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Mathematical Methods

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Mathematical Methods Mathematical Methods Li-Chung Wang December 10, 2011 Abstract Mathematical training is to teach students the methods of removing obstacles, especially the standard ones that will be used again and again. The more methods one has learned, the higher one’s skill and the more chances one may do some creative works in methematics. In this paper, we try to classify all the mathematical methods. Mathematical methods are too many to count. They look too diversified to manage. If we attempt to classify them by luck, it would be difficult as though we were lost in a maze and tried to get out. However, if we caredully study a method’s origins, development and functions, we may find some clues about our task by following along their texture. Studying methods helps us analyze patterns. In order to make a method’s essence outstanding, the setting must be simple. Using a method is like producing a product: A method can be divided into three stages: input, process, and output. Suppose we compare Method A with Method B. If the input of Method B is more than or equal to that of Method A and if both the process and the output of Method B are better than those of Method A, then we say that Method B is more productive than Method A. Here are some examples. Cases when the input is increased: The method given in the general case can be applied to specific cases. However, specific cases contain more resources, there may be more effective methods available for specific cases. Our goal is to seek the most effective method in each case. Strategies for improving the output: When we fomulate a theorem, the conclusion should be as strong as possible. When we seek a solution, the solution should be specific and precise. The execution of the solving plan should be finished perfectly and the solution should be expressed in closed form if possible. Strategies for improving the process: When we introduce a definition, the definition should be accessible in a finite number of steps; the target should be reached quickly, directly and effectively. When we present a proof or theory, we should avoid repetition, orgnize it in a consistent and systematic way, make it leaner and simpler. The qualities of process can be roughly divided into following categories: Accessibility, functions (reduction [reduction from a large number of cases to a smaller one], analogy, effectiveness, simplicity, directness, flexibility, fully utilizing resources, hitting multiple targets with one shot, avoiding contradictions, avoiding repetition and unnecessary com- plications, expanding the scope of application without loss of efficiency), structurization (insightfulness, essence, flowcharts or networks). Structurization may help us systematically operate and deeply under- stand the complicated cases. Like two neighboring colors in a rainbow, there is no clear dividing line between two qualities. The method of weakening a hypothesis is a method of reducing the input for a given output. The concept of almost everywhere can often ruin the delicacy of the method of weakening a hypothesis. For physical methods, we discuss physical interpretations and physical proofs. As we go to a more advanced level and widen our consideration, new physical meanings of mathematical equations continue to develop and meanings of equations become richer and more delicate. Nonetheless the mean- ings in older theories are still well-preserved in a newer theory. Guided by a theorem’s physical meaning, one may develop a better strategy to prove it. The ideal physical proof is the one each of whose step has a pertinent physical interpretation. The development of physical methods shows the tendency toward such an ideal. For example, the mathematical formulation of the second law of thermodynamics leads to a criterion for integrability of Pfaffian forms. Sometimes, a mathematical proof requires advanced 1 and complicated knowledge and a long argument; we may easily get lost. In contrast, a physical proof may lay bare the key idea with one penetrating remark. A physical proof is usually more direct than a geometric proof. Physical and geometric proofs provide richer meanings, insights, and interesting stories than analytic proofs. A proper physics model can be a natural guide to the study of PDEs. Improvements of classical methods: a remedy rather than a thorough revamp is all we need; this introduction mode based on needs may make the key to improvement most outstanding. A hyperbolic paraboloid has center at (∞,∞,∞) because Cartesian coordinates lack ability to distinguish infinities of all directions; if we use spherical coordinates instead, there will be no common midpoint for the chords through the origin. If the set of centers is empty and we allow a point involving ∞ to be its element due to a tool abuse, then all the theorems to which the false existence of elements leads will be meaningless. Only after a construction is tailored to our needs can it solve the problem effectively. An index set must be chosen properly: one more candidate would be too many and one less would be too few. The Dirac delta generalized function should not be treated as a function; we provide an easy way to bridge the gap between a function and a generalized function. A good theorem should provide complete information. The Ritz method is an effective tool for studying Sturm–Liouville Problems. In order to effectively solve a problem, we must quickly understand the circumstance with the minimum effort, and then directly attack the heart of the matter. Find extrema with subsidiary conditions. Compare physics proofs with mathematics proofs. Dis- tribution theory is a new theory that we create to avoid the contradiction that the domain of a function contains a point whose function value cannot be defined. How do we deal with a problem that may easily cause us to commit errors? “Generalized orientations: I. When studying a generalized definition, we should understand its primi- tive version, its entire process of revolution, and the reason for the necessity of generalization; if we proceed directly toward the most general version in axiomatic approaches, its setting usually requires a more strange language and less familiar structures which may blur the essential idea, and the algorithm to check the definition usually becomes less effective; thus, an improper approach to generalized definitions may easily lead to an empty formality and make it difficult for us to see the advantages of generalized definitions over the primitive version; providing several non-trivial examples alone is not enough. II. A. Being positively oriented, having the same orientation and orientation preserving can be vewed as the same concept. B. Having pictures in mind for geometric theorems.” “Using formulas in a table without care may easily result in mistakes. One is under the impression that once the solution form is obtained, the actual solution is determined. This is not so. If the resulting func- tion is multivalued and the formula fails to indicate which value to choose, then the formula would be useless. One should find a delicate method to determine the correct value. If one uses such a unfinished formula in a proof, then the proof would be incorrect. Such a mistake is often difficult to detect.” “How do we detect errors in a textbook? When I find an error, the first response is usually to refuse to accept this fact and try to rationalize the opposite viewpoint. After all, there are many authors who have not found it incorrect after copying it. Nevertheness, I try to remember this odd experience so that I can easily find a reason when a problem occurs afterwards. However, this “rationalization” actually conceals a mistake. The reason why we fail to detect an error is that we have not gone far enough to forsee its consequences. Errors cannot withstand tests. Soon or later they will be detected. Even if an error may not be detected at the first checkpoint in application, it can hardly survive at the second one. If we consider an error true, then the world would fall into pieces as if Pandora’s box were opened. I became so frustrated that I had to choose the other option: the error is not necessarily true. Then I found a counterexample. I could omit some details and still make this paragraph logical, but this would destroy the evidence of true experience and eliminate the track of the natural thought for solving a problem.” “Contour integrals for special functions: I. When we deal with a contour integral for a special function, all we have to do is to choose a point on the contour and assign a possible value to its argument. II. The only purpose of detailed discussion about branch points is to tell us that if we want to choose a point and its argument properly to facilitate calculations, we must consider branch points first.” “Tying up loose ends” “The finishing touch: Providing a solution to a problem alone is not enough; the author should tell the 2 readers from where the solution comes. This way can bring the readers to an advantageous point for a bird’s-eye view of the circumstance.” “Musket to kill a butterfly: Example [Differentiating under an integral]. The modern method attacks directly toward the goal by using therems flexibly. A complex measure need not distinguish a compact integral contour from a noncompact one. A single proof is good enough for dealing with both compact and noncompact cases. Furthermore, the proof is free from complex analysis except for using the defini- tion of analytic functions. In contrast, the classical method must follow a formal, tedious, and inflexible procedure. In order to ensure the finiteness of a contour integral, the Borel measure must distinguish a compact integral contour from a noncompact one.
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