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sin3 θ 2
30 Notice that we replaced both x in the integrand and y in the inner limit of integration
with the appropriate cylindrical expressions.
There is a second way to generalize the concept of polar coordinates to three dimensions, which
is simultaneously more natural and more complicated. Just as the polar coordinate system in
R2 use the distance from the origin and one angle, the spherical coordinate system in R3 uses
the distance from the origin and two angles. The distance ρ is measured directly from the
origin to the point (in R3 ), so ρ = x2 + y 2 + z 2 , ρ ∈ [0, ∞). The angle θ is deﬁned exactly the same way as the angle θ we use for cylindrical coordinates, so θ ∈ [0, 2π ). Meanwhile, the
angle φ is the angle away from the positive z -axis; for this we need only φ ∈ [0, π ] (since the
farthest we can get away from the positive z -axis is the negative z -axis). See Figure 7 (we’ve
included the polar coordinates r and θ for reasons you’ll see in a moment). Actually, spherical
coordinates won’t be a completely new concept for you; the idea is almost identical to the
5 Figure 7:
usage of latitude and longitude to identify locations on the surface of the earth. The angle θ
corresponds exactly to the measurement of longitude (with the Greenwich meridian deﬁning
the xz -plane, as 0◦ ), while the angle φ diﬀers from the measurement of latitude only in that
the reference point is the “north pole” instead of the equator; if we measured latitude using
φ then the values would range from 0◦ at the north pole to 180◦ at the south pole, instead of
from 90◦ N to 90◦ S. The other diﬀerence is a simple one; the coordinate ρ allows us to specify
locations which don’t lie exactly on the earth’s surface, but above or below it.
For the conversion formulas, notice from Figure 7 that we can relate the spherical coordinates easily to the cylindrica...
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