DIFFERENTIATION OF A FUNCTION OF TWO VARIABLES. 101 



130. If x and y denote quantities such that either of 

 them may be considered to change without affecting the other, 

 they are called independent variables, and any quantity u, the 

 value of which depends on the values of x and y, is called a 

 "function of the independent variables x and y" 



du d'-u d*u . 



-j- , -T-3, -73, ..., denote the successive differential co- 



efficients of u, taken on the supposition that x alone varies ; 



du d?u d j u , ,. . 



-y- > -j~s > TS , > denote the successive differential co- 



dy dtf dy* 



efficients of u, taken on the supposition that y alone varies. 



131. If u be a function of the independent variables x 

 and y, then -y- will also be generally a function of x and y. 



Hence we may have occasion for its differential coefficient 

 with respect to x or y. The former is denoted by 



da?' 



as already stated ; the latter is denoted by 



, du 



d ~j~ 

 dx 



dy > 



which is abbreviated into ^ , . 



dydx 



Again, both -^ 2 and j .- will be generally functions 



of both x and y. These may require to be differentiated with 

 respect to x or y. Hence we use such symbols as 



d?u d?u , d?u 



dydx*' dxdydx' dy^dx* 



the meaning of which may be gathered from the preceding 



remarks. For example. -= ^ r- implies the performance 

 dxdydx 



of three operations : we are to differentiate u with respect 



