HP 50G, User Manual

Page: 469 / 887

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
.
When calculating an integral using such transformation, the expression to use
is , 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
With this result, integrals in polar coordinates are written as
⎟⎟
⎟
⎟
⎠
⎞
⎜⎜
⎜
⎜
⎝
⎛
∂
∂
∂
∂
∂
∂
∂
∂
= =
v
y
u
y
v
x
u
x
J J
det ) det( | |
∫∫ ∫∫
=
'
| | )] , ( ), , ( [ ) , (
R R
dudv J v u y v u x dydx y x
φ φ
r
r
r
y
r
y
x
r
x
J =
⋅
⋅ −
=
∂
∂
∂
∂
∂
∂
∂
∂
=
) cos( ) sin(
) sin( ) cos(
| |
θ θ
θ θ
θ
θ