Page:Encyclopædia Britannica, Ninth Edition, v. 13.djvu/27

INFINITESIMAL CALCULUS (3) If y = a"- 1 log a-, prove that ~^ = i^ dx n x (4) If T/ = COS (a sin- 1 .*,-), prove that (5) If y=a cos nx + b sin nx, prove that Partial Differentiation. 28. We have hitherto treated of functions of a single independent variable solely. The principles established so far apply equally to the case of functions of two or more independent variables. For example, in the equation u = ax- + 2bxy + c?/ 2 , the variables x and y may be capable of change independently of each other ; and if we suppose x to vary, y remaining constant, the corresponding differential coefficient of u is represented by du -, , ,- , and we have ax = 2ax + 2by . dx In the same case if we suppose y to vary, x being unchanged, the corresponding differential coefficient is represented by , and we ay have ~ dy In general, if u be a function of two variables, x and y, represented by the equation u=-<f>(x, y), we have two differential coefficients du d<f>(x , ?/) . du d<t>(x , - or v , , and -=- or , dy dy dx These are called the partial differential coefficients of the function, with regard to x and y respectively. They are usually written and -3- , and are plainly determined in the same manner as in dx dy the case of a single variable. 29. These new functions ^ and admit of being treated in like dx dy manner. Thus the partial differential coefficient of , taken with dx , du 1~ respect to x, y being supposed unchanged, is represented by or by ; likewise its differential coefficient with respect to y is represented by ; and so on. dydx 30. It can be seen without difficulty that , du , du fy ^ a u a u . dx ay dxdy dydx dy dx In fact signifies the limit to which - dydx Ax and Ay diminish beyond limit. Au ^<p(x + Ax , y) - <f>(x , y) Ax Ax In like manner, Au approaches as Again Au A Ax , y + Ay)-<f>(x + Ax,, Ax Ax Ay , y + Ay) - 4>(x + Ax , y) - <f>(x , y + Ay) + <p(x , _y ) AxAy has the same value. Accordingly Ax It is easily seen that the limits of the two expressions must be equal, and hence we infer d*u _ oPu dydx dxdy 17 31. In general, if u be a function of several independent variables a 1 !, x zt . . . x, wo obtain n partial differential coefficients of the first order, denoted by du du du du In like manner, the partial differential coefficients of the second order are represented by d?u d 2 u d?u j 3 ~j T~ i j o > etc., and so on. We have, as in the former case, between each pair of variables 32. In the equation u=*<t>(x,y if we consider x and y to increase simultaneously, then, if Au represents the total increment of u, we have , y + Ay)-<t>(x, Ax . AX yAy)-<t>(x, y) , y + Ay) - <p(x , ?/ - - A?/ If now we suppose Ax and Ay to diminish indefinitely, and repre sent the corresponding differentials by du, dx, dy, we have in the limit 7 ^ U rr ^ U rl dx dy This is called the total differential of u, and it is readily seen that it is equal to the sum of the partial differentials arising from the separate increments in x and y. The same principle plainly holds in a function of any number of variables. 33. If u = <f> (v, w), where v and w are both functions of x, then by the preceding it is readily seen that du du dv du dw dx dv dx dw dx and similarly for any number of functions. 34. The principles of total and partial differentiation admit of simple illustration in plane and in spherical trigonometry. For, in either a plane or a spherical triangle, we may regard any three 1 of the parts a, b, c, A, B, C as being independent variables, and each of the others as a function of the three so chosen. For instance, in a plane triangle, if the sides a and b and the contained angle C be taken as the independent variables, we have C = Qs* ~r~ 2tCtU COS (_/ ! dc a-b cos C = cos D ; hence da likewise -^ = 038 A , = sin B , db dC . . dc = cos B da + cos A db + a sin B dC. c Again, to find - , we have b sin A = a si hence, regarding a and b as constant, we have &cos A c =, acos (A + C) /I + ^j = - a dA a cos B = a sin (A + C) ; In like manner we have, in the same case, c?A_sin B dA sin A db^ .:dA-. da c db c Again, in a spherical triangle, cos c = cos a cos b + sin a sin b cos C. From this we obtain sin B . sin A a cos B da-- db dC. da = cos A , c -^- = sin a sin B , do dC dc cos B da + cos A db + sin a sin B dC. This, and the preceding, also admit of a simple geometrical demonstration, by drawing the triangle and comparing the small increments in each case. 35. Again, since from any equation in spherical trigonometry another can be deduced by aid of the polar triangle, we get from the preceding dC = - cos b dA - cos a cB + sin A sin b dc. Corresponding formula? are obtained by an interchange of letters. 1 The case of the throe angles of a plane triangle is excepted, as they arc equivalent to but two independent data. XIII. 3