
q = 0
0 ≤ r ≤
p
0 ≤ r ≤
p
(a)
(b)
Figure 4.25.
Change of variable Example 4.8.7.
which transforms from polar coordinates to Cartesian coordinates. The Jacobian
matrix
cos
q
-
r
sin
q
Ê
Ë
ˆ
¯
¢
()
=
gr
,
q
sin
q
r
cos
q
has determinant r. Therefore, if
Y
Ã
R
2
is a region in polar coordinate space and
X
=
g(
Y
) is the corresponding region in Cartesian space, then equation (4.24) turns into
Ú
(
)
Ú
(
)
f x y dxdy
,
=
f r
cos , sin
q
r
q
rdrd
q
.
X
Y
As a special case, consider the region
Y
shown in Figure 4.25(b), which is bounded
by the lines r =p, r =q, and q=0. This region is mapped by g into the region
X
shown
in Figure 4.25(a). If we were asked to compute an integral of a function f defined on
the complicated region
X
, then, using our change of coordinate map g, we can trans-
late the problem into computing a simple iterated integral over the region
Y
, that is,
(
)
p
r
Ú
(
)
Ú
Ú
(
)
f x y dxdy
,
=
r
f r
cos , sin
q
r
q
d
q
dr
.
X
0
0
Theorem 4.8.6 proves two well-known facts. It demonstrates the geometric
meaning of the determinant of a linear transformation as the factor by which volumes
are changed. It also proves a formula for computing the volume of a parallelotope in
R
n
(Corollary 4.8.9 below).
Let T :
R
n
Æ
R
n
be a linear transformation. If
A
Ã
R
n
4.8.8. Corollary.
is an open
bounded subset, then
(
()
)
=
()
( )
vol T
A
det
T vol
A
.
Proof.
The definition of volume, Theorem 4.8.6, and the fact that the derivative of
T is constant imply that
