Jacobian Of Coordinate Transformation; Double Integral In Polar Coordinates - HP F2226A - 48GII Graphic Calculator User Manual

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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
∫∫
∫∫
φ
(
,
)
=
φ
is
x
y
dydx
R
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
|
J
|
With this result, integrals in polar coordinates are written as
x
u
J
J
|
det(
)
det
y
u
[
(
,
),
(
,
| )]
|
x
u
v
y
u
v
J
dudv
x
x
cos(
θ
)
r
r
θ
y
y
sin(
θ
)
r
r
θ
x
v
.
y
v
, where R' is the region R
sin(
θ
)
r
cos(
θ
)
Page 14-9

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