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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
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
JJ det)det(||
∫∫∫∫
=
'
||)],(),,([),(
RR
dudvJvuyvuxdydxyx
φφ
r
r
r
y
r
y
x
r
x
J =
⋅
⋅−
=
∂
∂
∂
∂
∂
∂
∂
∂
=
)cos()sin(
)sin()cos(
||
θθ
θθ
θ
θ