Page 155 - Foundations Of Differential Calculus

138 7. On the Differentiation of Functions of Two or More Variables

1st differential 2nd differential 3rd differential

with respect to x alone y alone z alone
2xy dx 2xdxdy 4xz dx dy dz
2
2
2
2
a − z 2 a − z 2 (a − z ) 2
with respect to x alone z alone y alone
2xy dx 4xyz dx dz 4xz dx dy dz
2
2
2
2
2
a − z 2 (a − z ) 2 (a − z ) 2
with respect to y alone x alone z alone
2
x dy 2xdxdy 4xz dx dy dz
2
2
2
2
a − z 2 a − z 2 (a − z ) 2
with respect to y alone z alone x alone
2
2
x dy 2x zdy dz 4xz dx dy dz
2
2
2
2
2
a − z 2 (a − z ) 2 (a − z ) 2
with respect to z alone x alone y alone
2
2x yz dz 4xyz dx dz 4xz dx dy dz
2
2
2
2
2
2
(a − z ) 2 (a − z ) 2 (a − z ) 2
with respect to z alone y alone x alone
2 2
2x yz dz 2x zdy dz 4xz dx dy dz
2
2
2
2
2
2
(a − z ) 2 (a − z ) 2 (a − z ) 2
From this example it is clear that no matter in what order the variables are
taken, after the three differentiations we always have the same expression
4xz dx dy dz .
2
2
(a − z ) 2
238. Just as after three differentiations we arrived at the same expres-
sion, so we detect some agreement after the second differentiation. Among
these each expression occurs twice. It is clear that those formulas with the
same differentials are equal to each other, and the third differentials are all
equal to each other because they all have the same differentials dx dy dz.
From this we conclude that if V is a function of howsoever many variables
x,y,z,v,u,... and V is differentiated successively the number of times re-
quired so that always only one quantity is variable, then as often as we
arrive at expressions with the same differentials, those expressions will be
equal to each other. Thus, after two differentiations we find an expression
Zdx dy where in one only x is variable and in the other only y is variable,
no matter which is first or second. In a similar way there are six ways by