1.1. LOCATING POINTS IN SPACE
11
To invert these relations, we note that, since
ρ
≥
0 and 0
≤
φ
≤
π
by
convention,
z
and
r
completely determine
ρ
and
φ
:
ρ
=
√
r
2
+
z
2
θ
=
θ
φ
= arccos
z
ρ
.
(1.8)
The ambiguities in spherical coordinates are the same as those for
cylindrical coordinates: the origin has
ρ
= 0 and both
θ
and
φ
arbitrary;
any other point on the
z
axis (
φ
= 0 or
φ
=
π
) has arbitrary
θ
, and for
points oF the
z
axis,
θ
can (in principle) be augmented by arbitrary even
multiples of
π
.
Thus, the point
P
with cylindrical coordinates
r
= 4
θ
=
5
π
6
z
= 4
has spherical coordinates
ρ
= 4
√
2
θ
=
5
π
6
φ
=
π
4
.
Combining Equations (
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
 Fall '08
 ALL
 Calculus

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