analysis
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What is the main focus of analysis in mathematics?
Analysis deals with continuous change and processes likelimits,differentiation, andintegration. It originated from the study of continuous change and has applications in sciences, finance, economics, and sociology.
Who were the key figures in the discovery of calculus?
Isaac Newton andGottfried Wilhelm Leibniz independently discovered differential and integralcalculus in the late 17th century. Their work laid the foundation for modern analysis.
What is the fundamental theorem of calculus?
Thefundamental theorem of calculus linksintegration anddifferentiation, showing that the derivative of an integral of a function is the original function. It was discovered by Newton and Leibniz.
How did the concept of limits evolve in analysis?
The concept oflimits became formalized in the 19th century byKarl Weierstrass, who provided a precise definition that resolved previous logical issues with infinitesimals and continuity.
What is the significance of complex numbers in analysis?
Complex numbers extendreal numbers and allow for the development of complex analysis, which has applications in aerodynamics, fluid mechanics, and electric power generation. Complex numbers are used to solve equations that real numbers cannot.
analysis, a branch ofmathematics that deals withcontinuous change and with certain general types of processes that have emerged from the study of continuous change, such as limits,differentiation, andintegration. Since the discovery of thedifferential andintegral calculus byIsaac Newton andGottfried Wilhelm Leibniz at the end of the 17th century, analysis has grown into an enormous and central field of mathematical research, with applications throughout the sciences and in areas such as finance, economics, and sociology.
The historical origins of analysis can be found in attempts to calculate spatial quantities such as the length of a curvedline or the area enclosed by a curve. These problems can be stated purely as questions of mathematical technique, but they have a far wider importance because they possess a broad variety of interpretations in the physical world. The area inside a curve, for instance, is of direct interest in land measurement: how many acres does an irregularly shaped plot of land contain? But the same technique also determines the mass of a uniform sheet of material bounded by some chosen curve or the quantity of paint needed to cover an irregularly shapedsurface. Less obviously, these techniques can be used to find the total distance traveled by a vehicle moving at varying speeds, the depth at which a ship will float when placed in the sea, or the total fuelconsumption of a rocket.
Similarly, the mathematical technique for finding a tangent line to a curve at a given point can also be used to calculate the steepness of a curved hill or the angle through which a moving boat must turn to avoid a collision. Less directly, it is related to the extremely important question of the calculation of instantaneous velocity or other instantaneous rates of change, such as the cooling of a warm object in a cold room or thepropagation of a disease organism through a human population.
This article begins with a brief introduction to the historical background of analysis and to basic concepts such as number systems, functions,continuity,infinite series, and limits, all of which are necessary for an understanding of analysis. Following this introduction is a full technical review, from calculus to nonstandard analysis, and then the article concludes with a complete history.
Historical background
Bridging the gap between arithmetic and geometry
Mathematics divides phenomena into two broad classes,discrete andcontinuous, historically corresponding to the division betweenarithmetic andgeometry. Discrete systems can be subdivided only so far, and they can be described in terms of whole numbers 0, 1, 2, 3, …. Continuous systems can be subdivided indefinitely, and their description requires the real numbers, numbers represented by decimal expansions such as 3.14159…, possibly going on forever. Understanding the true nature of suchinfinite decimals lies at the heart of analysis.
The distinction between discrete mathematics and continuous mathematics is a central issue for mathematical modeling, the art of representing features of the natural world in mathematical form. The universe does not contain or consist of actual mathematical objects, but many aspects of the universe closely resemble mathematical concepts. For example, the number two does not exist as a physical object, but it does describe an important feature of such things as human twins and binary stars. In a similar manner, the real numbers provide satisfactory models for a variety of phenomena, even though no physical quantity can be measured accurately to more than a dozen or so decimal places. It is not the values of infinitely many decimal places that apply to the real world but the deductive structures that they embody and enable.
Analysis came into being because many aspects of the natural world can profitably be considered as being continuous—at least, to an excellent degree of approximation. Again, this is a question of modeling, not of reality. Matter is not truly continuous; if matter is subdivided into sufficiently small pieces, then indivisible components, or atoms, will appear. But atoms are extremely small, and, for most applications, treating matter as though it were acontinuum introduces negligibleerror while greatly simplifying the computations. For example, continuum modeling is standard engineering practice when studying the flow of fluids such as air or water, the bending of elastic materials, the distribution or flow ofelectric current, and the flow of heat.
Discovery of the calculus and the search for foundations
Two major steps led to the creation of analysis. The first was the discovery of the surprising relationship, known as thefundamental theorem of calculus, between spatial problems involving the calculation of some total size or value, such as length, area, or volume (integration), and problems involving rates of change, such as slopes oftangents and velocities (differentiation). Credit for the independent discovery, about 1670, of the fundamental theorem of calculus together with the invention of techniques to apply thistheorem goes jointly toGottfried Wilhelm Leibniz andIsaac Newton.
While the utility ofcalculus in explaining physical phenomena was immediately apparent, its use ofinfinity in calculations (through the decomposition of curves, geometric bodies, and physical motions into infinitely many small parts) generated widespread unease. In particular, the Anglican bishopGeorge Berkeley published a famous pamphlet,The Analyst; or, A Discourse Addressed to an Infidel Mathematician (1734), pointing out that calculus—at least, as presented by Newton and Leibniz—possessed serious logical flaws. Analysis grew out of the resulting painstakingly close examination of previously loosely definedconcepts such asfunction and limit.
Newton’s and Leibniz’s approach to calculus had been primarily geometric, involving ratios with “almost zero” divisors—Newton’s “fluxions” and Leibniz’s “infinitesimals.” During the 18th century calculus became increasingly algebraic, as mathematicians—most notably the SwissLeonhard Euler and the Italian FrenchJoseph-Louis Lagrange—began to generalize the concepts of continuity and limits from geometric curves and bodies to more abstract algebraic functions and began to extend these ideas to complex numbers. Although these developments were not entirely satisfactory from a foundational standpoint, they were fundamental to the eventual refinement of a rigorous basis for calculus by the FrenchmanAugustin-Louis Cauchy, the BohemianBernhard Bolzano, and above all the GermanKarl Weierstrass in the 19th century.