19 - Calculus of Several Variables

Summary

Preliminary Remarks

During the eighteenth and nineteenth centuries, Niklaus I Bernoulli, Clairaut, Euler, Fontaine, Lagrange, Gauss, Green, Cauchy, Ostrogradski, Jacobi, and others laid the foundations for the calculus of several variables. Most of this work was done in the context of mathematical physics. Not surprisingly, however, it is possible to see traces of this subject in the earliest work of both Newton and Leibniz. In his study of calculus conducted in May 1665, Newton expressed the curvature of f(x, y) = 0 in terms of homogenized first- and second-order partial derivatives of f(x, y). He even worked out a notation for these partial derivatives, though he did not make use of it in any of his other work. Newton used partial derivatives as a computational device without a formal definition. Again, in his calculus book of 1670–71, Newton described the method for finding the fluxional equation of f(x, y) = 0, identical to the equation f_x ẋ + f_y ẏ = 0, where the dot notation denoted the derivative with respect to some parameter.

Gottfried Leibniz (1646–1716), whose name was written Leibnitz in most of the early literature, published two papers, in 1792 and 1794, in which he developed a procedure to find the envelope of a family of curves: a curve touching, at each of its points, a member of the given family. He showed that the methods of calculus could be extended to a family of curves by treating the parameter defining the family as a differentiable quantity.