Just as the polar coordinate system in r2 use the

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Unformatted text preview: 5 ￿1 ￿ ￿ ￿￿ ￿ sin3 θ ￿ 2 r￿ 11 1 1 ￿ ￿ = =. ￿ ￿ 30 50 23 5 30 Notice that we replaced both x in the integrand and y in the inner limit of integration with the appropriate cylindrical expressions. Spherical Coordinates 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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This document was uploaded on 03/30/2014.

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