INFINITESIMAL CALCULUS Hence dv dy dv dy dv d9 dr dr di) dx dx dtf dx dy d9 dr dr do dv dx* dv dx d0 dr dr dO dv dij dr dy dx "drd9 d 2 v In like manner - , $ can be deduced, but their general values dx* dy* are too complicated for insertion here. A case which commonly arises is in the transformation from rectangular to polar coordinates. In this case, we have x*=r cos , y = r sin , and dv n dv sin dv -j- cos 0^ ~ j^ dx dr r d9 dv n dv cos dv -j- = sin 0-3- + - -j-;- dy dr r d9 d-v ( d sin df .dv sin dv tleuce -,,= cos 0-r - r^ cos 9-. -*: 1 dx* dr r d0/ dr r dQ J d 2 v 2 sin 9 cos dv d 2 v 2 9 dv sin 2 d 2 v The corresponding value of is got by substituting -- - instead dy* & of in the last equation. Hence wo easily find d*v d?v d*v 1 dv 1 d?v dy? dy * dr 2 r dr r 2 dff 2 95. Another important case, which is of extensive application in geometry, is that of linear transformations. i Let us consider the case of three variables, and suppose a; = aX + SY + cZ, y = a X + b Y + c Z , s = a"X + &"Y + c"Z , dv dv then , dv ,,dv dX. " dx T l dy dz dv .dv 7 , dv . ,.dv -rn. = b -T- + b -j- -f V -r- , d dx dy dz dv _ dv ,dv ,,dv Tnr == ^ ~r T ^ T r ~z aZ dx dy dz d?v ( d , d o-=- -I- a -j-
dx dy
4* v , o = a?-r-., + 2aa + a d-v -^ dz dxdy dxdz ,d*v d?v dydz Again, if we suppose x , y , z to be transformed by a similar sub stitution, i.e., then, if any function u = <p(x, y, z) transform into ^> 1 (X, Y, Z), we shall have <fr(x + kx , y + fcy , z + fc; ) = 1 (X + &X , Y + &Y , Z + kZ ). If these be expanded, and like powers of k at both sides be equated, we have fid . d . d f v, d , T . d . d , (tf -r + tf-r- +sf-r J* - x Sv + Y j^ + z TV ) u
dx J dy dz) dK dY dZJ
jt^ 2 _ /v Y -^. dz ) rfX d Consequently the functions x c -4- +if -4- + z ^- , &c., are unaltered dx dy dz by linear transformation. These functions have important geo metrical relations with the original function. Many applications of these principles will be found in Salmon s Higher Plane Curves, as also in his Geometry of Three Dimensions. A few additional examples are added for illustration. (1) If a; = tan 0, d"y 2% dy y _ 7 * 1 i -.9 -7-. *"~ 7~1 -.OV} ^ transforms into dx* 1 + x 2 dx (2) If z be a function of x and y, and u=px + qy-z, prove that when p and q are taken as independent variables we have du dw d?u t d 2 u -s d*u r dp ==x> Jq^V* ~<ffi~rt-s ^ = where p, q, r, s, t denote the partial differential coefficients of z with respect to x and y, of the first and second orders. (3) In the linear transformations in 95 the determinant (ab c") is called the modulus of transformation, and the transformation is said to be orthogonal when In this case the determinant d 2 ii d-u d-u dx 2 dxdy dxdz d"*u d 2 u cPu dxdy dy 3 dydz d 2 u d?io d 2 it dxdz dydz dz 3 is unaltered by the transformation. Jacobians. 96. We now proceed to a short treatment of a remarkable class of determinants first studied by Jacobi (De determinant^ us func- tionalibus, Crelle, 1841), in developing important generalizations of the fundamental principles of the differential and integral calculus. If u. 2 , u 3 , u n be functions of n independent variables then the following determinant du l au i dx. 2 dx 3 du. 2 dx 3 dll n du n was called a functional determinant by Jacobi. Sucli determinants are now more usually known as Jacobians, a designation introduced by Professor Sylvester, who largely developed their properties, and gave numerous applications of them in higher algebra, as also in curves and surfaces. The preceding determinant is frequently represented by the abridged notation O*(M! , u z . . . u n ) d(x 1 , x. 2 . . . X H ) The following discussion, for brevity, is limited for the most part to the case of three variables, but it can be readily extended to any number. 97. Altering the notation, we suppose u, v, in to represent func tions of three independent variables, a; y, z ; then (Bertram!, Liouville s Journal, 1851), if we attribute to each variable an in finitely small increment, there will result a corresponding increment for each of the functions. If now we choose arbitrarily a number of different systems of increments, there will result a corresponding number of systems of increments for the functions. Accordingly, representing the increments of x by d^x , d.jc , d 3 x , and similarly for the other variables, we shall have du du du du du dy du du dz Consequently, by the fundamental rule determinants, we shall have for the multiplication of du du du diX, d-,y, diZ dx dy dz
- J
d. 2 x, d. 2 y, d. 2 z x dv dv dv dx dy dz dw dw dw d 3 x, d 3 y, d. A z dx dy dz Let the first determinant be represented by (A), the second by (B), and the third, or Jacotian, by i, and we get J = --. That is to say, the Jacobian is the ratio of the determinant of the system of infinitesimal increments of the functions to that of the incre ments of the variables. This may be regarded as a generalization of the definition of the derived function in the case of a single variable. 98. Again, when the functions u, v, w are connected by any rela tion their Jacobian vanishes. For suppose u, v, w to be connected by an equation F(w, v, ) = ,
