数学英语翻译

数学英语（下）

3.1.1

1.调和函数，即拉普拉斯方程的调解，在数学、物理学、工程学的许多领域中扮演着一个至关重要的角色。

Harmonic functions—the solutions of Laplace‟s equation—play a crucial role in many areas of mathematics, physics, and engineering.

2.本书的主要目的是让学习调和函数更容易些。

The main purpose of our text, then, is to make learning about harmonic functions easier.

3.唯一的准备条件是在实分析、复分析以及泛函分析的基本理论上有坚实的基础。

The only prerequisite for the book is a solid foundation in real and complex analysis, together with some basic results from functional analysis.

4.为完整起见，我们包含了一些低年级研究生课程中经常忽略的分析学的专题。For completeness, we include some topics in analysis that frequently slip through the cracks in a beginning graduate student‟s curriculum, such as real-analytic functions.

5.本书的作者除了写这本书，还开发了一个软件包去处理调和数论中出现的许多表达式。

In addition to writing the text, the authors have developed a software package to manipulate many of the expressions that arise in harmonic function theory.

6.附录B告诉读者如何免费得到我们的软件包。

Appendix B explains how readers can obtain our software package free of charge.

3.1.2

1.在过去三十年间拓扑向量空间领域取得了许多进展。

During the last three decades much progress has been made in the field of topological vector spaces.

2.在本书中作者打算写出这些出现在拓扑向量空间、序拓扑向量空间、拓扑基和拓扑代数中的反例。

In this book the author has attempted to give due recognition to the authorship of most of the counterexamples presented in this book.

3.对于本书中的大部分反例，作者不打算给出其原作者的规属认定。

He makes no attempt to give due recognition to the authorship of most of the counterexamples presented in this book.

4.本书假定读者已掌握一份拓扑学的知识，读者可参见B[18]了解一份拓扑学相关的知识。

It is assumed that the reader is familiar with general topology. The reader may refer to B[18] for information about general topology.

5.在本书末的参考文献中，参考书与论文分别开来，参考文献中的书籍记为B[]，论文记为P[]。

The books and papers are listed separately in the bibliography at the end of the book. Any reference to a book is indicated by writing B[] and to a paper by P[].

3.2.4

1.一个集合S上的关系指的是S到它自身的一种对应，例如“亲戚”就是所有的人组成的集合上的一种关系（如果能提供一份完整的家族谱的话）。

By a relation on a set S we mean a correspondence of S with itself. E.g., …being related‟ is a relation on the set of all humans (provided that we are equipped with an exhaustive genealogy).

2.最后这一点说明，我们总必须确定我们正在讨论的集合。

This last point illustrates that we must always specify the set on which we are operating.

3.S上的一个满足反身性、对称性以及传递性的关系被称为等价关系。

A relation on S which is reflexive, symmetric and transitive is called an equivalence on S.

3.3.2

1.非零非单位元的整数P，如果它的所有因子是+P和单位元，那么就称它为素数。An integer p that is not zero or a unit is said to be prime if its only divisors are + p and