Jacobian Of Coordinate Transformation; Double Integral In Polar Coordinates - HP 50g User Manual

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
∫∫
( , )
x y dydx
is
R
expressed in (u,v) coordinates.

Double integral in polar coordinates

To transform from polar to Cartesian coordinates we use x(r, ) = r cos , and
y(r, ) = r sin . Thus, the Jacobian of the transformation is
With this result, integrals in polar coordinates are written as
J
| | det(
∫∫
[ ( , ),
x u v
R '
x x
r
| J |
y y
r
x
⎛
⎜
u
⎜
J
) det
y
⎜
⎜
u
⎝
( , | )] |
y u v J dudv
cos( ) r
sin( ) r
x
⎞
⎟
v
⎟
.
y
⎟
⎟
v
⎠
, where R' is the region R
sin( )
r
cos( )
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