The radius of Earth is 40004000 mi, so ρ=4000.ρ=4000. The intersection of the prime meridian and the equator lies on the positive x-axis. Movement to the west is then described with negative angle measures, which shows that θ=−83°,θ=−83°, Because Columbus lies 40°40° north of the equator, it lies 50°50° south of the North Pole, so φ=50°.φ=50°. In spherical coordinates, Columbus lies at point (4000,−83°,50°).(4000,−83°,50°).

The solid situated in the first octant with a vertex at the origin and enclosed by a cube of edge length a,a, where a>0a>0

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The formulas to convert from spherical coordinates to rectangular coordinates may seem complex, but they are straightforward applications of trigonometry. Looking at Figure 2.98, it is easy to see that r=ρsinφ.r=ρsinφ. Then, looking at the triangle in the xy-plane with rr as its hypotenuse, we have x=rcosθ=ρsinφcosθ.x=rcosθ=ρsinφcosθ. The derivation of the formula for yy is similar. Figure 2.96 also shows that ρ2=r2+z2=x2+y2+z2ρ2=r2+z2=x2+y2+z2 and z=ρcosφ.z=ρcosφ. Solving this last equation for φφ and then substituting ρ=r2+z2ρ=r2+z2 (from the first equation) yields φ=arccos(zr2+z2).φ=arccos(zr2+z2). Also, note that, as before, we must be careful when using the formula tanθ=yxtanθ=yx to choose the correct value of θ.θ.

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Use the following figure as an aid in identifying the relationship between the rectangular, cylindrical, and spherical coordinate systems.

[T] The “bumpy sphere” with an equation in spherical coordinates is ρ=a+bcos(mθ)sin(nφ),ρ=a+bcos(mθ)sin(nφ), with θ∈[0,2π]θ∈[0,2π] and φ∈[0,π],φ∈[0,π], where aa and bb are positive numbers and mm and nn are positive integers, may be used in applied mathematics to model tumor growth.

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The point with rectangular coordinates (1,−3,5)(1,−3,5) has cylindrical coordinates approximately equal to (10,5.03,5).(10,5.03,5).

Point RR has cylindrical coordinates (5,π6,4)(5,π6,4). Plot RR and describe its location in space using rectangular, or Cartesian, coordinates.

Spherical coordinates are useful in analyzing systems that have some degree of symmetry about a point, such as the volume of the space inside a domed stadium or wind speeds in a planet’s atmosphere. A sphere that has Cartesian equation x2+y2+z2=c2x2+y2+z2=c2 has the simple equation ρ=cρ=c in spherical coordinates.

The Cartesian coordinate system provides a straightforward way to describe the location of points in space. Some surfaces, however, can be difficult to model with equations based on the Cartesian system. This is a familiar problem; recall that in two dimensions, polar coordinates often provide a useful alternative system for describing the location of a point in the plane, particularly in cases involving circles. In this section, we look at two different ways of describing the location of points in space, both of them based on extensions of polar coordinates. As the name suggests, cylindrical coordinates are useful for dealing with problems involving cylinders, such as calculating the volume of a round water tank or the amount of oil flowing through a pipe. Similarly, spherical coordinates are useful for dealing with problems involving spheres, such as finding the volume of domed structures.

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In geography, latitude and longitude are used to describe locations on Earth’s surface, as shown in Figure 2.104. Although the shape of Earth is not a perfect sphere, we use spherical coordinates to communicate the locations of points on Earth. Let’s assume Earth has the shape of a sphere with radius 40004000 mi. We express angle measures in degrees rather than radians because latitude and longitude are measured in degrees.

In the cylindrical coordinate system, a point in space (Figure 2.89) is represented by the ordered triple (r,θ,z),(r,θ,z), where

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For the following exercises, the rectangular coordinates (x,y,z)(x,y,z) of a point are given. Find the cylindrical coordinates (r,θ,z)(r,θ,z) of the point.

In each of the following situations, we determine which coordinate system is most appropriate and describe how we would orient the coordinate axes. There could be more than one right answer for how the axes should be oriented, but we select an orientation that makes sense in the context of the problem. Note: There is not enough information to set up or solve these problems; we simply select the coordinate system (Figure 2.105).

For the following exercises, the equation of a surface in rectangular coordinates is given. Find the equation of the surface in spherical coordinates. Identify the surface.

For the following exercises, the spherical coordinates of a point are given. Find its associated cylindrical coordinates.

Find the latitude and longitude of Rio de Janeiro if its spherical coordinates are (4000,−43.17°,102.91°).(4000,−43.17°,102.91°).

Let’s consider the differences between rectangular and cylindrical coordinates by looking at the surfaces generated when each of the coordinates is held constant. If cc is a constant, then in rectangular coordinates, surfaces of the form x=c,x=c, y=c,y=c, or z=cz=c are all planes. Planes of these forms are parallel to the yz-plane, the xz-plane, and the xy-plane, respectively. When we convert to cylindrical coordinates, the z-coordinate does not change. Therefore, in cylindrical coordinates, surfaces of the form z=cz=c are planes parallel to the xy-plane. Now, let’s think about surfaces of the form r=c.r=c. The points on these surfaces are at a fixed distance from the z-axis. In other words, these surfaces are vertical circular cylinders. Last, what about θ=c?θ=c? The points on a surface of the form θ=cθ=c are at a fixed angle from the x-axis, which gives us a half-plane that starts at the z-axis (Figure 2.91 and Figure 2.92).

As when we discussed conversion from rectangular coordinates to polar coordinates in two dimensions, it should be noted that the equation tanθ=yxtanθ=yx has an infinite number of solutions. However, if we restrict θθ to values between 00 and 2π,2π, then we can find a unique solution based on the quadrant of the xy-plane in which original point (x,y,z)(x,y,z) is located. Note that if x=0,x=0, then the value of θθ is either π2,3π2,π2,3π2, or 0,0, depending on the value of y.y.

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In the xy-plane, the right triangle shown in Figure 2.89 provides the key to transformation between cylindrical and Cartesian, or rectangular, coordinates.

For the following exercises, the spherical coordinates (ρ,θ,φ)(ρ,θ,φ) of a point are given. Find the rectangular coordinates (x,y,z)(x,y,z) of the point.

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Conversion from cylindrical to rectangular coordinates requires a simple application of the equations listed in Conversion between Cylindrical and Cartesian Coordinates:

When we expanded the traditional Cartesian coordinate system from two dimensions to three, we simply added a new axis to model the third dimension. Starting with polar coordinates, we can follow this same process to create a new three-dimensional coordinate system, called the cylindrical coordinate system. In this way, cylindrical coordinates provide a natural extension of polar coordinates to three dimensions.

In the Cartesian coordinate system, the location of a point in space is described using an ordered triple in which each coordinate represents a distance. In the cylindrical coordinate system, location of a point in space is described using two distances (randz)(randz) and an angle measure (θ).(θ). In the spherical coordinate system, we again use an ordered triple to describe the location of a point in space. In this case, the triple describes one distance and two angles. Spherical coordinates make it simple to describe a sphere, just as cylindrical coordinates make it easy to describe a cylinder. Grid lines for spherical coordinates are based on angle measures, like those for polar coordinates.

The use of cylindrical coordinates is common in fields such as physics. Physicists studying electrical charges and the capacitors used to store these charges have discovered that these systems sometimes have a cylindrical symmetry. These systems have complicated modeling equations in the Cartesian coordinate system, which make them difficult to describe and analyze. The equations can often be expressed in more simple terms using cylindrical coordinates. For example, the cylinder described by equation x2+y2=25x2+y2=25 in the Cartesian system can be represented by cylindrical equation r=5.r=5.

[T] Use a CAS to graph the region between elliptic paraboloid z=x2+y2z=x2+y2 and cone x2+y2−z2=0.x2+y2−z2=0. Then describe the region in cylindrical coordinates.

The latitude of Columbus, Ohio, is 40°40° N and the longitude is 83°83° W, which means that Columbus is 40°40° north of the equator. Imagine a ray from the center of Earth through Columbus and a ray from the center of Earth through the equator directly south of Columbus. The measure of the angle formed by the rays is 40°.40°. In the same way, measuring from the prime meridian, Columbus lies 83°83° to the west. Express the location of Columbus in spherical coordinates.

Use the second set of equations from Conversion between Cylindrical and Cartesian Coordinates to translate from rectangular to cylindrical coordinates:

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Use the equations in Converting among Spherical, Cylindrical, and Rectangular Coordinates to translate between spherical and cylindrical coordinates (Figure 2.100):

San Francisco is located at 37.78°N37.78°N and 122.42°W.122.42°W. Assume the radius of Earth is 40004000 mi. Express the location of San Francisco in spherical coordinates.

For the following exercises, the equation of a surface in rectangular coordinates is given. Find the equation of the surface in cylindrical coordinates.

A cylindrical shell of height 1010 determined by the region between two cylinders with the same center, parallel rulings, and radii of 22 and 5,5, respectively

There are actually two ways to identify φ.φ. We can use the equation φ=arccos(zx2+y2+z2).φ=arccos(zx2+y2+z2). A more simple approach, however, is to use equation z=ρcosφ.z=ρcosφ. We know that z=6z=6 and ρ=22,ρ=22, so

Notice that these equations are derived from properties of right triangles. To make this easy to see, consider point PP in the xy-plane with rectangular coordinates (x,y,0)(x,y,0) and with cylindrical coordinates (r,θ,0),(r,θ,0), as shown in the following figure.

If this process seems familiar, it is with good reason. This is exactly the same process that we followed in Introduction to Parametric Equations and Polar Coordinates to convert from polar coordinates to two-dimensional rectangular coordinates.

For the following exercises, the equation of a surface in spherical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface.

For the following exercises, the cylindrical coordinates of a point are given. Find its associated spherical coordinates, with the measure of the angle φφ in radians rounded to four decimal places.

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The rectangular coordinates (x,y,z)(x,y,z) and the cylindrical coordinates (r,θ,z)(r,θ,z) of a point are related as follows:

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Plot the point with spherical coordinates (8,π3,π6)(8,π3,π6) and express its location in both rectangular and cylindrical coordinates.

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Washington, DC, is located at 39°39° N and 77°77° W (see the following figure). Assume the radius of Earth is 40004000 mi. Express the location of Washington, DC, in spherical coordinates.

A spherical shell determined by the region between two concentric spheres centered at the origin, of radii of aa and b,b, respectively, where b>a>0b>a>0

Cylindrical and spherical coordinates give us the flexibility to select a coordinate system appropriate to the problem at hand. A thoughtful choice of coordinate system can make a problem much easier to solve, whereas a poor choice can lead to unnecessarily complex calculations. In the following example, we examine several different problems and discuss how to select the best coordinate system for each one.

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If a point has cylindrical coordinates (r,θ,z),(r,θ,z), then these equations define the relationship between cylindrical and spherical coordinates.

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We choose the positive square root, so r=10.r=10. Now, we apply the formula to find θ.θ. In this case, yy is negative and xx is positive, which means we must select the value of θθ between 3π23π2 and 2π:2π:

For the following exercises, the rectangular coordinates (x,y,z)(x,y,z) of a point are given. Find the spherical coordinates (ρ,θ,φ)(ρ,θ,φ) of the point. Express the measure of the angles in degrees rounded to the nearest integer.

[T] Use a CAS to graph in spherical coordinates the “ice cream-cone region” situated above the xy-plane between sphere x2+y2+z2=4x2+y2+z2=4 and elliptical cone x2+y2−z2=0.x2+y2−z2=0.

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For the following exercises, the cylindrical coordinates (r,θ,z)(r,θ,z) of a point are given. Find the rectangular coordinates (x,y,z)(x,y,z) of the point.

Let the center of Earth be the center of the sphere, with the ray from the center through the North Pole representing the positive z-axis. The prime meridian represents the trace of the surface as it intersects the xz-plane. The equator is the trace of the sphere intersecting the xy-plane.

The point with cylindrical coordinates (4,2π3,−2)(4,2π3,−2) has rectangular coordinates (−2,23,−2)(−2,23,−2) (see the following figure).

As we did with cylindrical coordinates, let’s consider the surfaces that are generated when each of the coordinates is held constant. Let cc be a constant, and consider surfaces of the form ρ=c.ρ=c. Points on these surfaces are at a fixed distance from the origin and form a sphere. The coordinate θθ in the spherical coordinate system is the same as in the cylindrical coordinate system, so surfaces of the form θ=cθ=c are half-planes, as before. Last, consider surfaces of the form φ=c.φ=c. The points on these surfaces are at a fixed angle from the z-axis and form a half-cone (Figure 2.99).

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In the spherical coordinate system, a point PP in space (Figure 2.97) is represented by the ordered triple (ρ,θ,φ)(ρ,θ,φ) where

Plot the point with spherical coordinates (2,−5π6,π6)(2,−5π6,π6) and describe its location in both rectangular and cylindrical coordinates.