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2.7 Cylindrical and spherical coordinates  (Page 3/11)

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Convert point ( −8 , 8 , −7 ) from Cartesian coordinates to cylindrical coordinates.

( 8 2 , 3 π 4 , −7 )

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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 x 2 + y 2 = 25 in the Cartesian system can be represented by cylindrical equation r = 5 .

Identifying surfaces in the cylindrical coordinate system

Describe the surfaces with the given cylindrical equations.

  1. θ = π 4
  2. r 2 + z 2 = 9
  3. z = r
  1. When the angle θ is held constant while r and z are allowed to vary, the result is a half-plane (see the following figure).
    This figure is the first quadrant of the 3-dimensional coordinate system. There is a plane attached to the z-axis, dividing the x y-plane with a diagonal line. The angle between the x-axis and this plane is pi/4.
    In polar coordinates, the equation θ = π / 4 describes the ray extending diagonally through the first quadrant. In three dimensions, this same equation describes a half-plane.
  2. Substitute r 2 = x 2 + y 2 into equation r 2 + z 2 = 9 to express the rectangular form of the equation: x 2 + y 2 + z 2 = 9 . This equation describes a sphere centered at the origin with radius 3 (see the following figure).
    This figure is a sphere. It has the z-axis through the center vertically. The point of intersection with the z-axis and the sphere is (0, 0, 3). There is also the y-axis through the center of the sphere horizontally. The intersection of the sphere and the y-axis is the point (0, 3, 0).
    The sphere centered at the origin with radius 3 can be described by the cylindrical equation r 2 + z 2 = 9 .
  3. To describe the surface defined by equation z = r , is it useful to examine traces parallel to the xy -plane. For example, the trace in plane z = 1 is circle r = 1 , the trace in plane z = 3 is circle r = 3 , and so on. Each trace is a circle. As the value of z increases, the radius of the circle also increases. The resulting surface is a cone (see the following figure).
    This figure is the 3-dimensional coordinate system. It has an elliptic cone with the z-axis down the center. The two cones, one right side up, the other upside down, meet at the origin.
    The traces in planes parallel to the xy -plane are circles. The radius of the circles increases as z increases.
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Describe the surface with cylindrical equation r = 6 .

This surface is a cylinder with radius 6 .
This figure is a right circular cylinder. It is upright with the z-axis through the center. It is on top of the x y plane.

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Spherical coordinates

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 ( r and z ) and an angle measure ( 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.

Definition

In the spherical coordinate system    , a point P in space ( [link] ) is represented by the ordered triple ( ρ , θ , φ ) where

  • ρ (the Greek letter rho) is the distance between P and the origin ( ρ ≠ 0 ) ;
  • θ is the same angle used to describe the location in cylindrical coordinates;
  • φ (the Greek letter phi) is the angle formed by the positive z -axis and line segment O P — , where O is the origin and 0 ≤ φ ≤ π .
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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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