FORM计算机代数系统快速入门与应用

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"FORM 计算机代数系统学习教程" FORM 是一个开源的计算机代数系统，专为处理大规模、复杂的符号计算任务而设计。它的特点是代码简洁且性能高效，与市面上主流的计算机代数系统相比，FORM 在特定领域展现出独特的优势。在FORM的基础学习中，首先会接触到“FORM for Pedestrians”这份教程，由Andr`e Heck撰写，旨在引导初学者入门。教程涵盖了FORM的基本概念和操作，以及一系列示例来帮助理解其工作原理和功能。 1.1章节介绍了如何开始使用FORM，包括系统的特点和第一个示例。FORM的特点在于它的高速度和处理高维度问题的能力。初学者通过简单的例子，可以了解如何在FORM中执行基本的计算。 1.2章节深入讨论了FORM中的变量类型，包括函数、向量和索引、张量等。函数部分讲解了如何定义和使用函数，向量和索引部分则阐述了如何处理多维数据结构，而张量部分则涉及到了更高级的数学对象。 1.3章节揭示了FORM的内部工作机制，让学习者对FORM的执行流程有更深入的理解。这部分内容可能包括解析、优化和输出等关键步骤。 1.4章节则是一系列FORM应用实例，如求和、Levi-Civita张量、外积运算、行列式计算、Gram行列式、图论中的图可实现性问题等。这些例子涵盖了广泛的数学领域，帮助学习者掌握在实际问题中运用FORM的技巧。接下来，教程进入2.0章，重点是模式匹配。2.1章节介绍了如何进行替换操作，2.2章节讲解了模式对象，2.3章节涉及替换规则中的模式，2.4章节则探讨了更为复杂的模式应用。在每个子章节中，都配有练习题以巩固学习，并且通过具体的例子演示了如何进行多项式替换、计算斐波那契数列、对多项式进行微分、进行坐标变换以及处理Levi-Civita张量等操作。总而言之，FORM作为一个强大的计算机代数系统，提供了一整套工具来进行高级数学计算。通过这份详尽的教程，学习者可以逐步掌握FORM的使用，从而解决符号计算中的各种挑战。无论是在物理学、工程学还是纯数学领域，FORM都能成为研究和教学的强大助手。

Local F4 = t;

ToVector t,v;

Print;

.end

F1 =

v(1)*v(i)*v(j)*v(k);

F2 =

F3 =

u.v^2*v.v;

F4 =

1.2.4 Exercises

1. Check how FORM handles the summation convention for the following expressions.

(i) a

(ii) a

(iii) a

(iv) δ

2. Demonstrate with FORM the following equalities.

(i) a

= a

(ii) (a

+ a

= 2a

3. Let a be an antisymmetric tensor of rank two . Demonstrate with FORM the following properties.

(i) a

= 0 for any vector x.

(ii) the tensor b of rank two deﬁned by the contraction b

= a

is symmetric.

4. There are three ways to control the printing of powers of functions:

FunPowers nofunpowers;

FunPowers commutingonly;

FunPowers allfunpowers;

Find out by experimentation what the statements actually do and check also how they aﬀect the

printing of powers of tensors .

1.3 Inner Workings of FORM

When w riting or studying FORM programs, it is useful to have at least some idea of what FORM internally

does. First, you need to know what objects it actually manipulates. The answer is that FORM works with

expressions that are sums of terms; each term consisting of a rationa l coeﬃcient times a product of factors ,

possibly to some power. The fac tors can be symbols or more complicated structures such as functions or

tensors. The expressions that can b e manipulated are called active expressions. There can be many active

expressions at the same time.

The method of operation of FORM is as follows.

1. The preprocessor rea ds from the input stream and changes the input on a purely textual level into code

that can be interpreted by the compiler. The preprocessor has variables, loops, conditional statements,

and oﬀers the possibility to include other ﬁles or deﬁne procedures. Pre pro c e ssor instructions always

start with the sharp symbol (#).

2. The compiler interprets the pr epared code and translates it into machine code.

3. The compiler is halted when the prepr ocessor reads a module instruction, which starts with a period

and marks the end of a module.

4. FORM runs the machine code: it applies the ope rations speciﬁed in the module to the active ex-

pressions, one by one, and works out brackets. For each expression, FORM applies the requested

operations in the order as speciﬁed in the module to the ﬁrst term of the expression (possibly gener-

ating new terms ) a nd brings the resulting terms into a standard form. It does the same thing to the

second term, and so on until all terms have been processed.

5. FORM so rts the results and calls this the input for the next module.

So, you see that FORM sequentially processes expressions term by term. This mode of operation means

that FORM has no operations that use more than one term at the s ame time. For example, a substitution

rule like a + b → c canno t be expr e ssed as such. You will have to use tricks such as the replacement r ule

a → c −b. It also means that there is no factorizatio n built into FORM because the whole expression must

be taken into account for this mathematical operation. To summarize in one sentence:

All operations in FORM are local to one term of an active expression.

1.4 Some FORM Examples

The examples in this section are to acquaint you with FORM and to show some of its built-in facilities.

1.4.1 Summation

FORM provides you of course with tools to compose an expression. For example, to obtain

i=0

you can

enter the following:

Symbols x,i;

Local expr = sum_( i, 0, 5, x^i/fac_(i) );

Print;

.end

expr =

1 + x + 1/2*x^2 + 1/6*x^3 + 1/24*x^4 + 1/120*x^5;

1.4.2 Levi-Civita Tensor

The Levi-Civita tensor or permutation tensor ǫ

...i

plays an impo rtant role in tensor calculus. When the

indices range fr om 1 to n, it is deﬁned as

...i







0 if two indices are the same

1 if (i

, i

, . . . , i

) is a even permutation of (1, 2, . . . , n)

−1 if (i

, i

, . . . , i

) is an odd permutation of (1, 2, . . . , n)

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