Analysis by Its History pp 271-350 | Cite as

# Calculus in Several Variables

Chapter

## 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 by a

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

*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).## Keywords

Partial Derivative Differentiable Function Triangle Inequality Inverse Image High Derivative## Copyright information

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