1.7: Cylindrical and Spherical Coordinates
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Learning Objectives
- Convert from cylindrical to rectangular coordinates.
- Convert from rectangular to cylindrical coordinates.
- Convert from spherical to rectangular coordinates.
- Convert from rectangular to 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.
Cylindrical 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.
Definition: The Cylindrical Coordinate System
In the cylindrical coordinate system , a point in space (Figure \(\PageIndex{1}\)) is represented by the ordered triple \((r,θ,z)\), where
- \((r,θ)\) are the polar coordinates of the point’s projection in the \(xy\)-plane
- \(z\) is the usual \(z\)- coordinate in the Cartesian coordinate system
In the \(xy\)-plane, the right triangle shown in Figure \(\PageIndex{1}\) provides the key to transformation between cylindrical and Cartesian, or rectangular, coordinates.
Conversion between Cylindrical and Cartesian Coordinates
The rectangular coordinates \((x,y,z)\) and the cylindrical coordinates \((r,θ,z)\) of a point are related as follows:
These equations are used to convert from cylindrical coordinates to rectangular coordinates.
- \(x=r\cos θ\)
- \(y=r\sin θ\)
- \(z=z\)
These equations are used to convert from rectangular coordinates to cylindrical coordinates
- \(r^2=x^2+y^2\)
- \(\tan θ=\dfrac{y}{x}\)
- \(z=z\)
As when we discussed conversion from rectangular coordinates to polar coordinates in two dimensions, it should be noted that the equation \(\tan θ=\dfrac{y}{x}\) has an infinite number of solutions. However, if we restrict \(θ\) to values between \(0\) and \(2π\), then we can find a unique solution based on the quadrant of the \(xy\)-plane in which original point \((x,y,z)\) is located. Note that if \(x=0\), then the value of \(θ\) is either \(\dfrac{π}{2},\dfrac{3π}{2},\) or \(0\), depending on the value of \(y\).
Notice that these equations are derived from properties of right triangles. To make this easy to see, consider point \(P\) in the \(xy\)-plane with rectangular coordinates \((x,y,0)\) and with cylindrical coordinates \((r,θ,0)\), as shown in Figure \(\PageIndex{2}\).
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 \(c\) is a constant, then in rectangular coordinates, surfaces of the form \(x=c, y=c,\) or \(z=c\) 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 \(z=c\) are planes parallel to the \(xy\)-plane. Now, let’s think about surfaces of the form \(r=c\). 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 \(θ=c\)? The points on a surface of the form \(θ=c\) are at a fixed angle from the \(x\)-axis, which gives us a half-plane that starts at the \(z\)-axis (Figures \(\PageIndex{3}\) and \(\PageIndex{4}\)).
Example \(\PageIndex{1}\): Converting from Cylindrical to Rectangular Coordinates
Plot the point with cylindrical coordinates \((4,\dfrac{2π}{3},−2)\) and express its location in rectangular coordinates.
Solution
Conversion from cylindrical to rectangular coordinates requires a simple application of the equations listed in Conversion between Cylindrical and Cartesian Coordinates:
\[\begin{align*} x &=r\cos θ=4\cos\dfrac{2π}{3}=−2 \\[4pt] y &=r\sin θ=4\sin \dfrac{2π}{3}=2\sqrt{3} \\[4pt] z &=−2 \end{align*}. \nonumber \]
The point with cylindrical coordinates \((4,\dfrac{2π}{3},−2)\) has rectangular coordinates \((−2,2\sqrt{3},−2)\) (Figure \(\PageIndex{5}\)).
Exercise \(\PageIndex{1}\)
Point \(R\) has cylindrical coordinates \((5,\frac{π}{6},4)\). Plot \(R\) and describe its location in space using rectangular, or Cartesian, coordinates.
- Hint
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The first two components match the polar coordinates of the point in the \(xy\)-plane.
- Answer
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The rectangular coordinates of the point are \((\frac{5\sqrt{3}}{2},\frac{5}{2},4).\)
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.
Example \(\PageIndex{2}\): Converting from Rectangular to Cylindrical Coordinates
Convert the rectangular coordinates \((1,−3,5)\) to cylindrical coordinates.
Solution
Use the second set of equations from Conversion between Cylindrical and Cartesian Coordinates to translate from rectangular to cylindrical coordinates:
\[\begin{align*} r^2 &= x^2+y^2 \\[4pt] r &=±\sqrt{1^2+(−3)^2} \\[4pt] &= ±\sqrt{10}. \end{align*}\]
We choose the positive square root, so \(r=\sqrt{10}\).Now, we apply the formula to find \(θ\). In this case, \(y\) is negative and \(x\) is positive, which means we must select the value of \(θ\) between \(\dfrac{3π}{2}\) and \(2π\):
\[\begin{align*} \tan θ &=\dfrac{y}{x} &=\dfrac{−3}{1} \\[4pt] θ &=\arctan(−3) &≈5.03\,\text{rad.} \end{align*}\]
In this case, the z -coordinates are the same in both rectangular and cylindrical coordinates:
\[ z=5. \nonumber \]
The point with rectangular coordinates \((1,−3,5)\) has cylindrical coordinates approximately equal to \((\sqrt{10},5.03,5).\)
Exercise \(\PageIndex{2}\)
Convert point \((−8,8,−7)\) from Cartesian coordinates to cylindrical coordinates.
- Hint
-
\(r^2=x^2+y^2\) and \(\tan θ=\frac{y}{x}\)
- Answer
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\((8\sqrt{2},\frac{3π}{4},−7)\)
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 \(r=5\).
Example \(\PageIndex{3}\): Identifying Surfaces in the Cylindrical Coordinate System
Describe the surfaces with the given cylindrical equations.
- \(θ=\dfrac{π}{4}\)
- \(r^2+z^2=9\)
- \(z=r\)
Solution
a. When the angle \(θ\) is held constant while \(r\) and \(z\) are allowed to vary, the result is a half-plane (Figure \(\PageIndex{6}\)).
b. 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 \(3\) (Figure \(\PageIndex{7}\)).
c. 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 (Figure \(\PageIndex{8}\)).
Exercise \(\PageIndex{3}\)
Describe the surface with cylindrical equation \(r=6\).
- Hint
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The \(θ\) and \(z\) components of points on the surface can take any value.
- Answer
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This surface is a cylinder with radius \(6\).
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, the 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: spherical coordinate system
In the spherical coordinate system , a point \(P\) in space (Figure \(\PageIndex{9}\)) 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 \(\overline{OP}\), where \(O\) is the origin and \(0≤φ≤π.\)
By convention, the origin is represented as \((0,0,0)\) in spherical coordinates.
HOW TO: Converting among Spherical, Cylindrical, and Rectangular Coordinates
Rectangular coordinates \((x,y,z)\), cylindrical coordinates \((r,θ,z),\) and spherical coordinates \((ρ,θ,φ)\) of a point are related as follows:
Convert from spherical coordinates to rectangular coordinates
These equations are used to convert from spherical coordinates to rectangular coordinates.
- \(x=ρ\sin φ\cos θ\)
- \(y=ρ\sin φ\sin θ\)
- \(z=ρ\cos φ\)
Convert from rectangular coordinates to spherical coordinates
These equations are used to convert from rectangular coordinates to spherical coordinates.
- \(ρ^2=x^2+y^2+z^2\)
- \(\tan θ=\dfrac{y}{x}\)
- \(φ=\arccos\left(\dfrac{z}{\sqrt{x^2+y^2+z^2}}\right).\)
Convert from spherical coordinates to cylindrical coordinates
These equations are used to convert from spherical coordinates to cylindrical coordinates.
- \(r=ρ\sin φ\)
- \(θ=θ\)
- \(z=ρ\cos φ\)
Convert from cylindrical coordinates to spherical coordinates
These equations are used to convert from cylindrical coordinates to spherical coordinates.
- \(ρ=\sqrt{r^2+z^2}\)
- \(θ=θ\)
- \(φ=\arccos\left(\dfrac{z}{\sqrt{r^2+z^2}}\right)\)
The formulas to convert from spherical coordinates to rectangular coordinates may seem complex, but they are straightforward applications of trigonometry. Looking at Figure \(\PageIndex{10}\), it is easy to see that \(r=ρ \sin φ\). Then, looking at the triangle in the \(xy\)-plane with r as its hypotenuse, we have \(x=r\cos θ=ρ\sin φ \cos θ\). The derivation of the formula for \(y\) is similar. Figure \(\PageIndex{10}\) also shows that \(ρ^2=r^2+z^2=x^2+y^2+z^2\) and \(z=ρ\cos φ\). Solving this last equation for \(φ\) and then substituting \(ρ=\sqrt{r^2+z^2}\) (from the first equation) yields \(φ=\arccos\left(\dfrac{z}{\sqrt{r^2+z^2}}\right)\). Also, note that, as before, we must be careful when using the formula \(\tan θ=\dfrac{y}{x}\) to choose the correct value of \(θ\).
As we did with cylindrical coordinates, let’s consider the surfaces that are generated when each of the coordinates is held constant. Let \(c\) be a constant, and consider surfaces of the form \(ρ=c\). Points on these surfaces are at a fixed distance from the origin and form a sphere. The coordinate \(θ\) in the spherical coordinate system is the same as in the cylindrical coordinate system, so surfaces of the form \(θ=c\) are half-planes, as before. Last, consider surfaces of the form \(φ=c\). The points on these surfaces are at a fixed angle from the \(z\)-axis and form a half-cone (Figure \(\PageIndex{11}\)).
Example \(\PageIndex{4}\): Converting from Spherical Coordinates
Plot the point with spherical coordinates \((8,\dfrac{π}{3},\dfrac{π}{6})\) and express its location in both rectangular and cylindrical coordinates.
Solution
Use the equations in Converting among Spherical, Cylindrical, and Rectangular Coordinates to translate between spherical and cylindrical coordinates (Figure \(\PageIndex{12}\)):
\[ \begin{align*} x &=ρ\sin φ\cos θ \\[4pt] &=8 \sin\left(\dfrac{π}{6}\right) \cos\left(\dfrac{π}{3}\right) \\[4pt] &= 8\left(\dfrac{1}{2}\right)\dfrac{1}{2} \\[4pt] &=2 \\[4pt] y &=ρ\sin φ\sin θ \\[4pt] &= 8\sin\left(\dfrac{π}{6}\right)\sin\left(\dfrac{π}{3}\right) \\[4pt] &= 8\left(\dfrac{1}{2}\right)\dfrac{\sqrt{3}}{2} \\[4pt] &= 2\sqrt{3} \\[4pt] z &=ρ\cos φ \\[4pt] &= 8\cos\left(\dfrac{π}{6}\right) \\[4pt] &= 8\left(\dfrac{\sqrt{3}}{2}\right) \\[4pt] &= 4\sqrt{3} \end{align*}\]
The point with spherical coordinates \((8,\dfrac{π}{3},\dfrac{π}{6})\) has rectangular coordinates \((2,2\sqrt{3},4\sqrt{3}).\)
Finding the values in cylindrical coordinates is equally straightforward:
\[ \begin{align*} r&=ρ \sin φ \\[4pt] &= 8\sin \dfrac{π}{6} \\[4pt] &=4 \\[4pt] θ&=θ \\[4pt] z&=ρ\cos φ\\[4pt] &= 8\cos\dfrac{π}{6} \\[4pt] &= 4\sqrt{3} .\end{align*}\]
Thus, cylindrical coordinates for the point are \((4,\dfrac{π}{3},4\sqrt{3})\).
Exercise \(\PageIndex{4}\)
Plot the point with spherical coordinates \((2,−\frac{5π}{6},\frac{π}{6})\) and describe its location in both rectangular and cylindrical coordinates.
- Hint
-
Converting the coordinates first may help to find the location of the point in space more easily.
- Answer
-
Cartesian: \((−\frac{\sqrt{3}}{2},−\frac{1}{2},\sqrt{3}),\) cylindrical: \((1,−\frac{5π}{6},\sqrt{3})\)
Example \(\PageIndex{5}\): Converting from Rectangular Coordinates
Convert the rectangular coordinates \((−1,1,\sqrt{6})\) to both spherical and cylindrical coordinates.
Solution
Start by converting from rectangular to spherical coordinates:
\[ \begin{align*} ρ^2 &=x^2+y^2+z^2=(−1)^2+1^2+(\sqrt{6})^2=8 \\[4pt] \tan θ &=\dfrac{1}{−1} \\[4pt] ρ&=2\sqrt{2} \text{ and }θ=\arctan(−1)=\dfrac{3π}{4}. \end{align*}\]
Because \((x,y)=(−1,1)\), then the correct choice for \(θ\) is \(\frac{3π}{4}\).
There are actually two ways to identify \(φ\). We can use the equation \(φ=\arccos\left(\dfrac{z}{\sqrt{x^2+y^2+z^2}}\right)\). A more simple approach, however, is to use equation \(z=ρ\cos φ.\) We know that \(z=\sqrt{6}\) and \(ρ=2\sqrt{2}\), so
\(\sqrt{6}=2\sqrt{2}\cos φ,\) so \(\cos φ=\dfrac{\sqrt{6}}{2\sqrt{2}}=\dfrac{\sqrt{3}}{2}\)
and therefore \(φ=\dfrac{π}{6}\). The spherical coordinates of the point are \((2\sqrt{2},\dfrac{3π}{4},\dfrac{π}{6}).\)
To find the cylindrical coordinates for the point, we need only find \(r\):
\(r=ρ\sin φ=2\sqrt{2}\sin(\dfrac{π}{6})=\sqrt{2}.\)
The cylindrical coordinates for the point are \((\sqrt{2},\dfrac{3π}{4},\sqrt{6})\).
Example \(\PageIndex{6}\): Identifying Surfaces in the Spherical Coordinate System
Describe the surfaces with the given spherical equations.
- \(θ=\dfrac{π}{3}\)
- \(φ=\dfrac{5π}{6}\)
- \(ρ=6\)
- \(ρ=\sin θ \sinφ\)
Solution
a. The variable \(θ\) represents the measure of the same angle in both the cylindrical and spherical coordinate systems. Points with coordinates \((ρ,\dfrac{π}{3},φ)\) lie on the plane that forms angle \(θ=\dfrac{π}{3}\) with the positive \(x\)-axis. Because \(ρ>0\), the surface described by equation \(θ=\dfrac{π}{3}\) is the half-plane shown in Figure \(\PageIndex{13}\).
b. Equation \(φ=\dfrac{5π}{6}\) describes all points in the spherical coordinate system that lie on a line from the origin forming an angle measuring \(\dfrac{5π}{6}\) rad with the positive \(z\)-axis. These points form a half-cone (Figure \(\PageIndex{14}\)). Because there is only one value for \(φ\) that is measured from the positive \(z\)-axis, we do not get the full cone (with two pieces).
To find the equation in rectangular coordinates, use equation \(φ=\arccos\left(\dfrac{z}{\sqrt{x^2+y^2+z^2}}\right).\)
\[ \begin{align*} \dfrac{5π}{6} &=\arccos\left(\dfrac{z}{\sqrt{x^2+y^2+z^2}}\right) \\[4pt] \cos\dfrac{5π}{6}&=\dfrac{z}{\sqrt{x^2+y^2+z^2}} \\[4pt] −\dfrac{\sqrt{3}}{2}&=\dfrac{z}{\sqrt{x^2+y^2+z^2}} \\[4pt] \dfrac{3}{4} &=\dfrac{z^2}{x^2+y^2+z^2} \\[4pt] \dfrac{3x^2}{4}+\dfrac{3y^2}{4}+\dfrac{3z^2}{4} &=z^2 \\[4pt] \dfrac{3x^2}{4}+\dfrac{3y^2}{4}−\dfrac{z^2}{4} &=0. \end{align*}\]
This is the equation of a cone centered on the \(z\)-axis.
c. Equation \(ρ=6\) describes the set of all points \(6\) units away from the origin—a sphere with radius \(6\) (Figure \(\PageIndex{15}\)).
d. To identify this surface, convert the equation from spherical to rectangular coordinates, using equations \(y=ρ\sin φ\sin θ\) and \(ρ^2=x^2+y^2+z^2:\)
\(ρ=\sin θ \sin φ\)
\(ρ^2=ρ\sin θ\sin φ\) Multiply both sides of the equation by \(ρ\).
\(x^2+y^2+z^2=y\) Substitute rectangular variables using the equations above.
\(x^2+y^2−y+z^2=0\) Subtract \(y\) from both sides of the equation.
\(x^2+y^2−y+\dfrac{1}{4}+z^2=\dfrac{1}{4}\) Complete the square.
\(x^2+\left(y−\dfrac{1}{2}\right)^2+z^2=\dfrac{1}{4}\). Rewrite the middle terms as a perfect square.
The equation describes a sphere centered at point \((0,\dfrac{1}{2},0)\) with radius \(\dfrac{1}{2}\).
Exercise \(\PageIndex{5}\)
Describe the surfaces defined by the following equations.
- \(ρ=13\)
- \(θ=\dfrac{2π}{3}\)
- \(φ=\dfrac{π}{4}\)
- Hint
-
Think about what each component represents and what it means to hold that component constant.
- Answer a
-
This is the set of all points \(13\) units from the origin. This set forms a sphere with radius \(13\).
- Answer b
-
This set of points forms a half plane. The angle between the half plane and the positive \(x\)-axis is \(θ=\dfrac{2π}{3}.\)
- Answer c
-
Let \(P\) be a point on this surface. The position vector of this point forms an angle of \(φ=\dfrac{π}{4}\) with the positive \(z\)-axis, which means that points closer to the origin are closer to the axis. These points form a half-cone.
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 \(x^2+y^2+z^2=c^2\) has the simple equation \(ρ=c\) in spherical coordinates.
In geography, latitude and longitude are used to describe locations on Earth’s surface, as shown in Figure \(\PageIndex{16}\). 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 \(4000\) mi. We express angle measures in degrees rather than radians because latitude and longitude are measured in degrees.
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.
Example \(\PageIndex{7}\): Converting Latitude and Longitude to Spherical Coordinates
The latitude of Columbus, Ohio, is \(40°\) N and the longitude is \(83°\) W, which means that Columbus is \(40°\) 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 \(40°\). In the same way, measuring from the prime meridian, Columbus lies \(83°\) to the west. Express the location of Columbus in spherical coordinates.
Solution
The radius of Earth is \(4000\)mi, so \(ρ=4000\). 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 \(θ=−83°\), Because Columbus lies \(40°\) north of the equator, it lies \(50°\) south of the North Pole, so \(φ=50°\). In spherical coordinates, Columbus lies at point \((4000,−83°,50°).\)
Exercise \(\PageIndex{6}\)
Sydney, Australia is at \(34°S\) and \(151°E.\) Express Sydney’s location in spherical coordinates.
- Hint
-
Because Sydney lies south of the equator, we need to add \(90°\) to find the angle measured from the positive \(z\)-axis.
- Answer
-
\((4000,151°,124°)\)
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.
Example \(\PageIndex{8}\): Choosing the Best Coordinate System
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 \(\PageIndex{17}\)).
- Find the center of gravity of a bowling ball.
- Determine the velocity of a submarine subjected to an ocean current.
- Calculate the pressure in a conical water tank.
- Find the volume of oil flowing through a pipeline.
- Determine the amount of leather required to make a football.
Solution
- Clearly, a bowling ball is a sphere, so spherical coordinates would probably work best here. The origin should be located at the physical center of the ball. There is no obvious choice for how the \(x\)-, \(y\)- and \(z\)-axes should be oriented. Bowling balls normally have a weight block in the center. One possible choice is to align the \(z\)-axis with the axis of symmetry of the weight block.
- A submarine generally moves in a straight line. There is no rotational or spherical symmetry that applies in this situation, so rectangular coordinates are a good choice. The \(z\)-axis should probably point upward. The \(x\)- and \(y\)-axes could be aligned to point east and north, respectively. The origin should be some convenient physical location, such as the starting position of the submarine or the location of a particular port.
- A cone has several kinds of symmetry. In cylindrical coordinates, a cone can be represented by equation \(z=kr,\) where \(k\) is a constant. In spherical coordinates, we have seen that surfaces of the form \(φ=c\) are half-cones. Last, in rectangular coordinates, elliptic cones are quadric surfaces and can be represented by equations of the form \(z^2=\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}.\) In this case, we could choose any of the three. However, the equation for the surface is more complicated in rectangular coordinates than in the other two systems, so we might want to avoid that choice. In addition, we are talking about a water tank, and the depth of the water might come into play at some point in our calculations, so it might be nice to have a component that represents height and depth directly. Based on this reasoning, cylindrical coordinates might be the best choice. Choose the \(z\)-axis to align with the axis of the cone. The orientation of the other two axes is arbitrary. The origin should be the bottom point of the cone.
- A pipeline is a cylinder, so cylindrical coordinates would be best the best choice. In this case, however, we would likely choose to orient our \(z\)-axis with the center axis of the pipeline. The \(x\)-axis could be chosen to point straight downward or to some other logical direction. The origin should be chosen based on the problem statement. Note that this puts the \(z\) -axis in a horizontal orientation, which is a little different from what we usually do. It may make sense to choose an unusual orientation for the axes if it makes sense for the problem.
- A football has rotational symmetry about a central axis, so cylindrical coordinates would work best. The \(z\)-axis should align with the axis of the ball. The origin could be the center of the ball or perhaps one of the ends. The position of the \(x\)-axis is arbitrary.
Exercise \(\PageIndex{7}\)
Which coordinate system is most appropriate for creating a star map, as viewed from Earth (see the following figure)?
How should we orient the coordinate axes?
- Hint
-
What kinds of symmetry are present in this situation?
- Answer
-
Spherical coordinates with the origin located at the center of the earth, the \(z\)-axis aligned with the North Pole, and the \(x\)-axis aligned with the prime meridian
Key Concepts
- In the cylindrical coordinate system, a point in space is represented by the ordered triple \((r,θ,z),\) where \((r,θ)\) represents the polar coordinates of the point’s projection in the \(xy\)-plane and z represents the point’s projection onto the \(z\)-axis.
- To convert a point from cylindrical coordinates to Cartesian coordinates, use equations \(x=r\cos θ, y=r\sin θ,\) and \(z=z.\)
- To convert a point from Cartesian coordinates to cylindrical coordinates, use equations \(r^2=x^2+y^2, \tan θ=\dfrac{y}{x},\) and \(z=z.\)
- In the spherical coordinate system, a point \(P\) in space is represented by the ordered triple \((ρ,θ,φ)\), where \(ρ\) is the distance between \(P\) and the origin \((ρ≠0), θ\) is the same angle used to describe the location in cylindrical coordinates, and \(φ\) is the angle formed by the positive \(z\)-axis and line segment \(\overline{OP}\), where \(O\) is the origin and \(0≤φ≤π.\)
- To convert a point from spherical coordinates to Cartesian coordinates, use equations \(x=ρ\sin φ\cos θ, y=ρ\sin φ\sin θ,\) and \(z=ρ\cos φ.\)
- To convert a point from Cartesian coordinates to spherical coordinates, use equations \(ρ^2=x^2+y^2+z^2, \tan θ=\dfrac{y}{x},\) and \(φ=\arccos\left(\dfrac{z}{\sqrt{x^2+y^2+z^2}}\right)\).
- To convert a point from spherical coordinates to cylindrical coordinates, use equations \(r=ρ\sin φ, θ=θ,\) and \(z=ρ\cos φ.\)
- To convert a point from cylindrical coordinates to spherical coordinates, use equations \(ρ=\sqrt{r^2+z^2}, θ=θ,\) and \(φ=\arccos\left(\dfrac{z}{\sqrt{r^2+z^2}}\right).\)
Glossary
- cylindrical coordinate system
- a way to describe a location in space with an ordered triple \((r,θ,z),\) where \((r,θ)\) represents the polar coordinates of the point’s projection in the \(xy\)-plane, and z represents the point’s projection onto the \(z\)-axis
- spherical coordinate system
- a way to describe a location in space with an ordered triple \((ρ,θ,φ),\) where \(ρ\) is the distance between \(P\) and the origin \((ρ≠0), θ\) is the same angle used to describe the location in cylindrical coordinates, and \(φ\) is the angle formed by the positive \(z\)-axis and line segment \(\bar{OP}\), where \(O\) is the origin and \(0≤φ≤π\)
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5: Arithmetic in Base Ten Last updated Save as PDF Page ID 35075 Illustrative Mathematics OpenUp Resources Section 26: Warming Up to Decimals 26.1: Using Decimals in a Shopping Context Section 27: Adding and Subtracting Decimals 27.1: Using Diagrams to Represent Addition and Subtraction 27.2: Adding and Subtracting Decimals with Few Non-Zero Digits 27.3: Adding and Subtracting Decimals with Many Non-Zero Digits Section 28: Multiplying Decimals 28.1: Decimal Points in Products 28.2: Methods for Multiplying Decimals 28.3: Using Diagrams to Represent Multiplication 28.4: Calculating Products of Decimals Section 29: Dividing Decimals 29.1: Using the Partial Quotients Method 29.2: Using Long Division 29.3: Dividing Numbers that Result in Decimals 29.4: Dividing Decimals by Whole Numbers 29.5: Dividing Decimals by Decimals Section 30: Let's Put It to Work 30.1: Using Operations on Decimals to Solve Problems 30.2: Making and Measuring Boxes


5.7: Biological and Psychological Positivism
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- Last updated
- Save as PDF
- Alison S. Burke, David Carter, Brian Fedorek, Tiffany Morey, Lore Rutz-Burri, & Shanell Sanchez
- Southern Oregon University via OpenOregon
Trait theories assume there are fundamental differences that differentiate criminals from non-criminals. These differences can be discovered through scientific investigations. Additionally, many early biological and psychological theories used hard determinism, which implies people with certain traits will be criminals.
Cesare Lombroso was a trained medical doctor in Italy when he had an epiphany. As he was performing autopsies on Italian prisoners, he started to believe many of these men had different physical attributes compared to law-abiding people and that these differences were biologically inherited. In 1876, five years after Darwin’s claim about some humans might be evolutionary reversions, Lombroso wrote The Criminal Man . [1] Lombroso claimed 1/3 of all offenders were born criminals who were atavistic (evolutionary throwbacks).
He identified a list of physical features he believed to deviate from the “normal” population. These included an asymmetrical face, monkey-like ears, large lips, receding chin, twisted nose, long arms, skin wrinkles, and many more. Lombroso believed he could identify criminals simply by the way they physically looked. Even though his theory was widely rejected years later, it served as an example of the first attempt to explain criminal behavior scientifically.
A few decades after Lombroso’s theory, Charles Goring took Lombroso’s ideas about physical differences and added mental deficiencies too. In The English Convict , Goring claimed there were statistical differences in physical attributes and mental defects. The focus on mental qualities led to a new kind of biological positivism – the Intelligence Era. Alfred Binet, who created the Intelligence Quotient Test, believed intelligence was dynamic and could change. He wanted to identify youths who were not performing well in school. Unfortunately, H.H. Goddard, like many Americans at the time, believed intelligence was innate and static. That is, intelligence was fixed and could not change. Goddard gave IQ tests to sort people and those who scored too low were institutionalized, deported, or sterilized. He was an early advocate to sterilize those who were mentally deficient, especially “morons,” who were just smart enough to blend in with the normal population. In 1927, the United States Supreme Court in Buck v. Bell allowed the use of sterilization.
Even after Lombroso, Goring, and Goddard, contemporary research reveals intelligence is at least as critical as race and social class for predicting delinquency (Hirschi & Hindelang, 1977). [2] However, how we measure intelligence and how we define intelligence are based on our preconceived assumptions of intelligence. For example, is intelligence inherited? Is it related to the dominant culture? Or is it based more on the person’s environment? Each has a least some element of truth.
Modern biological theorists have revealed that biology plays a role in our behavior, but we cannot say how much or even how so. Twin studies and adoption studies examined the nature versus nurture debate. Both play a role in our behavior. It is more nature and nurture. Perhaps the question ought to be “how do our biological differences interact with our sociological differences?” There is not a crime gene per se, but some genetic variations are correlated with anti-social behaviors. However, those with genetic variations are not necessarily criminal because of genetics. Though it puts the individual at risk for such behaviors, a caring and supportive environment often mitigates the impact of the genetic code.
Proximate causes, such as neurotransmitters, hormones, the central nervous system, and the autonomic nervous system, have links to aggressive behavior too. However, many of these explanations have several possible causal paths. For example, we know people with higher testosterone levels engage in more aggressive behavior, but when people engage in aggressive behavior, their testosterone levels increase. We do not know which causes which.
What about criminal personalities? What about sociopaths and psychopaths? The Gluecks (1950) determined there was no real criminal personality; instead, there are some interrelated personality characteristics that were clustered together. [3] Even after giving personality tests to criminals and non-criminals, there does not seem to be any logical relevance to understanding the causes of crime. However, there have been correlations between certain personality traits and criminal behavior. For example, impulsivity, lack of self-control, inability to learn from punishment, and low empathy have all been linked to criminal behaviors.
Consequently, none of these personality characteristics are criminal in and of themselves. The real danger is when a person has many of these personality characteristics. Capsi et al. (1994) found that constraint and negative emotionality, two super traits that contain a number of different characteristics, were “robust correlates of delinquency” (p. 185). [4]
In summary, researchers have been able to say that our biology and personality play a role in criminal behaviors, but we cannot say how much or to what degree. The characteristics of our social environment interact with our biology and personality. Human behavior is quite complex and it is difficult to determine the true causality of human actions.
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Lombroso, C. (1876). The criminal man.
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Hirschi, T., & Hindelang, M.J. (1977). Intellegence and delinquency: A revisionist review.
American Sociological Review, 42
, 572-587.
↵
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Glueck, S., & Glueck, E. (1950).
Unraveling juvenile delinquency
. Cambridge, MA: Harvard University Press.
↵
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Capsi, A., Moffitt, T.E., Silva, P.A., Stouthamer-Loeber, M., Krueger, R.F., & Schmutte, P.S. (1994). Personality and crime: Are some people crime prone? Replications of the personality-crime relationship acrass countries, genders, races, and methods.
Criminology, 32
(2).
↵


2.15: Solving Single-Step Equations
Learning Outcome
- Use the addition, subtraction, multiplication, and division properties to solve single-step equations
We will begin solving equations below, but first we need to understand what a solution to an equation looks like. The purpose in solving an equation is to find the value or values of the variable that make each side of the equation the same — it is the answer to the puzzle.
Solution of an Equation
A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.
Determine whether a number is a solution to an equation.
- Substitute the number for the variable in the equation.
- Simplify the expressions on both sides of the equation.
-
Determine whether the resulting equation is true.
- If it is true, the number is a solution.
- If it is not true, the number is not a solution.
In the following example, we will show how to determine whether a number is a solution to an equation that contains addition and subtraction. You can use this idea to check your work later when you are solving equations.
EXAMPLE
Determine whether \(y=\Large\frac{3}{4}\) is a solution for \(4y+3=8y\).
Solution:
| \(4y+3=8y\) | |
| Substitute \(\color{red}{\Large\frac{3}{4}}\) for \(y\) | \(4(\color{red}{\Large\frac{3}{4}}\normalsize)+3\stackrel{\text{?}}{=}8(\color{red}{\Large\frac{3}{4}})\) |
| Multiply. | \(3+3\stackrel{\text{?}}{=}6\) |
| Add. | \(6=6\quad\checkmark\) |
Since \(y=\Large\frac{3}{4}\) results in a true equation, \(\Large\frac{3}{4}\) is a solution to the equation \(4y+3=8y\).
Now it is your turn to determine whether a fraction is the solution to an equation.
TRY IT
Let’s use a model to help understand how the process of solving an equation is like solving a puzzle. An envelope represents the variable – since its contents are unknown – and each counter represents one.
Suppose a desk has an imaginary line dividing it in half. We place three counters and an envelope on the left side of desk, and eight counters on the right side of the desk. Both sides of the desk have the same number of counters, but some counters are hidden in the envelope. Can you tell how many counters are in the envelope?
What steps are you taking in your mind to figure out how many counters are in the envelope? Perhaps you are thinking “I need to remove the \(3\) counters from the left side to get the envelope by itself. Those \(3\) counters on the left match with \(3\) on the right, so I can take them away from both sides. That leaves five counters on the right, so there must be \(5\) counters in the envelope.”
What algebraic equation is modeled by this situation? Each side of the desk represents an expression and the center line takes the place of the equal sign. We will call the contents of the envelope \(x\), so the number of counters on the left side of the desk is \(x+3\). On the right side of the desk are \(8\) counters. We are told that \(x+3\) is equal to \(8\) so our equation is \(x+3=8\).
\(x+3=8\)
Let’s write algebraically the steps we took to discover how many counters were in the envelope.
| \(x+3=8\) | |
| First, we took away three from each side. | \(x+3\color{red}{--3}=8\color {red}{--3}\) |
| Then we were left with five. | \(x=5\) |
Now let’s check our solution. We substitute \(5\) for \(x\) in the original equation and see if we get a true statement.
\(x+3=8\)
\(\color{red}{5}+3=8\)
\(8=8\quad\checkmark\) Our solution is correct. Five counters in the envelope plus three more equals eight.
example
Write an equation modeled by the envelopes and counters, and then solve the equation:
Solution
| On the left, write \(x\) for the contents of the envelope, add the \(4\) counters, so we have \(x+4\). | \(x+4\) |
| On the right, there are \(5\) counters. | \(5\) |
| The two sides are equal. | \(x+4=5\) |
| Solve the equation by subtracting \(4\) counters from each side. |
We can see that there is one counter in the envelope. This can be shown algebraically as:
\(x+4=5\)
\(x+4\color{red}{--4}=5\color{red}{--4}\)
\(x=1\)
Substitute \(1\) for \(x\) in the equation to check.
\(x+4=5\)
\(\color{red}{1}+4=5\)
\(5=5\quad\checkmark\)
Since \(x=1\) makes the statement true, we know that \(1\) is indeed a solution.
try it
Write the equation modeled by the envelopes and counters, and then solve the equation:
[reveal-answer q=”7845″]Show Answer[/reveal-answer]
[hidden-answer a=”7845″]
\(x+1=7\)
\(x=6\)
[/hidden-answer]
Write the equation modeled by the envelopes and counters, and then solve the equation:
[reveal-answer q=”9034″]Show Answer[/reveal-answer]
[hidden-answer a=”9034″]
\(x+3=4\)
\(x=1\)
[/hidden-answer]
Our puzzle has given us an idea of what we need to do to solve an equation. The goal is to isolate the variable by itself on one side of the equation. In the previous examples, we used the Subtraction Property of Equality, which states that when we subtract the same quantity from both sides of an equation, we still have equality.
Subtraction Property of Equality
For all real numbers \(a,b\), and \(c\), if \(a=b\), then \(a-c=b-c\).
In all the equations we have solved so far, a number was added to the variable on one side of the equation. We used subtraction to “undo” the addition in order to isolate the variable.
But suppose we have an equation with a number subtracted from the variable, such as \(x - 5=8\). We want to isolate the variable, so to “undo” the subtraction we will add the number to both sides.
We use the Addition Property of Equality, which says we can add the same number to both sides of the equation without changing the equality. Notice how it mirrors the Subtraction Property of Equality.
Addition Property of Equality
For all real numbers \(a,b\), and \(c\), if \(a=b\), then \(a+c=b+c\).
When you add or subtract the same quantity from both sides of an equation, you still have equality.
We introduced the Subtraction Property of Equality earlier by modeling equations with envelopes and counters. The image below models the equation \(x+3=8\).
The goal is to isolate the variable on one side of the equation. So we “took away” \(3\) from both sides of the equation and found the solution \(x=5\).
Some people picture a balance scale, as in the image below, when they solve equations.
The quantities on both sides of the equal sign in an equation are equal, or balanced. Just as with the balance scale, whatever you do to one side of the equation you must also do to the other to keep it balanced.
In the following examples we review how to use Subtraction and Addition Properties of Equality to solve equations. We need to isolate the variable on one side of the equation. You can check your solutions by substituting the value into the original equation to make sure you have a true statement.
EXAMPLE
Solve: \(x+11=-3\)
[reveal-answer q=”190834″]Show Answer[/reveal-answer]
[hidden-answer a=”190834″]
Solution:
To isolate \(x\), we undo the addition of \(11\) by using the Subtraction Property of Equality.
| \(x+11=-3\) | ||
| Subtract 11 from each side to “undo” the addition. | \(x+11\color{red}{-11}=-3\color{red}{-11}\) | |
| Simplify. | \(x=-14\) | |
| Check: | \(x+11=-3\) | |
| Substitute \(x=-14\). | \(\color{red}{-14}+11\stackrel{\text{?}}{=}-3\) | |
| \(-3=-3\quad\checkmark\) |
Since \(x=-14\) makes \(x+11=-3\) a true statement, we know that it is a solution to the equation.
[/hidden-answer]
Now you can try solving an equation that requires using the addition property.
TRY IT
In the original equation in the previous example, \(11\) was added to the \(x\), so we subtracted \(11\) to “undo” the addition. In the next example, we will need to “undo” subtraction by using the Addition Property of Equality.
EXAMPLE
Solve: \(m - 4=-5\)
[reveal-answer q=”604060″]Show Answer[/reveal-answer]
[hidden-answer a=”604060″]
Solution:
| \(m-4=-5\) | ||
| Add 4 to each side to “undo” the subtraction. | \(m-4\color{red}{+4}=-5\color{red}{+4}\) | |
| Simplify. | \(m=-1\) | |
| Check: | \(m-4=-5\) | |
| Substitute \(m=-1\). | \(\color{red}{-1}+4\stackrel{\text{?}}{=}-5\) | |
| \(-5=-5\quad\checkmark\) | ||
| The solution to \(m - 4=-5\) is \(m=-1\) |
[/hidden-answer]
Now you can try using the addition property to solve an equation.
TRY IT
In the following video, we present more examples of solving equations using the addition and subtraction properties.
You may encounter equations that contain fractions or decimals — especially in financial applications — so let’s practice solving those in the following problems.
TRY IT
TRY IT
[ohm_question]156990[/ohm_question]
All of the equations we have solved so far have been of the form \(x+a=b\) or \(x-a=b\). We were able to isolate the variable by adding or subtracting the constant term. Now we’ll see how to solve equations that involve division.
We will model an equation with envelopes and counters.
Here, there are two identical envelopes that contain the same number of counters. Remember, the left side of the workspace must equal the right side, but the counters on the left side are “hidden” in the envelopes. So how many counters are in each envelope?
To determine the number, separate the counters on the right side into \(2\) groups of the same size. So \(6\) counters divided into \(2\) groups means there must be \(3\) counters in each group (since \(6\div2=3\)).
What equation models the situation shown in the figure below? There are two envelopes, and each contains \(x\) counters. Together, the two envelopes must contain a total of \(6\) counters. So the equation that models the situation is \(2x=6\).
We can divide both sides of the equation by \(2\) as we did with the envelopes and counters.
\(\Large{\frac{2x}{\color{red}{2}}=\frac{6}{\color{red}{2}}}\)
\(x=3\)
We found that each envelope contains \(\text{3 counters.}\) Does this check? We know \(2\cdot 3=6\), so it works. Three counters in each of two envelopes does equal six.
Another example is shown below.
Now we have \(3\) identical envelopes and \(\text{12 counters.}\) How many counters are in each envelope? We have to separate the \(\text{12 counters}\) into \(\text{3 groups.}\) Since \(12\div 3=4\), there must be \(\text{4 counters}\) in each envelope.
The equation that models the situation is \(3x=12\). We can divide both sides of the equation by \(3\).
\(\Large{\frac{3x}{\color{red}{3}}=\frac{12}{\color{red}{3}}}\)
\(x=4\)
Does this check? It does because \(3\cdot 4=12\).
Division Property of Equality
For all real numbers \(a,b,c\), and \(c\ne 0\), if \(a=b\), then \(\Large\frac{a}{c}\normalsize =\Large\frac{b}{c}\).
Stated simply, when you divide both sides of an equation by the same quantity, you still have equality.
Remember, the goal is to “undo” the operation on the variable. In the example below the variable is multiplied by \(4\), so we will divide both sides by \(4\) to “undo” the multiplication.
example
Solve: \(4x=-28\)
Solution:
To solve this equation, we use the Division Property of Equality to divide both sides by \(4\).
| \(4x=-28\) | |
| Divide both sides by 4 to undo the multiplication. | \(\Large\frac{4x}{\color{red}4}\normalsize =\Large\frac{-28}{\color{red}4}\) |
| Simplify. | \(x =-7\) |
| Check your answer. | \(4x=-28\) |
| Let \(x=-7\). Substitute \(-7\) for x. | \(4(\color{red}{-7})\stackrel{\text{?}}{=}-28\) |
| \(-28=-28\) |
Since this is a true statement, \(x=-7\) is a solution to \(4x=-28\).
Now you can try to solve an equation that requires division and includes negative numbers.
try it
Now, consider the equation \(\Large\frac{x}{4}\normalsize=3\). We want to know what number divided by \(4\) gives \(3\). So to “undo” the division, we will need to multiply by \(4\). The Multiplication Property of Equality will allow us to do this. This property says that if we start with two equal quantities and multiply both by the same number, the results are equal.
Multiplication Property of Equality
For all real numbers \(a,b,c\), if \(a=b\), then \(ac=bc\).
Stated simply, when you multiply both sides of an equation by the same quantity, you still have equality.
Previously we learned how to “undo” multiplication by dividing. How do you think we “undo” division?
Next, we will show an example that requires us to use multiplication to undo division.
example
Solve: \(\Large\frac{a}{-7}\normalsize =-42\)
[reveal-answer q=”399032″]Show Answer[/reveal-answer]
[hidden-answer a=”399032″]
Solution:
Here \(a\) is divided by \(-7\). We can multiply both sides by \(-7\) to isolate \(a\).
| \(\Large\frac{a}{-7}\normalsize =-42\) | |
| Multiply both sides by \(-7\). |
\(\color{red}{-7}( \Large\frac{a}{-7}\normalsize)=\color{red}{-7}(-42)\)
\(\Large\frac{-7a}{-7}\normalsize=294\) |
| Simplify. | \(a=294\) |
| Check your answer. | \(\Large\frac{a}{-7}\normalsize=-42\) |
| Let \(a=294\). | \(\Large\frac{\color{red}{294}}{-7}\normalsize\stackrel{\text{?}}{=}-42\) |
| \(-42=-42\quad\checkmark\) |
[/hidden-answer]
Now see if you can solve a problem that requires multiplication to undo division. Recall the rules for multiplying two negative numbers — two negatives give a positive when they are multiplied.
try it
As you begin to solve equations that require several steps you may find that you end up with an equation that looks like the one in the next example, with a negative variable. As a standard practice, it is good to ensure that variables are positive when you are solving equations. The next example will show you how.
example
Solve: \(-r=2\)
[reveal-answer q=”388033″]Show Answer[/reveal-answer]
[hidden-answer a=”388033″]
Solution:
Remember \(-r\) is equivalent to \(-1r\).
| \(-r=2\) | |
| Rewrite \(-r\) as \(-1r\). | \(-1r=2\) |
| Divide both sides by \(-1\). | \(\Large\frac{-1r}{\color{red}{-1}}\normalsize =\Large\frac{2}{\color{red}{-1}}\) |
| Simplify. | \(r=-2\) |
| Check. | \(-r=2\) |
| Substitute \(r=-2\) | \(-(\color{red}{-2})\stackrel{\text{?}}{=}2\) |
| Simplify. | \(2=2\quad\checkmark\) |
[/hidden-answer]
Now you can try to solve an equation with a negative variable.
try it
The next video includes examples of using the division and multiplication properties to solve basic equations.


By far, sex and gender has been one of the most socially significant social factors in the history of the world and the United States. Sex is one's biological classification as male or female and is set into motion at the moment the sperm fertilizes the egg. Sex can be precisely defined at the genetic level with XX being female and XY being male. Believe it or not, there are very few sex differences based on biological factors. Does this surprise you? Many of my students say “but what about that whole opposite sex argument?” Truth is, biologically there is no opposite sex.  For the sake of argument, ignore the reproductive differences and you basically see taller, stronger, and faster males. The real difference is the reproductive body parts, their function, and corresponding hormones. The average US woman has about 2 children in her lifetime. She also experiences a monthly period. Other than that and a few more related issues listed in Table
\(\PageIndex{1}\)
, reproductive roles are a minor difference in the overall daily lives women, yet so very much importance has been placed on these differences throughout history.
Table \(\PageIndex{1}\) : Known Biological Sex Differences
Female
Male
Reproductive
Vagina
Penis
Uterus
Testicles
Ovaries
Scrotum
Breast Development
Breast Dormant
Cyclic Hormones
-
Other
Shorter
Taller
Less aggression - Testosterone
More aggression - Testosterone
Runs a bit slower
Runs a bit faster
Less upper body strength
More upper body strength
Live years longer (7 years in developed countries)
Live shorter lives (3 years shorter worldwide)
We have much more in common than differences. In table
\(\PageIndex{2}\)
you see a vast list of similarities common to both men and women. Every major system of the human body functions in very similar ways to the point that health guidelines, disease prevention and maintenance, and even organ transplants are very similar and guided under a large umbrella of shared guidelines. True, there are medical specialists in treating men and women, but again the similarities outweigh the differences. Today you probably ate breakfast, took a shower, walked in the sunlight, sweated, slept, used the bathroom, was exposed to germs and pathogens, grew more hair and finger nails, exerted your muscles to the point that they became stronger, and felt and managed stress. So did every man and woman you know and in very similar ways.
Table \(\PageIndex{2}\) : Known Biological Sex Similarities
Sex Similarities
1. Digestive System
2. Respiratory System
3. Circulatory System
4. Lymphatic System
5. Urinary System
6. Musculoskeletal System
7. Nervous System
8. Endocrine System
9. Sensory System 10. Immune System
11. Urinary System
12. Integumentary System - Skin, Hair, and Nails
13.  Yes, you probably guessed correctly. Both males and females have all these hormones, plus many others, including testosterone.
Not only are males and females very similar, but science has shown that we truly are more female than male in biological terms. So, why the big debate of the battle of the sexes? Perhaps it's because of the impact of Gender is the cultural definition of what it means to be a man or a woman. Gender is cultural-based and varies in a thousand subtle ways across the many diverse cultures of the world. Gender has been shaped by political, religious, philosophical, language, tradition and other cultural forces for many years. To this day, in most countries of the world women are still oppressed and denied access to opportunities more than men and boys. This can be seen through many diverse historical documents.
When reading these documents, the most common theme of how women were historically oppressed in the world's societies is the omission of women as being legally, biologically, economically, and even spiritually on par with men. The second most common theme is the assumption that women were somehow broken versions of men (Google Aristotle's The Generation of Animals, Sigmund Freud's Penis Envy, or John Grey's Mars and Venus work).
Biology has disproven the belief that women are broken versions of men. In fact, the 23rd chromosome looks like XX in females and XY in males and the Y looks more like an X with a missing leg than a Y. Ironically, science has shown that males are broken or variant versions of females and the more X traits males have the better their health and longevity.
Debunking Myths About Women
In Table
\(\PageIndex{1}\)
you saw how females carry the lion share of the biological reproduction of the human race. Since history assumed that women were impaired because of their reproductive roles (men were not), societies have defined much of these reproductive traits as hindrances to activities. I found an old home health guide at an antique store in Ohio. I bought it and was fascinated that in 1898 the country's best physicians had very inaccurate information and knowledge about the human body and how it worked (See, if you can find one, The Book of Health A Practical Family Physician, 1898, by Robert W. Patton). Interestingly, pregnancy was considered “normal” within most circumstances while menstruation was seen as at type of disease process that had to be treated (back then most physicians were men and still are today). On pages 892-909 it refers to menstrual problems as being “unnatural” and normal only if “painless” and thus the patient should be treated rather than the “disease.” Indeed from a male scientific perspective in 1898, females and their natural reproductive cycles were problematic.
But, to the author, females were more fragile and vulnerable and should be treated more carefully than males especially during puberty. Patton states, “The fact is that the girl has a much greater physical and a more intense mental development to accomplish than the boy…” As for public education, he states that “The boy can do it; the girl can-sometimes…” He attributes most of the female sexual and reproductive problems to public school which is a byproduct of “women's rights, so called.”
He'd probably be stunned to see modern medicine's discoveries today. In our day, women are not defined as being inferior in comparison to men. But, in 1898, a physician (source of authority and scientific knowledge) had no reservations about stating the cultural norm in print, that women were considered broken in contrast to men.
Gender Socialization is the shaping of individual behavior and perceptions in such a way that the individual conforms to the socially prescribed expectations for males and females. One has to wonder what might have been different if all women were born into societies that valued their uniqueness and similarities in comparison to men. How much further might civilizations have progressed? It is wisdom to avoid the exclusion of any category of people-based on biological or other traits-from full participation in the development of knowledge and progress in society. In the history of the world, such wisdom has been ignored far too often.
Gender Roles as a Social Force
One can better understand the historical oppression of women by considering three social factors throughout the world's history, religion, tradition, and labor-based economic supply and demand. In almost all of the world's major religions (Christianity, Islam, Hinduism, Buddhism, Judaism, and many others) very clear distinctions have been made about Gender Roles are socialized expectations of what is normal, desirable, acceptable, and conforming for males and females in specific jobs or positions in groups and organizations over the life course. These gender roles have very specific meanings for the daily lives and activities of males and females who live under the religious cultures in nations throughout history and even in our day. The Book of Leviticus in the Judeao-Christian Old Testament has many biological rituals based specifically on Women's' hygiene. A close friend of mine performed her Master's thesis in Ancient Near East Studies on the reproductive hygiene rituals described in the book of Leviticus (see Is God a Respecter of Persons?: Another Look at the Purity Laws in Leviticus / Anne M. Adams, 2000 in BYU Library Holdings). In brief, she found no modern-day scientific support for these religious rituals on female's health nor on their reproduction. Her conclusion was that these were religious codes of conduct, not biologically-based scientifically beneficial codes.
Many ancient writings in religions refer to the flaws of females, their reproductive disadvantages, their temperament, and the rules that should govern them in the religious community. Please don't get me wrong, if it sounds like I'm bashing religious beliefs, I'm not. In fact many current religious doctrines have transformed as society's values of gender equality have emerged. I am also a fan of religious worship and participation in whatsoever religion a person chooses to follow. My point is that throughout history, religions were a dominant social force in many nations and the religious doctrines, like the cultural values, often placed women in a subjugated role to men and a number of different levels.
The second social force is tradition. Traditions can be and have been very harsh toward women. Look at Table 3 below which shows a scale of the outcomes of oppression toward women that have and currently do exist somewhere in the world. I have always found it remarkable that even though the average woman out lives the average man by 3 years worldwide and 7 years in developed countries, there are still a few countries where cultural and social oppression literally translates into shorter life expectancies for women.
Table \(\PageIndex{3}\) : Outcomes of the 10 Worse Forms of Oppression of Women - Worst to Least
Opression of Women
10 - Death from cultural and social oppression
1
(Various Countries)
9 - Sexual and other forms of slavery (Western Africa and Thailand)
8 - Maternal deaths (Sub-Sahara Africa and developing nations)
7 - Female Genital Mutilation (Mid-Africa about 120 million victims)
6 - Rape and sexual abuse (South Africa and United States are worst countries)
5 - Wage disparity (Worldwide)
4 - No/low education for females (Various degrees in most countries of the world)
3 - Denial of access to jobs and careers (many developing nations)
2 - Mandatory covering of females' bodies head to toe (Traditional countries, Muslim)
1 - Public demeaning of women (Still practiced, public and private)
1
www.prb.org World Population Data Sheet2008; pages 7-15. www.prb.org/pdf08/08WPDS_Eng.pdf (Niger, Zambia, Botswana and Namibia have lower death rates for women while Kenya, Zimbabwe, Afghanistan, and Micronesia have a tie between men and women's life expectancy-this even though in developing nations the average woman outlives the average man by 3 years)
Some cultural traditions are so harsh that females are biologically trumped by males-this by withholding nutrition, abandoning wife and daughters, abuse, neglect, violence, refugee status,, diseases, and complications of childbirth unsupported by the government. If you study this online looking at the Population Reference Bureau's many links and reports, you will find a worldwide concerted effort to persuade government, religious, and cultural leaders to shift their focus and efforts to nurture and protect women/females (www.PRB.org see also United Nations www.un.org ). Progress has already been made to some degree, but much change is still warranted because life, health and well-being are at stake for billions of women worldwide.
One of the most repugnant traditions in our world has been and is the sales of children/women into sexual and other forms of slavery. Countless civilizations that are still influential in our modern thought and tradition have sold girls and women the same way one might sell a horse or cow. It's estimated by a variety of organizations and sources that about 1 million women are currently forced into the sex slavery industry (boys are also sold and bought into slavery). India, Western Africa, and Thailand are some of the most notorious regions for this atrocity Google amnesty International, Sexual Slavery, PRB.org, United Nations, and search Wikipedia.org).  First, they allow it to occur as in the case of Thailand where it's a major draw of male tourists; and Second, they fail to police sexual slavery which is often criminal and/or organized crime in nature. The consequences to these girls and women are harsh at every level of human existence and is often connected to the spread of HIV and other communicable diseases.
Although pregnancy is not a disease it carries with it many health risks when governments fail to provide resources to expecting mothers before, during, and after delivery of their baby. Maternal Death is the death of a pregnant woman resulting from pregnancy, delivery, or recovery complications. Maternal deaths number in the hundreds of thousands and are estimated by the United Nations to be around _ million per year worldwide (See www.UN.org ). Typically very little medical attention is required to prevent infection, mediate complications, and assist in complications to mothers. To answer this problem one must approach it at the larger social level with government, health care systems, economy, family, and other institutional efforts. The Population Reference Bureau puts a woman's risk of dying from maternal causes at 1 in 92 worldwide with it being as low as 1 in 6,000 in developed countries and as high as 1 in 22 for the least developed regions of the world (See www.prb.org World Population Data Sheet 2008). The PRB reports “little improvement in maternal Mortality in developing countries (see page 3 of the Data Sheet).
Female Genital Mutilation is the traditional cutting, circumcision, and removal of most or all external genitalia of women for the end result of closing off some or part of the vagina until such time that the woman is married and cut open. In some traditions, there are religious underpinnings. In others, there are customs and rituals that have been passed down. In no way does the main body of any world religion condone or mandate this practice-many countries where this takes place are predominantly Muslim-yet local traditions have corrupted the purer form of the religion and its beliefs and female genital mutilation predates Islam (see Obermeyer, C.M. March 1999, Female Genital Surgeries: The Known and the Unknowable. Medical Anthropology Quaterly13, pages 79-106;p retrieved 5 December from www.anthrosource.net/doi/abs/...q.19<IP_ADDRESS> ). An analogy can be drawn from the Taliban which was extreme in comparison to most Muslims worldwide and which literally practiced homicide toward its females to enforce conformity. It should also be explained that there are no medical therapeutic benefits from female genital mutilation. Quite the contrary, there are many adverse medical consequence that result from it from ranging from pain, difficulty in childbirth, illness, and even death.
Many human rights groups, the United Nations, scientists, advocates, the United States, the World Health Organization, and other organizations have made aggressive efforts to influence the cessation of this practice worldwide. But, progress has come very slowly. Part of the problem is that women often perform the ritual and carry on the tradition as it was perpetrated upon them. In other words, many cases have women preparing the next generation for it and at times performing it on them.
 Rape is violence, motivated by men with power, anger, selfish, and sadistic issues. Rape is dangerous and destructive and more likely to happen in the United States than in most other countries of the world. There are 195 countries in the world today. The US typically is among the worst 5 percent in terms of rape (Yes, that means 95% of the world's countries are safer for women than the US). Consecutive studies performed by the United Nations Surveys on crime Trends and the Operations of Criminal Justice Systems confirm that South Africa is the most dangerous, crime-ridden nation on the planet in all crimes including rape (see
http://www.unodc.org/unodc/en/data-a...e-Systems.html
).
The world's histories with very few exceptions have recorded the pattern of sexually abusing boys, girls, and women. Slavery, conquest of war, kidnapping, assault, and other circumstances are the context of these violent practices. Online there is a Website at www.rainn.org which is a tremendous resources for knowledge and information especially about rape, assault, incest and issues relating to the United States. The United Nations reported that, “Women aged 15-44 are more at risk from rape and domestic violence than from cancer, motor accidents, war and malaria, according to World Bank data (Retrieved 5 December, 2008 from www.un.org/women/endviolence/docs/VAW.pdf, UNite To End Violence Against Women, Feb. 2008). The UN calls for a criminal Justice System response and for increased prioritization and awareness. Anything might help since almost every country of the world is struggling to prevent sexual violence and rape against its females.
Opportunities
Wage disparities between males and females is both traditional and labor-based economic supply and demand. Statistics show past and current discrepancies in lower pay for women. Diane White made a 1997 presentation to the United Nations General Assembly stated that “Today the wage disparity gap cost American women $250,000 over the course of their lives” (Retrieved 5 December from www.un.org/womenwatch/osagi/s...ne%20White.pdf. Indeed evidence supports her claim that women are paid less in comparison to men and their cumulated losses add up to staggering figures. The US Census Bureau reported in 2008 that US women earn 77 cents for every US man's $1 (See: American Community Survey . They also reported that in some places (Washington DC) and in certain fields (Computers and mathematical) women earn as much as 98 cents per a man's $1. At the worldwide level “As employees, women are still seeking equal pay with men. Closing the gap between women's and men's pay continues to be a major challenge in most parts of the world” (retrieved 5 Dec., 2008 from the UNstats.org from The World's Women 2005: Progress and Statistics
http://unstats.un.org/unsd/demograph..._4_Work_BW.pdf
; page 54).
The report also discussed the fact that about 60 countries have begun to keep statistics on informal (unpaid) work by women. Needless to say even though measuring paid and unpaid work of women is not as accurate as needed for world considerations, “Women contribute to development not only through remunerated work but also through a great deal of unremunerated work” (page 47).
Why the lower wages for women? The traditional definition of the reproductive roles of women as being “broken, diseased, or flawed” is part of the answer of wage disparity. The idea that reproductive roles interfere with the continuity of the workplace and the idea that women cannot be depended on plays heavily into the maltreatment of women. The argument can be made that traditional and economic factors have lead to the existing patterns of paying women less for their same education, experience, and efforts compared to men.
Efforts to provide formal education to females worldwide have escalated over the last few decades. The 2002 Kids Count International Data Sheet estimated rates as low as 11 percent of females in primary school in Somalia (retrieved 8 December , 2008 from www.prb.org/pdf/childrenwallchartfinal.pdf. A 1993 World Bank report made it very clear that females throughout the world were being neglected in receiving their formal educations when compared to males (see Subbarro, K. and Raney, L. 1993, “Social Gains from Female Education: A Cross-National Study”. World Bank Discussion Papers 194; retried from Eric ED 363542 on 8 December, 2008). In 1998 another example is found in efforts specific to Africa via the Forum of African Women Educationalists which focuses on governmental policies and practices for female education across the continent (retrieved 8 Dec 2008 from www.un.org/ecosocdev/geninfo/...x/114sped3.htm ). Literally hundreds of studies have since focused on other regions around and below the equator where education levels for females are much lower.
In 1999 it was reported by UNICEF that 1 billion people would never learn to read as children with 130 million school aged children (73 million girls) without access to basic education (retrieved, 8 Dec 2008 from www.unicef.org/sowc99/ ). Another UNICEF 2008 report clearly identifies the importance of educating girls who grow up to be mothers because of the tremendous odds that those educated mothers will ensure that their children are also formally educated ( see www.unicef.org/sowc08/docs/sowc08.pdf). In its statistical tables it shows that Somalia is now up to 22 percent for boys and girls in primary schools, yet in most countries females are still less likely to be educated ( see www.unicef.org/sowc08/docs/sowc08_table_1.pdf). The main point from UNICEF and many other formal reports is that higher formal education for females is associated with life, health, protection from crime and sexual exploitation, and countless other benefits, especially to females in the poorer regions of the world.
In the United States most females and males attend some form of formal education. After high school, many go to college. Even though the US numbers of 18-24 year old men are higher than women (www.USCensus.gov ) women are more likely to attend college based on percentages, 57% are Women (retrieved 8 December 2008 from www.usatoday.com/news/educati...ge-cover_x.htm ).
A projection from the National Center for Education Statistics projects a continuing trend up and through the year 2016 where about 58 percent of US college students will be female (retrieved 8 December, 2008 from “Projections of Education Statistics to 2016” nces.ed.gov/programs/projecti...2016/sec2c.asp ). By 2016 about 60 percent of graduated students will be females (see nces.ed.gov/programs/projecti...2016/sec4b.asp). These numbers reflect a strong and concerted push toward equality of opportunity for females in formal education that does date back over a century. The challenge is to avoid defining progress for US females in public and private education as having been made at the expense of males. That's much too simplistic.
They also reflect a change in the culture of breadwinning and the adult roles of males. Males and/or females who don't pursue a college degree will make less money than those who did. To make sense of this trend, many males have been identified as having a prolonged adolescence (even into their 30's), video game playing mentality, and a "live with your parents indefinitely" strategy until their shot at the labor force has passed them by. Others have pointed out the higher rates of learning disabilities in K-12, the relatively low percentage of K-5 teachers who are males, and the higher rate of male dropouts. Still others blame attention deficit and hyperactivity as part of the problem. Here is a truism about education in the US:
Higher education=higher pay=higher social prestige=higher income=higher quality of life.
Many countries of the world have neutralized the traditional, religious, and labor-force based biases against women and have moved to a merit-based system. Even in the US, there have been “men's wages, then women and children's wages (1/10th to 12/3rd of a man's). In a sense, any hard working, talented person can pursue and obtain a high-end job, including women. Communism broke some of these barriers early on in the 20th century, but the relatively low wages afforded those pursuing these careers somewhat offset the advances women could have made. In the US progress has come more slowly. Physicians are some of the brightest and best paid specialists in the world. Salaries tend to begin in the $100,000 range and can easily reach $500,000 depending on the specialty (see
http://www.allied-physicians.com/sal...n-salaries.htm
). Prior to 1970 most physicians were white and male, but things are slowly changing.
See Table
\(\PageIndex{4}\)
for trends between 1970 and 2006.
Table \(\PageIndex{4}\) : The Percentage of Physicians who are Male and Female
1
Year
% Male
% Female
1970
92.4%
7.6%
1980
88.4%
11.6%
1990
83.1%
16.9%
2000
76.3%
24.0%
2002
74.8%
25.2%
2003
74.2%
25.8%
2006
72.2%
27.8%
1
Retrieved from the American Medical Association 8 December, 2008 from “Table 1- Physicians By Gender (Excludes Students)” www.ama-assn.org/ama/pub/category/12912.html]
The upward trend shows a concerted effort to provide equal opportunity for females and males. Engineers have also seen a concerted effort to facilitate females into the profession. The Society of Women Engineers is a non-profit organization which helps support and recognize women as engineers (see
http://societyofwomenengineers.swe.org/index.php
). Look at Figure
\(\PageIndex{1}\)
below.
Computer-based careers are seeing striking gains in some areas for women who will be hired competitively based on merit. The same cannot be said for doctoral level employment in the more prestigious fields. In Figure
\(\PageIndex{2}\)
you can see 2005 estimates from the US National Science Foundation.
The first 6 fields are the highest paying fields to work in while social and psychological sciences are among the least paying. Women clearly dominate Psychology and nearly tie in social sciences and biology. True, at the doctoral level pay is higher than at the masters and bachelors levels, but the difference in engineering and psychology is remarkable at every level of education (see www.bls.gov/oes/current/oes_nat.htm#b00-0000 ).
The mandatory covering of females' bodies head to toe has been opposed by some and applauded by others. Christians, Hindus, and many other religious groups have the practice of covering or veiling in their histories. Yet, over the last 30 years, as fundamentalist Muslim nations and cultures have returned to their much more traditional way of life, Hijab, the Arabic word that means to cover or veil, has become more common. Often Hijab means modest and private in the day to day interpretations of the practice. For some countries it is a personal choice, while for others it becomes a crime not to comply. The former Taliban, punished such a crime with death (they also punished formal school of females and the use of makeup by death).
Many women's rights groups have brought public attention to this trend, not so much because the mandated covering of females is that oppressive, but because the veiling and covering is symbolic of the religious, traditional, and labor-forced patterns of oppression that have caused so many problems for women and continue to do so today.
I interviewed a retired OBGYN nurse who served as a training nurse for a mission in Saudi Arabia on a volunteer basis. She taught other local nurses from her 30 years of experience. Each and every day she was guarded by machine gun toting security forces everywhere she went. She was asked to cover and veil and did so. I asked her how she felt about that, given that her US culture was so relaxed on this issue.
“I wanted to teach those women and knew that they would benefit from my experience. I just had to do what I was told by the authorities,” she said.
“What would have happened if you had tried to leave the compound without your veil?” I asked.
“I suspect, I would have been arrested and shot.” She chuckles. “Not shot, perhaps, but If I did not comply, my training efforts would have been stopped and I would have been sent home.”
“So, you complied because of your desire to train the nurses?”
“That and the mothers and babies.” She answered.
(Interview with HB, 12 June, 2005)
The public demeaning of women has been acceptable throughout various cultures because publically demeaning members of society who are privately devalued and/or considered flawed fits the reality of most day-to-day interactions. Misogyny is the physical or verbal abuse and mistreatment of women. Verbal misogyny is unacceptable in public in most Western Nations today. With the ever present technology found in cell phones, video cameras, and security devices a person's private and public misogynistic language could easily be recorded and posted for millions to see on any number of Websites.
Perhaps, this fear of being found out as a woman-hater is not the ideal motivation for creating cultural values of respect and even admiration of women and men. As was mentioned above, most of the world historical leaders assumed that women were not as valuable as men and it has been a few decades since changes have begun. Yet, an even more sinister assumption has and does persist today, that women were the totality of their reproductive role, or Sex=Gender (Biology=Culture). If this were true then women would ultimately just be breeders of the species, rather than valued human beings they are throughout the world today.
An early pioneer and one of my personal heroines is an anthropologist named, Margaret Mead (1901-1978). Dr. Mead earned her Ph.D. under the direction of some of the best anthropologists of her day. But, she was a woman in a mostly male-dominated academic field. In my own readings of her works-her works are regularly quoted in many different disciplines today-I marvel that she successfully challenged the sexist and misogynistic notions established in academics at the time.
Bold Research on Gender
Mead's work entitled, Sex and Temperament in Three Primitive Societies (1935) became a major seminal work in the women's liberation movement and thereby in the redefinition of women in many Western Societies. Her observations of gender in three tribes, Arapesh, Mundugamor, and Tchambuli created a national discussion which lead many to reconsider the established Sex=Gender assumption. In these tribes she found the following:
Arapesh-both men and women displayed what we typically call feminine traits: sensitivity, cooperation, and low levels of aggression.
Mundugamor-both men and women were insensitive, uncooperative, and very aggressive. These were typical masculine traits at the time.
Tchambuli-women were aggressive, rational and capable and were also socially dominant. Men were passive assuming artistic and leisure roles.
Why then, Mead argued, if our reproductive roles determined our cultural and social opportunities were the gender definitions varied and unique among less civilized peoples? Were we not less civilized ourselves at one point in history and have we not progressed on a similar path the tribal people take? Could it be that tradition (culture) was the stronger social force rather than biology? Mead's work and her public influence helped to establish the belief that biology is only a part of the Sex and Gender question (albeit an important part). Mead established that Sex≠Gender. But, even with the harshest criticism launched against her works, her critics supported and even inadvertently reinforced the idea that biology shapes but cultures are more salient in how women and men are treated by those with power.
Misogyny is easier to perpetrate if one assumes the weakness, biological frailty, and perhaps even diminished capacity that women were claimed to have had. I personally witnessed the rise and fall of some who tried to persist in the traditional definition of women. Andrew Clay Silverstein (1957-present) was a nationally successful comedian who also played in a movie and TV show (although he recently appeared on Celebrity Apprentice). His career ended abruptly because of his harsh sexist themes which were being performed in an age of clarity and understanding about gender values. Mr. Clay failed to recognize the social change which surrounded him. We often overlook the change and the continuing problems ourselves. It is advantageous to you and I not to make the same mistake in our own career paths.
Professional and volunteer organizations have made concerted efforts to raise awareness of the English language and its demeaning language toward females. English as a derivative of German has many linguistic biases against women, non-whites, poor, and non-royalty. Raising awareness and discussing the assumptions within English or any other language has been part of the social transformation toward cultural and biological fairness and equality. If we understand how the words we use influence the culture we live in and how the value of that culture influence the way we treat one another, then we begin to see the importance of language on the quality of life.
The quality of life for women is of importance at many different levels in the world. As you've read through this chapter, you've probably noticed that much is yet to be accomplished worldwide. The United States has seen much progress. But, other nations continually rank the “world's best nation for women”. Many European countries far outrank the US for quality of Women's lives. In Fact, in 2008 the US ranked number 27th (retrieved 9 December, 2008 from www.nation.com.pk/pakistan-ne...omen-to-live/1 ).
The Global Gender Gap Index was developed to measure the quality of life for women between countries. It measures the gap between males and females in objective statistics that focus on equality. There are four pillars in the index which include economic participation and opportunity, educational attainment, political empowerment, and health and survival using 14 indicators from each countries national statistics. From 1998-2006, there was a reported net improvement for all countries (page 27).
When one considers the day-to-day lives of women in these national statistics, and perhaps more importantly in their personal lives, the concept of what women do as their contribution to the function of society becomes important. Instrumental Tasks are goal directed activities which link the family to the surrounding society, geared toward obtaining resources. This includes economic work, breadwinning, and other resource-based efforts. Expressive Tasks pertain to the creation and maintenance of a set of positive, supportive, emotional relationships within the family unit. This includes relationships, nurturing, and social connections needed in the family and society. Today, women do both and typically do them well.
Prior to the Industrial revolution both males and females combined their local economic efforts in homemaking. Most of these efforts were cottage industry-type where families used their children's' labor to make products they needed from soap, thread, fabric, butter, and many other products. When the factory model of production emerged in Western Civilizations, the breadwinner and homemaker became more distinct. The breadwinner is a parent or spouse who earns wages outside of the home and uses them to support the family. The homemaker is typically a women who occupies her life with mothering, housekeeping, and being a wife while depending heavily on the breadwinner.
What About Men?
In the past two decades a social movement referred to as The Men's Movement has emerged. The Men's Movement is a broad effort across societies and the world to improve the quality of life and family-related rights of men. Since the Industrial revolution, men have been emotionally exiled from their families and close relationships. They have become the human piece of the factory machinery (or computer technology in our day) that forced them to disconnect from their most intimate relationships and to become money-acquisition units rather than emotionally powerful pillars of their families.
Many in this line of thought attribute higher suicide rates, death rates, accident rates, substance abuse problems, and other challenges in the lives of modern men directly to the broad social process of post-industrial breadwinning. Not only did the Industrial revolution's changes hurt men, but the current masculine role is viewed by many as being oppressive to men, women, and children. Today a man is more likely to kill or be killed, to abuse, and to oppress others. Table
\(\PageIndex{5}\)
lists some of the issues of concern for those in the Men's Movement.
Table \(\PageIndex{5}\) : Concerns in the Men's Movement
Concerns of men
1. Life and health challenges
2. Emotional isolation
3. Sexual research and rights
4. Post-divorce/separation father's rights
5. False sex of physical abuse allegations
6. Early education challenges for boys
7. Declining college attendance
8. Protection from domestic abuse
9. Man-hating or bashing
10. Lack of support for fatherhood
11. Paternal rights and abortion
12. Affirmative action-sex and race
The list of concerns displays the quality of life issues mixed in with specific legal and civil rights concerns. Men's Movement sympathizers would most likely promote or support equality of rights for men and women. They are aware of the Male Supremacy Model, where males erroneously believe that men are superior in all aspects of life and that should excel in everything they do. They also concerns themselves with the Sexual Objectification of Women, where men learn to view women as objects of sexual consumption rather than as a whole person. Male Bashing is the verbal abuse and use of pejorative and derogatory language about men.
These and other concerns are not being aggressively supported throughout the world as are the women's rights and suffrage efforts discussed above. Most of the Men's Movement efforts are in Western Societies, India, and a handful of others.
Figure
\(\PageIndex{3}\)
below shows the transition in family gender roles over the course of the Industrial Revolution through to Post World War II. Families in Pre-Industrial Europe and the US were subsistence-based, meaning they spent much of their daily lives working to prepare food and other goods on a year-round basis. Men, women, children, and other family and friends succeeded because they all contributed to the collective good of the family economy.
The Industrial Revolution created the roles of breadwinners and homemakers. After the Industrial revolution was in full swing, women continued their subsistence work and remained homemakers while men continued in their breadwinning roles. After World War II, there was a social structural change where women began assuming the breadwinner role and became more and more common among the ranks of paid employees, especially beginning in 1960s-1980s. They had managed to remain homemakers, but men had not moved into the homemaking role to the same degree that women had moved into the breadwinning role. This creates a strong level of burden and expectation for US women who find themselves continuing to work outside the home for pay and inside the home for their informal domestic roles.
You will read later about intimacy and how it works between people. For now, suffice it to say that men often find a closer bond to their wife, children and other family members when they engage in domestic homemaking roles. Mundane family work is the activity that facilitates ongoing attachments and bonds among those who participate in it together.
Many couples today already share homemaking roles, just out of practical and functional need. They often find the co-homemaking/breadwinning role to be defined in a few typical styles. First, is the Tourist Husband style. The Tourist Husband is a visitor to the homemaking role who contributes the occasional assistance to his wife as a courtesy-much like a tourist might offer occasional assistance to their host. He often believes himself to be very generous since it is hers and not his role. Second, is the Assistant Homemaker where the husband looks to his wife for direction and for instruction on how to “help” her out in her homemaking role. Like one of the children, housework and homemaking task are the mother/wife's job and he helps if called upon.
Finally, there is the Co-homemaker husband who never “helps” his wife with homemaking task, but assumes that she and he equally share their breadwinning and homemaking responsibilities. The Co-homemaker husband is most likely to bond with his children, understand the daily joys and sorrows of all his individual family members, and feel a strong connection to his home and family (something Men's Movement advocates lament having lost).
Housework is one area of life that allows men to return to the intimacy and close familial influences they once enjoyed prior to the Industrial Revolution. Housework is mundane and repetitive. Yet, studies have shown that when men do housework with their children an emotional bonding process takes place and they create positive working memories together. I once heard it explained by a friend of mine who is a Clinical Psychologists. He suggested “turn off the TV, shut down the computer, unplug the games, take all phones of the hook. Then just try one hour of housework. Something magical will happen between parents and children as the boredom of housework begins to settle in-they begin to talk about things.” This is often true. Parents are much more interesting to children when all their friends and electronic distractions are removed. Children will open up while working with parents. And parents who avoid the urge to preach or make a speech, and who just talk to their children the way they might to their friends will find this very rewarding.
Listen carefully. I've said for 22 years that “men and children should never help their wife or mother with housework.” I truly mean this. If nearly 2/3rds of women work for pay, and if she has an average of 2 children, and if men truly respect and support their wife they will assume the responsibility as co-homemakers and not leave the burden solely upon her. They can't “help” her if it is their work too! It baffles me how husbands and wives even talk about work. She might ask, “I'm going out tonight can you babysit the children?” I say, “How can a man babysit his own children?” He doesn't. He just serves as a father to them while his wife is away.
With housework, it is his and his children's house too. Children and fathers who do housework together with their mothers find less stress for her and more closeness between family members. For women who come from traditional homes it is tempting to take on the role of housework police. She has in her mind what needs to be done and how it should appear once finished. To truly incorporate all members of the family in the housework, she often has to accept a clean house that may not exactly fit her ideals. Figure 7 shows a continuum of housework standards.
On the far left little to no housework gets done. The home is not clean nor is it attractive. Confusion is common in unclean households. No one has control over cleaning. On the far right the woman (wife, Mother, or Partner) does all the work and can keep the home precisely as she wants it to be. She may have to use coercion to get other family members to comply. One of my fellow professors is quoted as having said, “Do you want it clean or do you want us happy?”
Even if women in the sole home cleaning role find themselves capable of resisting the urge to manipulate other family members to join her in her ideal cleaning efforts, she keeps all the control and essentially becomes the ruler of house cleaning. Her children may grow up expecting to be cleaned up after by a woman. They may also feel emotionally disconnected from her. She can present a clean and attractive home but does not have the benefits of the bond that comes with all family members doing their part.
Today, the average US woman works for money outside the home. She has children or grandchildren and a male spouse or partner. If she chooses to share control and to accept a moderate level of hygiene and appearance, she can facilitate a group effort that includes all family members in the house cleaning work. This teaches children to learn how to work and work well with others. This also facilitates time spent together away from the distractions of technology where all members take responsibility for the home's care and maintenance. There is great potential for family unity and close bonds. The matriarch of the home often leads the family on this matter.


11.7D: Isotype Class Switching
- Describe the process of class switch recombination that results in changes in the antibody-heavy chain
Isotype Class Switching
Antibodies can come in different varieties, known as isotypes or classes. In placental mammals there are five antibody isotypes: IgA, IgD, IgE, IgG and IgM. They are each named with an “Ig” prefix that stands for immunoglobulin (another name for antibody) and differ in their biological properties, functional locations, and ability to deal with different antigens.
The antibody isotype of a B cell changes during cell development and activation. Immature B cells, which have never been exposed to an antigen, are known as naïve B cells and express only the IgM isotype in a cell surface bound form. B cells begin to express both IgM and IgD when they reach maturity; the co-expression of both of these immunoglobulin isotypes renders the B cell ‘mature’ and ready to respond to an antigen. B cell activation follows engagement of the cell-bound antibody molecule with an antigen, causing the cell to divide and differentiate into an antibody-producing cell, called a plasma cell. In this activated form, the B cell starts to produce antibody in a secreted form rather than a membrane-bound form. If these activated B cells encounter specific signaling molecules via their CD40 and cytokine receptors (both modulated by T helper cells), they undergo antibody class switching to produce IgG, IgA or IgE antibodies (from IgM or IgD) that have defined roles in the immune system.
Immunoglobulin class switching (or isotype switching, or isotypic commutation, or class switch recombination (CSR)) is a biological mechanism that changes a B cell’s production of antibody from one class to another; for example, from an isotype called IgM to an isotype called IgG. During this process, the constant region portion of the antibody-heavy chain is changed, but the variable region of the heavy chain stays the same (the terms “constant” and “variable” refer to changes or lack thereof between antibodies that target different epitopes). Since the variable region does not change, class switching does not affect antigen specificity. Instead, the antibody retains affinity for the same antigens, but can interact with different effector molecules. This allows different daughter cells from the same activated B cell to produce antibodies of different isotypes or subtypes (e.g. IgG1, IgG2 etc.).
Class switching occurs by a mechanism called class switch recombination (CSR) binding. Class switch recombination is a biological mechanism that allows the class of antibody produced by an activated B cell to change during a process known as isotype or class switching. During CSR, portions of the antibody-heavy chain locus are removed from the chromosome, and the gene segments surrounding the deleted portion are rejoined to retain a functional antibody gene that produces antibody of a different isotype. Double-stranded breaks are generated in DNA at conserved nucleotide motifs, called switch (S) regions, which are upstream from gene segments that encode the constant regions of antibody-heavy chains; these occur adjacent to all heavy chain constant region genes with the exception of the δ-chain. DNA is nicked and broken at two selected S-regions by the activity of a series of enzymes, including Activation-Induced (Cytidine) Deaminase (AID), uracil DNA glycosylase and apyrimidic/apurinic (AP)-endonucleases. The intervening DNA between the S-regions is subsequently deleted from the chromosome, removing unwanted μ or δ heavy chain constant region exons and allowing substitution of a γ, α or ε constant region gene segment. The free ends of the DNA are rejoined by a process called non-homologous end joining (NHEJ) to link the variable domain exon to the desired downstream constant domain exon of the antibody-heavy chain. In the absence of non-homologous end joining, free ends of DNA may be rejoined by an alternative pathway biased toward microhomology joins. With the exception of the μ and δ genes, only one antibody class is expressed by a B cell at any point in time.
Key Points
- The antibody isotype of a B cell changes during cell development and activation. Immature B cells have never been exposed to an antigen and are known as naïve B cells. B cells begin to express both IgM and IgD when they reach maturity and renders the B cell ‘mature’ and ready to respond to antigen.
- If activated B cells encounter specific signaling molecules via their CD40 and cytokine receptors (both modulated by T helper cells), they undergo antibody class switching to produce IgG, IgA or IgE antibodies that have defined roles in the immune system.
- During class switch recombination the constant region portion of the antibody-heavy chain is changed, but the variable region of the heavy chain stays the same; thus, class switching does not affect antigen specificity.
- The antibody retains affinity for the same antigens, but can interact with different effector molecules. This allows different daughter cells from the same activated B cell to produce antibodies of different isotypes or subtypes (e.g. IgG1, IgG2 etc. ).
Key Terms
- isotype : Antibodies can come in different varieties known as isotypes, which refer to the genetic variations or differences in the constant regions of the heavy and light chains of the antibody.
- class switch recombination : A biological mechanism that changes a B cell’s production of antibody from one class to another; for example, from an isotype called IgM to an isotype called IgG.