VARIATIONS 89 />= ( , Q=-r, A = , = -, T^-JJ. Hence / dp dq dr ds dt J X(lJ , /0 = 11 (N8z + Pdp + Qdq + RSr + S8s + TSt}dxdy (45). J W 3/o In the transformation of this expression we shall employ the fol lowing formulae., which are easily established, [ Xl f yi du, , /* P 1 , [ Xl / yi dy . .... I I dxdy = / dx I udy - I / u-j-dx (46), J *(K 3/0 dx 9 J 2/0 J xj y dx pi ryiffv pi /2/i and / / dxdy= I / Zirfas (47). J j>/ !/ dy J xJ i/o pi r2/i f ;5~ Hence we get / / P-^-dxdy J V 2/0 rta; d .-8zdxdy + ^dx [^PSzdy - j ^pf^Szdy, )x I x J j/o x y
- 1 l yi
pi nil dSz PI flhdO f* 1 l y also Q^dxdy=- ^dzdxdy + / QSzdx. J,Jy dy Jxjy.dy J 4/2/0 Next, to transform PI 2/0 .ft ft/, we have I , < dfe ; . PI pi rf / P -R-T-5 + dxdy- I , -B- ./ V 2/0 V **" <* rfa; / J V 2/<M /^ /" 3/1 T/WZ 7 P 1 / ;/1 T>^ 5 ~ d y , = / dx R~~dy- / R-^^dx, / r c J 2/ d* J ,-,/ j, d rf* r r i ryi/ts ^5 rf 2 ^, , p / d /rf^, , also / l --+ $ z fafiy l / i-Sz dxdy J .,J y,d* dx dx ) j x jydxdx J /*i. f yi dJK. . [*i /todR dy .. j = / dxl -j- 5^c?y - / / j- -^ 5s^z/. / ^ J yo dx J X J y<) dx dx Consequently, by subtraction, dx dxdx Again, in transforming the latter integral it is necessary to observe that ^ and y are functions of x, and accordingly that in that integral we have =-^-(82)-^ hence dx dx dx dy also But dx ^- -r ax dx - j^y.dyfdR dy dR . ~ ^T~ + J ( ~r~ + j T- l> hence we get dji- dxdx dx dy J + pi rvi d?5z Next, to transform I / S^r-dxdy. Here J x j yi> dxdy .(48). f* 1 ( yi ( r, d*Sz dSdSz, } PI pi rf / a< 3 7 / / ( "j" j" +T" T- )dxdy= I I I 8 -f- dxdy J x J yo dxdy dxdy) J X J yo dx dy J Vi dSzdy, S-r--/ dx, yo dydx Fip/dSdSz^.. d*S,, [l/todS, - also -.- -T- + SZ-T-J- }dxdy= / ~dzdx J x J ivo V dM dl J dxd yJ J xJ yo dx pipi/^^Ss . d S . hence by subtraction / / ( ,5 - - dz-rr- dxdy J :> yi> dj d> J dxd y) f 1 P O d8s , PI j y * /dS. ndte dy , = Sdy- I / l8z + S-j--f-)dx ......... 49. /V2/o d V J xJ y Q dx dydxj Finally, to transform / / T-^dxdy, we have ./ T,,./ ii n dy r0+ 5 i=r yi/di 2/oU dSz. . -j- }dxdi/= z, T-^-dx, dy 1 f Y 2" f r^ - TT fc V r( fy = f * / Y T ^ - ^ fc V- ( 5 )- J 2/ rfy- dr / J *J 2/ V rf y d y / a i Vdxdy o pi pi / dP dQ d*R drS d~T - (W- T---^ Jxjy dx dy r / J V y V y dy .(51). 28. In many applications the limits of x and y are determined from a single equation. Thus, suppose the integral U n Vdxdy to express the sum of the elements Vdxdy for all values of x and y which satisfy either <j>(x, y) >0 or <p(x, y) <0, then the limits are given by the equation <f>(x, y) 0. In this case the preceding value of dU becomes much simplified, for y , y t are determined from the equation 1 <(> ; and the extreme values of a- are found by eliminat ing?/ between the equations </> = and d<p/dy = Q. But these are the conditions that< = should have equal roots in y; we consequently infer that, when either xx or x=x lt we have y = 2/i- Hence we /*o nil PI ryi observe that / I udy=0 a.na / I udy = for all values of u, so J 2/0 J 2/o that the last four terms in the expression for dUin (51) disappear in this case. The methods and results of this and the preceding section can without difficulty be extended to triple and higher multiple integrals. 29. The method of application of the calculus of varia tions to the determination of maximum or minimum values of multiple definite integrals proceeds on the same prin ciples as those already considered in 9 for single integrals. We shall limit the discussion to a brief consideration cf the double integral U= II Vdxdy, in which F is a function of x, y, z, j), and q solely, and the limits are determined, as in 28, by an equation of the form <$>(x, #) = 0. Such problems readily admit of geometrical interpretation. Here (51) reduces to dP d $sj ; -T --- ^ zdxdy yo dx dy) ............ (52). XTT ou= Consequently, as in the case of single integrals, for a maximum or minimum value of 7 we must have _ dx dy and .(53), .(54). The former is in this case a partial differential equation of the second order, whose solution, whenever it can be obtained, consists of an equation between x, y, z having tyo arbitrary functions. The form of these functions in each case is to be determined by aid of (54) combined with the given limiting conditions of the problem. Thus, let us suppose the upper limit restricted to points in the surface z=f l (x, y), and let p = C -^, q = -/~; then 5 / z= / q Sy; and = q Sy . This gives (55). Substituting in (54), we get Again, along the limiting curve we have dz=pdx + qdy and dz=p dx + q dy; hence we get = -^ ^. Substituting in (56), dx q ~~ q we have along the limiting curve q) = ....... . ....... (57). 1 If the equation = gives for y several values, y , j/j, y, y 3 , 1/4, .... then d each integral is to be t XXIV. 12 . , , j, , 3 , 4, .... r pi r3 we substitute / r,ly= Vdy+ I Vdy+ . . ., and each integral is to be treated
s above. J J VQ J Vn