APEX Calculus

In short, cylindrical coordinates can be thought of as a combination of the polar and rectangular coordinate systems. One can identify a point \((x_0,y_0,z_0)\text{,}\) given in rectangular coordinates, with the point \((r_0,\theta_0,z_0)\text{,}\) given in cylindrical coordinates, where the \(z\)-value in both systems is the same, and the point \((x_0,y_0)\) in the \(xy\)-plane is identified with the polar point \(P(r_0,\theta_0)\text{;}\) see Figure 14.7.1. So that each point in space that does not lie on the \(z\)-axis is defined uniquely, we will restrict \(r\geq 0\) and \(0\leq \theta\leq 2\pi\text{.}\)

These identities, along with conversions related to spherical coordinates, are given later in Key Idea 14.7.15.

Setting each of \(r\text{,}\) \(\theta\) and \(z\) equal to a constant defines a surface in space, as illustrated in the following example.

Cylindrical coordinates are useful when describing certain domains in space, allowing us to evaluate triple integrals over these domains more easily than if we used rectangular coordinates.

Theorem 14.6.29 shows how to evaluate \(\iiint_Dh(x,y,z)\, dV\) using rectangular coordinates. In that evaluation, we use \(dV = dz\,dy\,dx\) (or one of the other five orders of integration). Recall how, in this order of integration, the bounds on \(y\) are “curve to curve” and the bounds on \(x\) are “point to point”: these bounds describe a region \(R\) in the \(xy\)-plane. We could describe \(R\) using polar coordinates as done in Section 14.3. In that section, we saw how we used \(dA = r\,dr\,d\theta\) instead of \(dA = dy\,dx\text{.}\)

Considering the above thoughts, we have \(dV = dz\big(r\,dr\,d\theta\big) = r\,dz\,dr\,d\theta\text{.}\) We set bounds on \(z\) as “surface to surface” as done in the previous section, and then use “curve to curve” and “point to point” bounds on \(r\) and \(\theta\text{,}\) respectively. Finally, using the identities given above, we change the integrand \(h(x,y,z)\) to \(h(r,\theta,z)\text{.}\)

This process should sound plausible; the following theorem states it is truly a way of evaluating a triple integral.

In short, spherical coordinates can be thought of as a “double application” of the polar coordinate system. In spherical coordinates, a point \(P\) is identified with \((\rho,\theta,\varphi)\text{,}\) where \(\rho\) is the distance from the origin to \(P\text{,}\) \(\theta\) is the same angle as would be used to describe \(P\) in the cylindrical coordinate system, and \(\varphi\) is the angle between the \(xy\)-plane and the ray from the origin to \(P\text{;}\) see Figure 14.7.13. So that each point in space that does not lie on the \(z\)-axis is defined uniquely, we will restrict \(\rho \geq 0\text{,}\) \(0 \leq \theta \leq 2\pi\) and \(-\pi/2 \leq \varphi \leq \pi/2\text{.}\)

The following Key Idea gives conversions to/from our three spatial coordinate systems.

Spherical coordinates are useful when describing certain domains in space, allowing us to evaluate triple integrals over these domains more easily than if we used rectangular coordinates or cylindrical coordinates. The crux of setting up a triple integral in spherical coordinates is appropriately describing the “small amount of volume,” \(dV\text{,}\) used in the integral.

Again, this development of \(dV\) should sound reasonable, and the following theorem states it is the appropriate manner by which triple integrals are to be evaluated in spherical coordinates.

This section has provided a brief introduction into two new coordinate systems useful for identifying points in space. Each can be used to define a variety of surfaces in space beyond the canonical surfaces graphed as each system was introduced.

However, the usefulness of these coordinate systems does not lie in the variety of surfaces that they can describe nor the regions in space these surfaces may enclose. Rather, cylindrical coordinates are mostly used to describe cylinders and spherical coordinates are mostly used to describe spheres. These shapes are of special interest in the sciences, especially in physics, and computations on/inside these shapes is difficult using rectangular coordinates. For instance, in the study of electricity and magnetism, one often studies the effects of an electrical current passing through a wire; that wire is essentially a cylinder, described well by cylindrical coordinates.

This chapter investigated the natural follow-on to partial derivatives: iterated integration. We learned how to use the bounds of a double integral to describe a region in the plane using both rectangular and polar coordinates, then later expanded to use the bounds of a triple integral to describe a region in space. We used double integrals to find volumes under surfaces, surface area, and the center of mass of lamina; we used triple integrals as an alternate method of finding volumes of space regions and also to find the center of mass of a region in space.

Integration does not stop here. We could continue to iterate our integrals, next investigating “quadruple integrals” whose bounds describe a region in 4-dimensional space (which are very hard to visualize). We can also look back to “regular” integration where we found the area under a curve in the plane. A natural analogue to this is finding the “area under a curve,” where the curve is in space, not in a plane. These are just two of many avenues to explore under the heading of “integration.”