a branch of mathematics that deals with continuous change and with certain general types of processes that have emerged from the study of continuous change, such as limits, differentiation, and integration. Since the discovery of the differential and integral calculus by Isaac Newton and Gottfried 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 curved line 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 shaped surface. 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 fuel consumption 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 the propagation 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.
Mathematics divides phenomena into two broad classes, discrete and continuous, historically corresponding to the division between arithmetic and geometry. 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 such infinite 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 a continuum introduces negligible error 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 of electric current, and the flow of heat.
The-transformation-of-a-circular-region-into-an-approximately-rectangularThe transformation of a circular region into an approximately rectangular region[Credits : Encyclopædia Britannica, Inc.]
Graphical-illustration-of-an-infinite-geometric-series-Clearly-the-sumGraphical illustration of an infinite geometric series[Credits : Encyclopædia Britannica, Inc.]
Graph-of-distance-traveled-versus-time-elapsed-for-the-motionGraph of distance traveled versus time elapsed for the motion of an automobile[Credits : Encyclopædia Britannica, Inc.]
Graph-of-a-function-Part-A-illustrates-the-general-ideaGraph of a function[Credits : Encyclopædia Britannica, Inc.]
An-illustration-of-the-difference-between-average-and-instantaneous-ratesAn illustration of the difference between average and instantaneous rates of change[Credits : Encyclopædia Britannica, Inc.]
A-curve-sketched-with-the-help-of-calculus-This-graphA curve sketched with the help of calculus[Credits : Encyclopædia Britannica, Inc.]
Integral-region-graph-The-shaded-region-bounded-by-the-verticalIntegral region graph[Credits : Encyclopædia Britannica, Inc.]
Graphical-illustration-of-the-fundamental-theorem-of-calculus-ddt-duGraphical illustration of the fundamental theorem of calculus: …[Credits : Encyclopædia Britannica, Inc.]
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