ALGEBRA 17 



1.374 Covariant property. If the variables xi, %,...., are transformed 

 by a linear substitution : 



Xi = an l + a2^2 + ...... + #tnn (fc = I, 2, . . . ., W) 



and the functions yi, y 2 , ..... , y of *i> ^ ....... > * become the functions 



T/I, 172, ..... , i?n of 1, 2, ....... , n : 



, i, , ..... , _ , ..... > . a .. 



~ 



or = a/ 



where | an \ is the determinant or modulus of the transformation. 

 For the Hessian, 



1.380 To change the variables in a multiple integral : 



/ = / ....... fF(yi, y*, ..... i yjdyidyt ..... dy n 



to new variables, #1, xz, . . . ., x n when yi, y 2 , ..... , y n are given functions 

 of xi, xz, ..... , x n : 



7= f ....... f fi y " y2 ' ..... ' y "i F 



J J d(*l, ^2, ..... , ^n) 



where F() is the result of substituting x\ t Xz, . . . ., x n for yi, y 2 , . . ., y n 

 in F(yi, y z , ..... , yO- 



PERMUTATIONS AND COMBINATIONS 



1.400 Given n different elements. Represent each by a number, i, 2, 3, , 



w. The number of permutations of the n different elements is, 



nPn = n\ 



e.g., n = 3 : 



(123), (132), (213), (231), (312), (321) =6 = 3! 



1.401 Given n different elements. The number of permutations in groups of 

 r (r<ri), or the number of r-permutations, is, 



P . " ! 

 nPr ~ (n-r)\ 



e.g., n = 4, r = 3 : 



(123) (132) (124) (142) (134) (143) (234) (243) (231) (213) (214) (241) (341) (314) 

 (312) (321) (324) (342) (412) (421) (431) (413) (423) (432) = 24 



