If R, be the principal radii of curvature1 at any P on S, then I C —; moreover, this RR' K is not affected by bending S in any way with out stretching or tearing— a beautiful Gaussian theorem of profound philosophic import. In this sense, an S that may be flattened out into a plane (a Developable) has 0-curvature; for such, RR' must become a • hence either R= a or R'== co . But RR'=-(1 + if)/ (rt — s'), where (Euler). Hence is the equa tion of Developables. For Applicables and fur ther illustrations, see SURFACES, THEORY OF.
The difficulty in dealing with Implicit Func tions (defined by unsolved equations) lies in the Existence-theorems, which can only be stated. I. Let F(x, y)+0 at (x., y.), and have 1st partial D's finite and continuous about (x., y,), and 0 at (x., y.) : then there is a y="y(x) that becomes y. for x = x., and satis fies identically F(x, y)— 0 in the vicinity, and is unique, and has a D, II. Quite similarly for F(x, y, 0, y), the last statement being: has two partial D's.
, .
= — F and so on for n variables.
Most generally (III) let F,, . . . , Fn be n func tions of m variables x, y, . . . , and n variables u, v, . , all the F's vanishing at (x., y., . . . , 140, v., .. .), all admitting partial D's in that vicinity, and J(FI, . . . Fn; st, ..)+-0(p. 176), at (x., y., . . . , . . .) : then there is a system of functions of m independents x, y, . . . that become uo, v., . . . at (x., yo, . . .), that satisfy identically all the F's in that vicinity, and that have partial D's.
Hence we have as the ordinary rule for find ing from F(x, 0: Differentiate F as to x regarding y as a function of x, as in mediate derivation, and solve the result as to yx.
To find now maximum or minimum y in F(x, y)=:), we have yx =0, F:=0; also, therefore, + = 0. If + 0, there is maximum for Fix and F, like-signed, minimum for F ...e and unlike-signed, and no determi nation for Fzz= O.* Often we seek (so-called) relative maximum and minimum of z= f(x, y) when F(x, O. The former equation is a surface S, the latter an intersecting surface determining a path over S,— we seek the peak and valley points in this path. Differentiating we find as the prime con dition, fx. — Fx= 0 = I (f, F; x, y), from which and F =0 we find the x and y that maxi . mize or minimize f.
More generally we seek maximum or mini mum of a function of (m + n) variables, f(x, y,...u, v,...), under n conditions Fi(x,y,...u,
v, . . .)= 0, ... Fn(x, y, . . .u, v, .. .):). Theo retically we might eliminate n variables u, v, ... leaving the other m independent ; it is better to let them remain considered as functions of the in independents, x, y, . . . Hence, on putting each partial D = 0, we get m equations which, with the n Fi= 0, ... Fn— 0, form (m+n) equa tions for finding (m + n) unknowns x, y, ... st, v, w, . . . To discriminate between maximum and minimum by the sign of d'f will now be tedious, but often geometrically or mechani cally unnecessary, Swifter and simpler is Lagrange's 'Method of Multipliers.) We form a new function, 0 (x, y, . . . u, v, . . .)= f(x, y, . . . u, v, . . .) F(x, y, ... u, v, . ..). Only so long as each F =0 will 0 = f identically for all values (under consideration) of the variables. We now de termine these 2's so as to make vanish simul taneously all the partial D's of 0 as to x, y, • . . u, v,... The ts conditions are rolled off from the u, v, . . . upon the n ?'s. We may proceed similarly in dealing with Envelopes, where (n + 1) parameters are connected lay n condi tions.
Transformation of Variables is often neces sary, like transformation of Co-ordinates. The formulae, simple at first, soon become highly complicated and we are led into the Theory of Substitutions, Invariants, Reciprocants and the like, which cannot be treated here.
,/-- Integration.— As the Differential Calculus is the doctrine of Limits of Quotients of Simul taneous Infinitesimal Differences, so the INTEGRAL CALCULUS is the doctrine of Limits of the Sums of Infinitesimal Products that increase in number while decreasing in size, both indefinitely. The type is the quadrature of an area (A) bounded by X, a curve y= f(x), and two end-ordinates, x= a and x = b. Cut it into is strips, .1,4, standing each on a dx; their sum is A; plainly Y dx>dA>ydx, Y being the greatest, y the least, ordinate stand ing on its own particular base dx. Since I ..4x= b— a, if y =f(x) be continuous, finite, one-valued throughout [a, b], each Y — y is an e, hence
— y)dx