Calculus in Several Variables

Abstract

Functions of several variables have their origin in geometry (e.g., curves depending on parameters (Leibniz 1694a)) and in physics. A famous problem throughout the 18th century was the calculation of the movement of a vibrating string (d’Alembert 1748, Fig. 0.1). The position of a string u(x, t) is actually a function of x, the space coordinate, and of t, the time. An important breakthrough for the systematic study of several variables, which occured around the middle of the 19th century, was the idea of denoting pairs (then n-tuples)

$$ {\text{(}}{{\text{x}}_1}{\text{,}}{{\text{x}}_2}{\text{) = :x}}\quad {\text{(}}{{\text{x}}_1}{\text{,}}{{\text{x}}_2}{\text{,}} \ldots {\text{,}}{{\text{x}}_n}{\text{) = :x}}$$

by a single letter and of considering them as new mathematical objects. They were called “extensive Grösse” by Grassmann (1844, 1862), “complexes” by Peano (1888), and “vectors” by Hamilton (1853).

The influence of physics in stimulating the creation of such mathematical entities as quaternions, Grassmann’s hypernumbers, and vectors should be noted. These creations became part of mathematics.

(M. Kline 1972, p. 791)