8—Multivariable Calculus 221The surfaces that have constant values of these coordinates are planes in rectangular coordinates; planes
and cylinders in cylindrical; planes, spheres, and cones in spherical. In every one of these cases the constant-
coordinate surfaces intersect each other at right angles, hence the name “orthogonal coordinate” systems. In
spherical coordinates I used the coordinateθas the angle from thez-axis andφas the angle around the axis. In
mathematics books these are typically reversed, so watch out for the notation. On the globe of the Earth,φis
the longitude andθthe latitude except that longitude goes± 180 ◦instead of zero to 2 π. Latitude is± 90 ◦from
the equator instead of zero toπfrom the pole.
The volume elements for these systems come straight from the drawings, just as the area elements do in
plane coordinates. In every case you can draw six surfaces, bounded by constant coordinates, and surrounding a
small box. Because these are orthogonal coordinates you can compute the volume of the box easily as the product
of its three edges.
In the spherical case, one side is∆r. Another side isr∆θ. The third side is notr∆φ; it isrsinθ∆φ. The
reason for the factorsinθis that the arc of the circle made at constantrand constantθis not in a plane passing
through the origin. It is in a plane parallel to thex-yplane, so it has a radiusrsinθ.
rectangular cylindrical spherical
volume dxdy dz r dr dθ dz r^2 sinθ dr dθ dφ
area dxdy r dθ dz or r dθ dr r^2 sinθ dθ dφExamples of Multiple Integrals
Even in rectangular coordinates integration can be tricky. That’s because you have to pay attention to the limits
of integration far more closely than you do for simple one dimensional integrals. I’ll illustrate this with two
dimensional rectangular coordinates first, and will choose a problem that is easy but still shows what you have to
look for.