Real Analysis
JohnEB
Leibniz and Newton are usually both credited with the invention of calculus.
Newton was the first to apply calculus to general physics and Leibniz developed
much of the notation used in calculus today. The basic insights that both Newton
and Leibniz provided were the laws of differentiation and integration, second
and higher derivatives, and the notion of an approximating polynomial series.
By Newton's time, the fundamental theorem of calculus was known.
This foundation is now called "real analysis".
http://en.wikipedia.org/wiki/Real_analysis
While some of the ideas of calculus had been 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 its basic principles. The development
of calculus was built on earlier concepts of instantaneous motion and area underneath curves.
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 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 of Elea gave several
famous examples of such paradoxes. Calculus provides tools, especially the limit and the
infinite series, which resolve the paradoxes.
Calculus
http://en.wikipedia.org/wiki/Calculus#Foundations
Interactive Real Analysis
http://www.mathcs.org/analysis/reals/index.html