Calc9 - MATH 104 study guide PDF

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MATH 104 study guide...

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Real Analysis (or, traditionally, the Theory of Functions of a Real Variable) a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real numbers, and continuity, smoothness and related properties of real-valued functions Mathematical Proof a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, this can be traced back to self-evident or assumed statements, known as axioms. These are examples of deductive reasoning and are distinguished from inductive or empirical arguments; this must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. Measure Theory (or, Measure) a collection of propositions to illustrate the principles of a subject invented by Lebesgue to define integrals of all but the most pathological (abnormal) functions. In mathematical analysis, a (this) on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, this is a generalization of the concepts of length, area, and volume. Distributions introduced by Schwartz to take the derivative of any function whatsoever. these are objects that generalize the classical notion of functions in mathematical analysis. They make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a ________ (adj. form) derivative. These are widely used in the theory of partial differential equations, where it may be easier to establish the existence of ______ (adj. form) solutions than classical solutions, or appropriate classical solutions may not exist

Non-Standard Analysis reformulates the calculus using a logically rigorous notion of infinitesimal numbers. originated in 1960s by Robinson. it uses technical machinery from mathematical logic to augment the real number system with infinitesimal and infinite numbers, as in the original Newton-Leibniz conception. The resulting numbers are called hyperreal numbers, and they can be used to give a Leibniz-like development of the usual rules of calculus Mathematical Logic a subfield of mathematics exploring the applications of formal reasoning conducted or assessed according to strict principles of validity to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. The unifying themes in this include the study of the expressive power of formal systems and the deductive power of formal proof systems Infinity (or, Infinite) (symbol: ∞) is an abstract concept describing something without any bound and is relevant in a number of fields, predominantly mathematics and physics. In mathematics, it is often treated as if it were a number (i.e., it counts or measures things: "an _____ number of terms") but it is not the same sort of number as natural or real numbers...