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微积分和数学分析引论-第1卷 节选
eface
During the latter part of the seventeenth century the new mathe-
matical analysis emerged as the dominating force in mathematics.
It is characterized by the amazingly successful operation with infinite
processes or limits. Two of these processes, differentiation and inte-
gration, became the core of the systematic Differential and Integral
Calculus, often simply called "Calculus," basic for all of analysis.
The importance of the new discoveries and methods was immediately
felt and caused profound intellectual excitement. Yet, to gain mastery
of the powerful art appeared at first a formidable task, for the avail-
able publications were scanty, unsystematic, and often lacking in
clarity. Thus, it was fortunate indeed for mathematics and science
in general that leaders in the new movement soon recognized the
vital need for writing textbooks aimed at making the subject ac-
cessible to a public much larger than the very small intellectual elite of
the early days. One of the greatest mathematicians of modern times,
Leonard Euler, established in introductory books a firm tradition .and
these books of the eighteenth century have remained sources of inspira-
tion until today, even though much progress has been made in the
clarification and simplification of the material.
After Euler, one author after the other adhered to the separation of
differential calculus from integral calculus, thereby obscuring a key
point, the reciprocity between differentiation and integration. Only in
1927 when the first edition of R. Courant's German Vorlesungen uber
Differential und Integralrechnung, appeared in the Springer-Verlag
was this separation eliminated and the calculus presented as a unified
subject.
From that German book and its subsequent editions the present
work originated. With the cooperation of James and Virginia McShaue
a greatly expanded and modified English edition of the "Calculus" wes
prepared and published by Blackie and Sons in Glasgow since 1934, and
distributed in the United States in numerous reprintings by Inter-
science-Wiley.
During the years it became apparent that the need of college and uni-
versity instruction in the United States made a rewriting of this work
desirable. Yet, it seemed unwise to tamper with the original versions
which have remained and still are viable.
Instead of trying to remodel the existing work it seemed preferable to
supplement it by an essentially new book in many ways related to the
European originals but more specifically directed at the needs of the
present and future students in the United States. Such a plan became
feasible when Fritz John, who had already greatly helped in the prepara-
tion of the first English edition, agreed to write the new book together
with R. Courant.
While it differs markedly in form and content from the original, it is
animated by the same intention: To lead the student directly to the
heart of the subject and to prepare him for active application of his
knowledge. It avoids the dogmatic style which conceals the motivation
and the roots of the calculus in intuitive reality. To exhibit the interac-
tion between mathematical analysis and its various applications and to
emphasize the role of intuition remains an important aim of this new
book. Somewhat strengthened precision does not, as we hope, inter-
fere with this aim.
Mathematics presented as a closed, linearly ordered, system of truths
without reference to origin and purpose has its charm and satisfies a
philosophical need. But the attitude
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