Advanced Calculus fi..

98 Advanced Calculus, Fifth EditionAs At approaches 0, Ax/ At and AylAt approach the derivatives d.r/dt and dyldt,respectively. while rl and 62 approach 0, since A.1- and Av approach 0. Hencethat is,as was to be proved.The three functions of t considered here-xh ([)]-have differentials= g(t). y = h(t). : = f[g(t).From (2.33). one concludes thatthat is. thatBut this is the same as (2.24). in whichdl and dy are A.Y and A?, arbitrary incrementsof independent variables. Thus (2.24) holds whether x and y are independent anddz is the corresponding differential or whether x and y, and hence z. depend on t.so that d.r, dv, dz are the differentials of these variables in terms oft.Similar reasoning applies to (2.34). Here u and v are the independent variableson which x, v, and z depend. The corresponding differentials areBut (2.34) givesa2ax + arn)A. + (--a.r az ava.r au ay au a.rav ayav 1dz = (-- +-- A]!Again (2.24) holds. Generalization of this to (2.35) permits one to conclude:THEOREMThe differential formula