Page 14-9
Jacobian of coordinate transformation
Consider the coordinate transformation x = x(u,v), y = y(u,v). The Jacobian of
this transformation is defined as
∂
∂
∂
∂
∂
∂
∂
∂
=
=
v
y
u
y
v
x
u
x
J
J
det
)
det(
|
|
.
When calculating an integral using such transformation, the expression to use
is
∫∫
∫∫
=
'
|
|
)]
,
(
),
,
(
[
)
,
(
R
R
dudv
J
v
u
y
v
u
x
dydx
y
x
φ
φ
, where R’ is the region R
expressed in (u,v) coordinates.
Double integral in polar coordinates
To transform from polar to Cartesian coordinates we use x(r,
θ
) = r cos
θ
, and
y(r,
θ
) = r sin
θ
. Thus, the Jacobian of the transformation is
r
r
r
y
r
y
x
r
x
J
=
⋅
⋅
−
=
∂
∂
∂
∂
∂
∂
∂
∂
=
)
cos(
)
sin(
)
sin(
)
cos(
|
|
θ
θ
θ
θ
θ
θ
With this result, integrals in polar coordinates are written as
