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
: 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
, 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
, 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?