# Calculus in Several Variables

• E. Hairer
• G. Wanner
Part of the Undergraduate Texts in Mathematics book series (UTM)

## 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).

## Keywords

Partial Derivative Differentiable Function Triangle Inequality Inverse Image High Derivative