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Mechanics

  • John Stillwell
Chapter
Part of the Undergraduate Texts in Mathematics book series (UTM)

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In Chapter 9 we introduced the concepts of derivative and integral geometrically, as tangents and areas respectively. Geometry was certainly an important source of calculus problems and concepts, but not the only one. From the beginning, mechanics was just as important. Mechanics is conceptually important because the derivative and the integral are inherent in the concept of motion: velocity is the derivative of displacement (with respect to time), and displacement is the integral of velocity. Also, mechanics was initially the only source of nonalgebraic curves; for example, the cycloid, which is generated by rolling a circle along a line. The “mechanical” curves spurred the development of calculus for the simple reason that they were not accessible to pure algebra. An even greater spur was the development of continuum mechanics, which studies the behavior of such things as flexible and elastic strings, fluid motion, and heat flow. Continuum mechanics involves functions of several variables, and their various derivatives, hence partial differential equations. Some of the most important partial differential equations, such as the wave equation and the heat equation, are clearly inseparable from their origins in continuum mechanics. Yet these very equations confronted mathematicians with basic questions in pure mathematics: for example, what is a function?

Keywords

Heat Equation Trigonometric Series Suspension Bridge Time Graph Biographical Note 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Diacu, F. and P. Holmes (1996). Celestial Encounters. Princeton, NJ: Princeton University Press.MATHGoogle Scholar
  2. Dostrovsky, S. (1975). Early vibration theory: physics and music in the seventeenth century. Arch. History Exact Sci. 14(3), 169–218.CrossRefMathSciNetGoogle Scholar
  3. Dugas, R. (1957). A History of Mechanics. Editions du Griffon, Neuchâtel, Switzerland. Foreword by Louis de Broglie. Translated into English by J. R. Maddox.MATHGoogle Scholar
  4. Engelsman, S. B. (1984). Families of Curves and the Origins of Partial Differentiation. Amsterdam: North-Holland Publishing Co.Google Scholar
  5. Sitnikov, K. (1960). The existence of oscillatory motion in the three-body problem. Soviet Physics Dokl. 5, 647–650.MATHMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of San FranciscoSan FranciscoUSA

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