VI 
PREFACE 
In the early days of modern mathematics no sharp distinction was 
made between the Differential and Integral Calculus, and the Calculus 
of Variations — it was all, the Infinitesimal Calculus. With the 
knowledge of the great delicacy, both in concepts and methods, of 
which the Calculus of Variations is capable, came to mathematicians 
an awe of this subject, which resulted in a certain aloofness ; the sub 
ject became a topic in the theory of functions of real variables. For 
the physicist, however, Hamilton’s Principle is indispensable, and he 
has been obliged to get together some account of the rudiments of the 
Calculus of Variations as best he may. Sufficient conditions for a 
maximum or a minimum of an integral do not interest him. He needs 
to know when a certain integral is stationary, and this condition de 
pends on the definition of a variation, 8x, 8U, etc. It is, therefore, 
essential that this definition be treated with care from the start, 
for it becomes increasingly complex as one proceeds. The Principle 
is applied to a variety of important problems in elastic vibrations. 
There is a chapter on the systematic treatment of differential equa 
tions. But what is far more important is the unsystematic treatment 
of differential equations, which permeates these two volumes on the 
Calculus, beginning with the chapter on Mechanics in the Introduction 
to the Calculus. I have, moreover, taken occasion in the present 
chapter to point out the inner meaning of a differential equation 
through the geometric picture of a field of infinitesimal vectors or an 
assemblage of surface elements, and have thus led up to the idea of 
the integrals as families of curves, or of surfaces generated by 
characteristic strips. 
As regards method, it sometimes happens that the naive use of in 
finitesimals, even when it cannot be directly justified, has suggestive 
heuristic value ; consider, for example, the transformation of multi 
ple integrals and the flux across a surface ; Chapter XII. In such 
cases, I have taken pain§ to conserve all that is helpful in these 
primitive conceptions, and have then supplemented them by proofs 
which meet our present standards of rigor. In this connection may 
also be cited (although it is not a question here of infinitesimals) the 
note on density and specific pressure or specific force ; Chap. Ill, § 14. 
A new form of the definition of a definite integral, simple or mul 
tiple, makes possible a simple and rigorous proof of the Fundamental 
Theorem of the Integral Calculus ; Chap. XII, §§ 1-3. 
There is a chapter on Vector Analysis, with applications to the 
proof of Stoke’s Theorem and the deduction of the Frenet formulas.