INFINITESIMAL CALCULUS 57 dx-i where 0! represents the value of <f>(x } , x, into a function of u lt it. 2 , . . . ii n . Moreover, by 102, the product . . a 1 ,,) when transformed is, in this case, the Jacobian of the original system of variables x. 2 , . . . x,, regarded as functions of the new variables. Accordingly, for dx t d,r . . . dx n we substitute dx l For instance, tJf*Vdxdy be transformed to new variables u, v, denoting by V, the value which V assumes, the double integral becomes where Again, if the coordinates of each point on a surface be given in terms of two independent variables u, v, to find the transformed expression for the superficial area dx Here dxdij becomes (x tl y v l/ux v ) dudv as before. Also, since, from the equation to the surface, z may be regarded as a function of x and y, we have , dz , dz dz x uz v - z u x . dz_ dx v - y u x v dy x ,,y v - y u x v Accordingly the transformed expression is ff{ (x u y v - y ux v )- + (y uz v - z ,,?/ ,) 2 + (z u x v - x ^rf } idudv . For example, the coordinates of any point on the ellipsoid o" "I j~*t <T ~ a* o* c~ may be represented by the equations x = a sine cos<f>, y = b sinfl sin 0, z = c cos0 ; liencc it can be shown that its total surface S is represented by sin 2 si sin 2 cos 2 iu which the integration with respect to 6 can be immediately effected. Again, the coordinates of any point on a sphere of radius a can be represented by the equations = a sin Vl - V, ?/ = sin <f> Vl - k * sin -fl , = a cos cos $, where This is obvious, since the sum of the squares of these expressions s a 2 . Accordingly x? ?/ -if x = - a " cos e cos
- 2 cos 3 e)
2 cos 2 + k 2 cos-9) - - L Hence we get - VI - tf sinV Vl - k - si Consequently, since the entire surface of the sphere is 4n-a 2 , we have _ o Jo Vl-1; 2 "siiiV Vl-i 2 siu^" 2 Ihe well-known general formula of Legendre, connecting com- ])lete elliptic functions of the first and second species, follows at once from this last result. 184. In the case of three variables, adopting a similar notation, the integral fffVdxdydz transforms into (*utf, - y ux 1 *} + y w (~ u x v - x u z t ) + x K (y u ~. , - z ,,y T ^ For example, in the general transformation from rectangular to polar coordinates we find, as already observed, that ?"sin 6drded<J> is to be substituted for the element of volume dxdydz. This is but a particular case of the general transformation given in 103. The preceding formula of transformation for three variables was given by Euler in 1769, and afterwards generalized by Lagrange in 1773. Jacobi appears, however, to have been the first to have es tablished the general transformation, in his memoir referred to in 96. The method of proof here adopted is that given by Bertram!. Ex. 1. In the case of linear transformations, viz., when + a. 2 u. . . . + a n u n , . . . +b n u n , we get where Ex. 2. If the Jacobian is u l 1<"2 ~ U 9 U 3 W_ 3 7^ ttj ! HJ 3 - itlttj, 1 .., < : v., The value of this determinant is easily seen to be 4 ; hence fff~V dx l dx 2 dx 3 transforms into ^fff^^du^du^du^ . Ex. 3. As an additional example we shall take Jacobi s method of establishing the fundamental formula of Eulerian integrals ( 153). Since - x x l - l dx, T(m) we have If now we transform by making x = uv, y = u are and oo , and those for v are and 1 ; also dx dy dx dy dv du du dv ~ * ,-OOx-i hence T(l}Y(m) = / I c - u u l + m - i ij- /0 *SQ f 1 i i
- m J Q
f l .1-1 S 185. In the more general case, where x lt x 2 . . . x n are not given explicitly in terms of u lt u. 2 . . . u,,, but are" connected with them by n equations of the form j iv.2 * t- n) == ^j -^ 2 1> *^9. *^MJ ^^1* ^2 * * * ^n) ^ = "
- HI- l*^-i *^2 t . *C|jj tt| j Ttg * . . ICji) v ) J. H^*^JJ 9?o ^nj ^^1) ^ *
we get, by 99, dx l dx l dx l du^ du 2 du n du, du du,, dx n dx n . A I "A, xiir. s
