45. Extension of Taylor’s theorem.Taylor’s theorem can be extended to functions of several variables. In the case of two variables the general formula, with a remainder after n terms, can be written most simply in the form
| ƒ(a + h, b + k) = ƒ(a, b) + dƒ(a, b) + | 1 | d2 ƒ(a, b) + ... |
| 2! |
| + | 1 | dn−1 ƒ(a, b) + | 1 | dn ƒ(a + θh, b + θk), |
| (n − 1)! | n! |
in which
| dr ƒ(a, b) = [ ( h | ∂ | + k | ∂ | )rƒ(x, y) ] | , | |
| ∂x | ∂y | x=a, y=b |
and
| dn ƒ(a + θh, b + θk) = [ ( h | ∂ | + k | ∂ | )nƒ(x, y) ] | . | |
| ∂x | ∂y | x=a+θh, y=b+θk |
The last expression is the remainder after n terms, and in it θ denotes some particular number between 0 and 1. The results for three or more variables can be written in the same form. The extension of Taylor’s theorem was given by Lagrange (1797); the form written above is due to Cauchy (1823). For the validity of the theorem in this form it is necessary that all the differential coefficients up to the nth should be continuous in a region bounded by x = a ± h, y = b ± k. When all the differential coefficients, no matter how high the order, are continuous in such a region, the theorem leads to an expansion of the function in a multiple power series. Such expansions are just as important in analysis, geometry and mechanics as expansions of functions of one variable. Among the problems which are solved by means of such expansions are the problem of maxima and minima for functions of more than one variable (see [Maxima] and [Minima]).
46. Plane curves.In treatises on the differential calculus much space is usually devoted to the differential geometry of curves and surfaces. A few remarks and results relating to the differential geometry of plane curves are set down here.