Mathematical Methods

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 complications, expanding the scope of application without loss of efficiency), structurization (insightfulness, essence, flowcharts or networks). Structurization may help us systematically operate and deeply understand 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 meanings 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