Calculus in Several Variables

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)

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

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).