Solving equations

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The problems in this volume are of various types. Some involve proofs or derivations, others involve calculating magnitudes, still others are designed merely to provoke thought. Several are short and may even appear trivial, whereas others are long and somewhat tedious. Some require that values of a specified quantity be found or that the relation between certain variables and/or parameters be deduced; others are hypotheses or statements to be proven. We hope that all emphasize important concepts.

Students (and others) try to solve problems in different ways. Some adopt a “road” that they hope will lead directly to the solution, others “mill around” with algebraic manipulations in the hope that they will stumble onto the solution, while still others start with what is to be proven and work backwards, manipulating quantities until they arrive at the proper starting point. In the process of solving a problem, some students mix up units or give answers without specifying the units, while many give more significant figures than the data warrant. This chapter attempts to help the student avoid such pitfalls.

The first step in a solution is to try to understand the problem; we should have a clear picture of what we wish to achieve, that is, what is the unknown? Secondly, we need to decide what quantities or factors must be considered in arriving at the correct answer. Thirdly, we must consider the available information from which the answer is to be deduced. Some of the available data may not be relevant. Finally, we must determine how the relations between the various factors can be used to get from the starting point to the final goal, that is, the path that must be followed.

We should plan how to obtain the solution. Often a diagram will help to understand interrelationships. Usually diagrams should be approximately to scale so that the relative magnitudes of different elements are preserved; nonessential and irrelevant elements should be omitted. Sometimes exaggerated or detailed diagrams are desirable. If a variable has a wide range of values, more than one diagram may be necessary.

Often we must introduce new symbols or notations; these should be defined or stated explicitly, not only to help avoid errors but also for the benefit of others trying to follow the derivations. Care must be taken to avoid using the same symbol to represent different quantities, for example, using to denote both location and offset. Wherever possible, a notation should suggest the nature of the quantity it represents, for example, using for amplitude, for velocity, for frequency, for time. Symbols should be kept as simple as possible. A single subscript should be related to the subgroup that it represents, for example, for apparent velocity. Double subscripts should generally be avoided if a feasible alternative exists; however, double subscripts are at times necessary or at least desirable, for example, in a double summation over both and or denoting the up-dip apparent velocity at the second interface. Symbols, especially subscripts, superscripts, and Greek letters, must be written clearly to avoid mistaking them for other symbols.

The chances of making an error in copying mathematical equations increases with the number and complexity of terms to be copied, so it may be advisable to use a symbol to represent a single complicated term or group of terms; for example, in problem 2.5, in problem 2.6, in problem 7.5b. However, this practice tends to hide individual symbols and terms so that their significance is obscured, hence should be used with caution.

Problems to be solved are of different kinds and require different approaches. “To solve” problems generally start with known or assumed relationships (hypotheses) and ask that we establish some other relationships (the conclusions). “Inductive” problems seek to discover new relationships based on a set of observations; solutions of this type are often probabilistic rather than rigorous, that is, the conclusion may be merely strongly suggested rather than proven exactly.

Some problems cannot be solved exactly by algebraic means but are amenable to numerical or graphical solution. The “accuracy” of a numerical solution usually can be as high as we wish, depending upon how much time we wish to spend; however, nothing is gained by attempting to achieve greater accuracy than that of the given data. The accuracy of a graphical solution depends on the scale of the graph and the care with which it is drawn. Graphical solutions often illustrate relationships and principles better than algebraic solutions so that we sometimes use them even where we may ultimately resort to an algebraic or numerical solution to achieve the desired accuracy.

In numerical and graphical solutions attention must be paid to the number of significant figures. The basic principle is that significant figures represent values to the accuracy of the measurement. The following examples illustrate the concept, the number of significant figures in the answers being four in all cases (note that is a significant figure except when it is used solely to indicate the magnitude).

Thus, in the first example, the digits to the right of the decimal point must be discarded (the second and third values being rounded to 21 and 30) because the first quantity is known only to the nearest unit. The following examples apply the same principle to subtraction, multiplication, and division. When rounding off a value which has the digit 5 at the right hand end, we can round upwards or downwards; however, people often use a rule such as rounding up or down so that the quantity ends in an even digit, for example, rounding 1.265 and 4.735 to become 1.26 and 4.74. To avoid accumulating error, we often carry an extra figure while working out a numerical solution and only round off to significant figures when we arrive at the answer.

Once we have solved a problem, we are not finished; we need to understand it in detail. What is of significance? What does it tell us that is of interest? What is the function of each component part? How will the relationships change as the parameters vary? Which input parameters have the greatest effects on the results and which are almost irrelevant? Under what circumstances will our conclusions not hold? What happens when parameters assume extreme values; does the solution “blow up”? Did we omit any important aspects? By examining our results critically we may convert a mere exercise into a worthwhile learning experience and develop an in-depth understanding of the subject. Sometimes an analysis will show that we did not actually solve the problem at all.

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