Seven Castles

Calculus

                                      

November 8, 2007

 

Dealing with limits and the differentiation and integration of functions of one or more variables, calculus ("pebble" or "little stone" in Latin) is a major area in mathematics "where infinitesimal data yields global information". It has widespread applications in science and engineering and is used to solve complex and expansive problems that cannot be solved by algebra alone. It builds on analytic geometry and mathematical analysis, and includes two major branches of differential and integral calculus, related by the fundamental theorem of calculus.

 

Applications of differential calculus include computations involving velocity and acceleration, the slope of a curve and optimization. Applications of integral calculus include computations involving area, volume, arc length, center of mass, work and pressure. More advanced applications include power series and Fourier series. Calculus can be used to compute the trajectory of a shuttle docking at a space station or the amount of snow in a driveway.

 

Calculus is also used to gain a more precise understanding of the nature of space, time and motion. For centuries, mathematicians and philosophers wrestled with paradoxes involving division by zero or sums of infinitely many numbers. These questions arise in the study of motion and area. The ancient Greek philosopher Zeno gave several famous examples of such paradoxes. Calculus provides tools, especially the limit and the infinite series, which resolve the paradoxes.

 

While some of the ideas of calculus were developed earlier, in Egypt, Greece, China, India, Iraq, Persia and Japan, the modern use of calculus began in Europe, during the 17th century, when Isaac Newton and Gottfried Wilhelm Leibniz built on the work of earlier mathematicians to introduce the basic principles of calculus. This work had a strong impact on the development of physics, which makes particular use of calculus; all concepts in classical mechanics are interrelated through calculus.

 

Calculus can be used in conjunction with other mathematical disciplines. For example, it can be used with linear algebra to find the "best fit" linear approximation for a set of points in a domain. In the realm of medicine, calculus can be used to find the optimal branching angle of a blood vessel so as to maximize flow. In analytic geometry, the study of graphs of functions, calculus is used to find high points and low points (maximums and minimums), slope, concavity and inflection points. In economics, calculus allows for the determination of maximal profit by providing a way to easily calculate both marginal cost and marginal revenue.

 

Calculus is a ubiquitous topic in most modern high schools and universities, and mathematicians around the world continue to contribute to its development.

【作者: zhangliping】【访问统计:】【2007年11月8日 星期四 05:11】【注册】【打印】