数学物理：科学家和工程师应用数学

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"mathematical physics" 是一本应用数学在科学家和工程师中的专业书籍，由 Bruce R. Kusse 和 Erik A. Westwig 共同撰写，已进入第二版。该书由 WILEY-VCH 出版社出版，旨在为科学研究和工程领域的专业人士提供数学物理学的基础知识。本书的内容可能涵盖了广泛的数学物理主题，包括但不限于： 1. 物理学的基本概念：书中可能会深入讲解物理学中的基本原理，如牛顿力学、热力学、电磁学和量子力学，以及它们与数学的紧密联系。 2. 微积分和分析：作为数学物理学的基础，微积分在描述物理现象中扮演关键角色。这可能涉及函数、极限、导数、积分、微分方程等概念，以及多元微积分在空间中的应用。 3. 线性代数：线性代数是现代物理中不可或缺的部分，尤其是在量子力学和统计力学中。书中可能包含向量、矩阵、线性方程组、特征值和特征向量等概念。 4. 概率与统计：在处理随机性和不确定性时，概率论和统计学的知识是必要的。这部分可能涵盖概率分布、随机过程、统计推断等内容。 5. 泛函分析：在高级物理中，泛函分析用于处理无限维空间中的问题，如希尔伯特空间和巴拿赫空间。这可能涉及到正交系统、算子理论和变分原理。 6. 数值方法：实际问题往往需要数值解法，因此书中可能包含有限差分、有限元方法、蒙特卡洛模拟等数值计算技术。 7. 近世代数和群论：这些抽象数学概念在对称性和守恒定律的研究中起到重要作用，特别是在粒子物理和固体物理中。 8. 偏微分方程：物理问题常常归结为求解偏微分方程，如波动方程、热传导方程和薛定谔方程等。 9. 控制理论和优化：对于工程师来说，理解和解决控制系统的数学模型以及优化问题是非常重要的，这可能涉及到拉格朗日乘子法和动态规划。通过阅读 "Mathematical Physics"，读者可以预期获得一个坚实的数学基础，以便更好地理解和解决科学和工程领域的复杂问题。作者 Bruce R. Kusse 和 Erik A. Westwig 的专业知识将为读者提供一个实用而全面的数学物理教育。

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A REVIEW

VECTOR AND

MATRIX ALGEBRA USING

SUBSCRIPTISUMMATION

CONVENTIONS

This chapter presents a quick review of vector and matrix algebra. The intent is not

to cover these topics completely, but rather use them to introduce subscript notation

and the Einstein summation convention. These tools simplify the often complicated

manipulations

linear algebra.

1.1

NOTATION

Standard, consistent notation is a very important habit to form in mathematics. Good

notation not only facilitates calculations but, like dimensional analysis, helps to catch

and correct errors. Thus, we begin by summarizing the notational conventions that

will be used throughout this

book,

as listed in Table

.l.

TABLE

1.1.

Notational Conventions

Symbol Quantity

real number

complex

number

vector component

matrix

tensor element

entire matrix

vector

basis vector

tensor

operator

- -

NOTATION

We refer to this combination of the subscript notation and the summation convention

subscripthummation notation.

Now imagine we want to write the simple vector relationship

This

equation is written in what we call

vector notation.

Notice how it does not

depend on a choice of coordinate system.

a particular coordinate system, we can

write the relationship between these vectors

terms of their components:

B3.

(1.7)

With subscript notation, these

three

equations can be written in a single line,

where the subscript

stands for any of the

three

values

(1,2,3).

you will see

in many examples at the end of

this

chapter, the use of the subscript/summation

notation can drastically simplify the derivation of many physical and mathematical

relationships. Results written in subscripthummation notation, however, are tied to

a particular coordinate system, and are often difficult to interpret. For these reasons,

we will convert our final results back into vector notation whenever possible.

matrix is a two-dimensional array of quantities that may or may not be associated

with a particular coordinate system. Matrices can

expressed

using

several different

types

notation. If we want to discuss a matrix

its entirety, without explicitly

specifying all its elements, we write it in

matrix notation

[MI.

If we do need

list out the elements of

[MI,

we can write them

a rectangular array inside a pair

brackets:

(1.9)

We call this

matrix array notation.

The individual element in the second row and

third column of

[MI

is written

M23.

Notice how the row of a given element is

always listed first, and the column second. Keep

mind, the array is not necessarily

square. This means that for the matrix in Equation

1.9,

does not have to equal

Multiplication between two matrices is only possible if the number of columns

in the premultiplier equals the number of rows

the postmultiplier. The result of

such a multiplication forms another matrix with the same number of rows as the

premultiplier and the same number of columns

the postmultiplier. For example,

the product between a

matrix

[MI

and a

matrix

[N]

forms the

matrix

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