INFINITESIMAL CALCULUS Consequently the Jucobitui of our .system is the continued pro- ,, du, did, did, duct of -j- , -r-^ , . . . -j- ft.i-j d,f., dx n In order to calculate - 1 it is necessary to express j as a func- ctej tion of ,7-j , u.,, . . . n ; and similarly for it., , i< 3 , &c. 103. For example, let it be required to find the Jacobian of the system a = r cos 9 L , ,r., = r sin 0j cos 6. 2 , ,- 3 = r siu 0j sin 0., i-os 3 , a;,,_i = r sin Q i sin 0., ... cos ;i -i, x u = r sin 0j sin 0., ... sin 6,,-i-. Here, squaring and adding, we get
- > i 2 + ;V 2 2 + + r 2 = 2
We shall employ this instead of the last equation of the system. Hence, adopting the conditions laid down in 102, we get -^ = - r sin 0, , ^ 2 = -r sin ^ sin 0., ,. .. &c., -^ = - . (/#! </0 a ^/ * Accordingly, the Jacobian of the system is r" sin" -10, sin"- -0., . . . sin0,,_i ~~lc7~ = ( _i)<i-i r n-i S iu"--0 1 sin"- 3 0., . . . sin0,,-2. 104. Again, suppose it, , u. 2 , - . . u,, to be the partial derived functions of a given function of the variables x^ , . ., , . . . x,, ; i.e., let (If nf Ui = i L t! -2 = -r ( i df ~ 7 j iv j y w ll F dx l dx , dx,, The Jacobian becomes d-f_ d-f d-f dff dxjdx.j dx^iXn d-f d-y __^!/_ dx. 2 dx l dx./ dx/lxu d ~f Such a determinant is called the Hessian of the function f(.i. , x. 2 , H)> after Hesse, who first introduced such determin ants into analysis, applying them in many investigations of funda mental importance in the theory of curves and surfaces. 105. Again, in the Jacobian d(X l , X. 2 . . . X H ) if the functions i/ 1 , y.,, . . . are fractions with the same deno minator, i.e., such that we have Hence 0, rf?! du z , W -T- 1 - - j -r . (Wj rt.J j o, r?j rZ rfM du KH , U -7 - Un , ax^ a.t j iix a " " From this, by elementary properties of determinants, we get ,, u i n+ i d (lli^y-^^JlA = d(x 1 ,x. 2 ... x,,) Hence du du 76 , U dx^ dXn di^ du } 11 1 , U dx l U dx~ n du,, did, 1l , U u ~J~T U ! . U n du du l dll n dx l dx,- dx 1 du du^ dlln dx n dXn dXn This latter determinant has been denoted by K(>i, u lt . . , it,,). It possesses interesting properties. For example, if -u, u lt ... are connected by any homogeneous relation, then K(, H! , . . . ii, t ) = 0. Tliis follows from 98, since the quantities y 1} ij. 2 , . . . y n are in this case connected by an equation. It is seen without difficulty that Jacobians and Hessians an; covariants. That is, if the functions be transformed by linear sub stitution ( 95), the Jacobian of the transformed functions is equal to the original Jacobian multiplied by the modulus of transforma tion ; and similarly the Hessian of the, transformed function is equal to that of the original function multiplied by the square of the modu lus. It can also be seen that, when the transformation is orthogonal, the Jacobian and Hessian are unaltered by the transformation. PART II. INTEGRAL CALCULUS. 106. The integral calculus may be said to have taken its origin from the -methods employed by Cavalieri, Wallis, and others, for the determination of the quadrature of curves and the cubature of surfaces. These methods, as we have seen, consisted in the division of the required area, or volume, into an indefinite number of thin slices, or "elements" ; and then from the law connecting their suc cessive values the sum of all the elements was determined or rather the limit" to which that sum approached when the number of elements was indefinitely increased. The processes thus employed were developed and reduced to a suitable notation by Newton and Leibnitz. Thus, adopting the more modern nomenclature, if <p(x) be a function of x which is finite for all values of x between the limits x and X, and if we suppose the interval X-a- divided into n parts, x l -x (} , x. 2 -x lt x 3 -x. 2 , . . . X -_], then, multiplying each element by the corresponding value of the function, i.e., 9> i ~ a o by 0( a o)> & c - > the sum S = (.>-! - .r ( ,)0(,r ) + (x., - r.Jtfa) ... + (X - a,">-i)0(a;-i) has, by elementary algebra, a finite value,, which may be repre sented by (X -a o)<(), where <() lies between the greatest and the least value <(>(x) admits of between the limits. If, now, we suppose the number of elements increased beyond limit, so that x 1 x a , a -j-a?!, &c., may be regarded as each becoming indefinitely small, then ultimately the value of S attains to a certain limit, which depends only on the form of the function <f>(x), and on the extreme values X and x . In this stage, introducing the sym bol of integration /, and adopting the notation / cj)(x)dx, instead J Jxo of S, we write r in which Olios between and 1. For greater simplicity, it is usual to suppose that the increments
- r i ~ a o> s -2 ~ x X - u 1 ,, _ i are all equal. In this case their common
value h is equal to the fraction ; and S becomes h {foo) +/(*o + V +/(*o + 2/0 + ... +/( X - 2/0 +/(X - h)} . Again f |x + (X - x ) > represents the mean value of f(x), as x proceeds by equal infinitesimal increments from the value a; to X. The application of the integral calculus to the solution of questions on mean or average values is founded on the result here given. Thus, denoting the mean value of <p(x), between the limits X and tf , by M0(.r), we have .- -_ - A. ,< 107. If in the definite interal </>(x)dx the upper limit X be
conceived to vary, x remaining constant, the integral itself will vary ; and if we replace X, regarded as variable, by x, the integral may be regarded as a new function, F(z), of x, determined by the equation F(j-) = /~ *0 W d.r = (x - .r ) [a- u + 0(s - ; )] . This function vanishes when X = SL Q ; . . F(A O ) = 0. Also, by the differential calculus ( 46) we have F(.t-) = (a: - xJF ^ + e(x - A- O )] . Consequently 0[.-o + 6(x - a;,,)] = F [^ + 6(x - a- )]. Again, making x=-x w we get 4>K)= F k>) ; XIII. <;
