Mathematical Competency Areas and the Sciences  

Since all scientific work is based upon experimentation, mathematics is often applied to the analysis of experimental data in the sciences.  In the INSS, as a result, the mathematical tools associated with the competencies are used primarily to uncover the behaviors of physical phenomena.  The INSS seeks to reinforce the application of the competencies in the sciences rather than to develop fundamental mathematical formalism.

 GRAPHING

 The goal of experimental science is to gather reliable information that can be used to either test models or formulate new models.  Of course, one is faced very quickly with the question of how to handle the information that has been obtained.  One of the most commonly used methods of "visualizing" experimental information is through the use of a graph.  The purpose of the graph is to display the information in a manner that is readily interpretable by a scientist.

 Typically, the type of graph that is most relevant to the INSS displays the results of two related quantities, which are called coordinates on the graph.  In general, when you carry out an experiment, you will adjust the apparatus to work in a specific way, and then measure the result of that adjustment.  In constructing a graph, the horizontal axis (x-axis) will be associated with the experimentally adjusted quantity (called the independent variable in mathematics).  The vertical axis (y-axis) will be associated with the corresponding experimental measurement (called the dependent variable in mathematics).  The quantities are displayed on a grid, upon which both coordinates are plotted.  For example, one may be interested in the position of a car traveling at 55 mi/hr at various times.  The data used for the plot is shown in Table 1.  In this case, the position of the car is represented on the vertical axis, while the time at which the position is measured is represented on the horizontal axis (Figure 1).    

Time (seconds)	Position (feet)
1.0	80
1.5	121
2.0	160
2.5	201
Table 1.  Position measurements at various times for a car moving at 55 mi/hr.  This information is plotted in Figure 1.			Figure 1.  Position vs time for an automobile traveling at 55 mi/hr.

THE PROPORTIONALITY

The mathematical laws which govern straight line graphs (as in Figure 1) are called proportionalities (or direct proportionalities).  Mathematical proportionalities are written in the form

In words, A is proportional to B means that if B is doubled, A is also doubled. If B is cut in half, A is also cut in half, etc. Many (but not all) laws of Physics can be written in this form of equation (1). Although equation (1) is fine for determining behaviors, but as stated this form is incomplete. The proportionality symbol can be replaced with an equal sign by introducing a constant,

The variables A and B and the constant are determined by the process under study.

THE GRAPHICAL INTERPRETATION

There is a convenient graphical interpretation of equation (2). Equation (2) defines a particular version of the equation of a line, often symbolized as

where the slope is defined as

The symbol ) means change in. We see, therefore, that if a proportionality is plotted, it will produce a straight line with a slope equal to the constant in equation (2).

For example, if the slope of the line in figure 1 can be easily calculated, and it is determined to be 80.4 ft/s = 55 mph.  As a result, the law governing the motion of the car is

as expected.

Notice that the proportionality does not imply that the constant of proportionality cannot be changed. Figure 2 represents the behavior of a moving object at two different (but constant) speeds. Both represent the same proportionality with different constants. As the physical situation changes, the constant of proportionality might also change. This is indicated by the different slopes in Figure 2.

	Figure 2.  Graphical depiction of the motions of two cars, one moving at 80.4 ft/s and the other at 40.2 ft/s.

INVERSE PROPORTIONALITIES

A relationship which is similar to the proportionality is the inverse proportionality. In this case,

or,

If B is doubled, A is cut in half. If B is tripled, A is reduced by one third. If, on the other hand, B is cut in half, then A is doubled,

since a denominator in a denominator can be moved to the numerator. As a result, A is doubled.

MORE COMPLICATED RELATIONSHIPS

Graphs of the type shown in Figure 1 have been exceptionally useful in carrying out the goal of scientists since the time of Galileo- the description of physical processes on the basis of mathematical equations.  Since it is believed that the data points are a partial representation of the underlying "law", it is often convenient to fit a curve to the points provided, as described above.  The information exhibits a behavior that is called linear (taken from the fact that the best fit curve to this information is a line).

It would be fortunate if all physical laws could be stated in terms of simple proportionalities.  Unfortunately, this is not the case.  A few examples of physically important behaviors that are not associated with simple proportionalities are shown in Figures 3 - 9.  One might guess, upon examination of these figures, that varying degrees of difficulty can be met in an attempt to extract a law from experimental information.  Thus, obtaining a law from Figures 3and 4 will probably be easier than trying to do the same with Figures 5 - 7.  One can expect that Figures 8 and 9 would be the most difficult of all.

Figure 3.  Nonlinear behavior of an object undergoing constant acceleration.		Figure 4.  Nonlinear behavior of the energy of a moving mass.  The mass of the object shown is 1 kg.
Figure 5.  Nonlinear behavior of the growth of the balance of a saving account drawing 10% interest annually.		Figure 6.  Nonlinear behavior of the division of cancer cells.
Figure 7.  Carbon Dioxide levels in the atmosphere over the last three centuries.		Figure 8.  The electrical activity of the human brain (EEG).
Figure 9.  Figure 9.  Speed of hurricane Gordon along the earth’s longitude.

NONLINEAR BEHAVIORS

In many cases, the laws of physics cannot be determined by simple proportionalities as discussed above.  Fortunately, there are fairly common nonlinear relationships that appear in many physical applications.  Many phenomena can be modeled by power law relationships, which are of the form

Where n is some constant.

Another very common behavior is associated with the exponential function, with relationship of the form

where a is a constant and e is called the exponential function.

UNCOVERING NONLINEAR RELATIONSHIPS

In order to uncover a non-linear law that governs a behavior, one can utilize the powers of the proportionality once again.  The determination of the law for the non-linear behavior is done by manipulating the data and then re-graphing the manipulated data.  The ultimate goal is to perform the manipulations in such a way that when the manipulated data is plotted, the resulting graph will be a straight line.

For example, consider Equation (6).  A graph of y vs x will not produce a straight line;  however, a graph of y vs xⁿ will produce a straight line, since, according to Equation (6), y is proportional to xⁿ.  The trick is that, in general, you do not know what the value of n is in a particular application.  Similarly Equation (7) indicates that a graph of y vs e^ax will also produce a straight line, since by definition, y is proportional to e^ax.  In this case, you will generally not know the value of the constant a.  Further, when presented with raw data, you do not really know what the general form of the law is in order to perform appropriate manipulations.  Fortunately, the area of mathematics provides some direction.

EDUCATED GUESSING

One could take raw data and make a good guess as to the form of the relationship involved.  The guess is used to manipulate the data set and graph this new data.  If the resulting graph is a straight line, one can compute the slope and construct the appropriate law. 

	Figure 10.  Manipulated data from Figure 4 for the energy of a moving object.

For example, consider Figure 4.  If one were to take the raw data and square all of the speeds plotted on the x-axis and then plot the energy vs speed squared, the resulting graph would be of the form shown in Figure 10.  The resulting graph is linear, and the slope of the line is .5 Joules/m²/s² .  A student who has completed SCI190 should recognize the rather odd unit of Joules/m²/s² as, in fact, a kg.  As a result, the law associated with the data is of the form

The student should also recognize that the energy depicted in the graph is kinetic energy, which, from lecture, is given by

For and object of mass .5 kg, equations (10) and (11) are identical.

If the behavior is expected to be exponential, then it might appear appropriate to plot y vs e^as.  Again, a straight line as a result of this manipulation would indicate that the law was an exponential.  Unfortunately, to do this properly would require knowledge of the value of a, which is not generally known to you.  The solution to this dilemma is to take the natural logarithm of the values on the y-axis rather than the exponential of the values on the x-axis.  Using the properties of the natural logarithm,

	Figure 11.  Manipulated data from Figure 6 on the division of cancer cells.

For convenience, the constant of proportionality has been chosen to be 1 in Equation (13).  As a result, a graph of the natural logarithm of the y-axis values vs x will produce a straight line of slope a.  Figure 11 shows the result of this manipulation on the cancel cell data.  The result is a straight line, with a slope of .693.

 MORE GENERAL MANIPULATIONS

It would appear as though there is a large number of guess that would have to be made in order to uncover physical law.  There are two manipulations that can be made on any data set that will easily allow a person to distinguish power law behavior from exponential.  Once again, the manipulations rely upon the properties of the natural logarithm.  The manipulation is summarized in Table 2.

MORE COMPLICATED BEHAVIORS

An examination of Figures 8 and 9 make a strong suggestion that these behaviors are not linear, power law, or exponential in their characteristics.  Very powerful techniques are required to uncover underlying laws.  Although these techniques are beyond the INSS, students will get the opportunity to examine the behaviors of a hurricane (Figure 9) in the SCI190 laboratory.