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See Article History Analysis, 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, differentiationand 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: 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, continuityinfinite 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 and continuoushistorically 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. 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.The essay “The Nightmare Life Without Fuel” by Isaac Asimov shows what will life be like when fuel has almost run out.
The scene is the United States of America in the future, at a time when fuel has run out. Essay on Analysis Of ' The Day Of The Locust ' - When Looking in goes Wrong The Longman Dictionary of English Language and Culture defined the American Dream as “the idea that the US is a place where everyone has the chance of becoming rich and successful.”But those principles have changed.
Somewhere among this and Nightmare Fuel is Isaac finding Guppy's head (a real cat's head at that) in one of Rebirth's trailers, where he holds it in his hands with despair.
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