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Copyright 2007 Samuel Dagan
dagan@post.tau.ac.il
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Orthogonality of polar coordinates
Orthogonality and element of area
in polar coordinates.
The origin of the figure is at (0,0)
©
Samuel Dagan
Orthogonality of polar
coordinates
φ0
Δφ
φ =
φ0
φ = φ0 +
Δφ
r = r0
r = r0 +
Δr
Δr
Δr
r0Δφ
x
y
Two lines of constant r (circular arcs),
and two with constant φ
(radial lines), intersect by
straight angles, defining
orthogonal coordinates. They bound a 4-sided geometrical
form.
Each of its radial sides is Δr
long,
and the shorter arc-side is
r0Δφ
long.
The other arc-side is
(r0
+ Δr)Δφ long.
In the case of short icrements Δr and
Δφ , one
can disregard any higher than first
approximation, and
this length becomes r0Δφ .
Consequenctly one obtains a
rectangle with an area of r0
ΔrΔφ ,
without using the Jacobian.
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