The Fundamental Theorems of the Differential Calculus
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At the present stage it is advisable to examine what modifications are necessary in §§ 19 — 23 to make them apply to functions of n variables where n>2. We have then n, instead of two, partial differential coefficients, fa,^, f^^, ... which, as in § 17, are assumed to be finite, and to appear linearly in the equation /(«!, Xi, ..., a;n)=f(ai, «2, ..., a„) + S (^^ - a,) (/a^ + e^), THE DIFFERENTIAL CALCULUS 31 the quantities ei, e^, ■■■, «« having zero as unique w-ple limit when the h's approach... zero in any manner whatever. As in § 18 we get a set of sufficient conditions for the existence of a differential at a certain point, demanding that one of the partial incrementary ratios should have an unique m-ple limit, one an unique (n- l)-ple limit, and so on, one an unique double limit and the last an unique simple limit, viz. the corresponding partial differential coefficient. This gives the corollary : — If at the point (ai, a^, ..., an) all the partial differential coefficients exist, all but one are continuous with respect to {xi, x^, all but one of those remaining with respect to (xi, x^, x^, and so on, finally the last one is continuous with respect to(xi,X2,---, Xn), thenf{xi, x^, .
What to read after The Fundamental Theorems of the Differential Calculus?
You can find similar books in the "Read Also" column, or choose other free books by W H William Henry Young to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
At the present stage it is advisable to examine what modifications are necessary in §§ 19 — 23 to make them apply to functions of n variables where n>2. We have then n, instead of two, partial differential coefficients, fa,^, f^^, ... which, as in § 17, are assumed to be finite, and to appear linearly in the equation /(«!, Xi, ..., a;n)=f(ai, «2, ..., a„) + S (^^ - a,) (/a^ + e^), THE DIFFERENTIAL CALCULUS 31 the quantities ei, e^, ■■■, «« having zero as unique w-ple limit when the h's approach... zero in any manner whatever. As in § 18 we get a set of sufficient conditions for the existence of a differential at a certain point, demanding that one of the partial incrementary ratios should have an unique m-ple limit, one an unique (n- l)-ple limit, and so on, one an unique double limit and the last an unique simple limit, viz. the corresponding partial differential coefficient. This gives the corollary : — If at the point (ai, a^, ..., an) all the partial differential coefficients exist, all but one are continuous with respect to {xi, x^, all but one of those remaining with respect to (xi, x^, x^, and so on, finally the last one is continuous with respect to(xi,X2,---, Xn), thenf{xi, x^, .
What to read after The Fundamental Theorems of the Differential Calculus?
You can find similar books in the "Read Also" column, or choose other free books by W H William Henry Young to read onlineMoreLess
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