COPYRIGHT NOTICE
Copyright 2007 Samuel Dagan
dagan@post.tau.ac.il
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Volume element of spherical coordinates
Volume element of spherical coordinates
The origin of the figure is at (0,0)
©
Samuel Dagan
Volume element of spherical
coordinates
z
// R surface
RΔθ
Rsin(θ+Δθ)
Δφ
// Theta surface
ΔR
(R+ΔR)sinθΔφ
RsinθΔφ
// R+dR surface
(R+ΔR)sin(θ+Δθ)Δφ
(R+ΔR)sinθΔφ
(R+ΔR)Δθ
// phi surface
ΔR
ΔR
RΔθ
(R+ΔR)Δθ
z
−z
+z
reset
A quarter of a sphere with radius R+ΔR
is centered at the origin (start of the
z axis).
A volume element is formed by
incrementing
the spherical coordinates of the
black dot
(R,θ,φ) by ΔR, Δθ
and Δφ,
yielding an
increase in the R, θ
and φ
directions,
forming a right-handed
coordinate system.
This can be verified by rotating
the element
about the z axis by the "±z", "fast",
and
"slow" buttons.
For a small enough element,
its volume can
be calculated from the
first approximation
of the length of its sides.
For that matter
(R+ΔR)Δθ = RΔθ
and
(R+ΔR)sin(θ+Δθ)Δφ
= Rsin(θ+Δθ)Δφ
=
(R+ΔR)sinθΔφ
= RsinθΔφ .
From the orthogonality
of the spherical
coordinates and by first
approximation, the
element becomes a cuboid with
volume
ΔR RΔθ RsinθΔφ
.
By the same argument the area of
a
spherical surface
element becomes
RΔθ RsinθΔφ
and that of a conical
surface -
ΔR RsinθΔφ
.
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view!
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