Page 149 - Foundations Of Differential Calculus

132 7. On the Differentiation of Functions of Two or More Variables
226. Now, in order to inquire into the general relationship between P and
Q, which constitute the differential Pdx + Qdy of any function V of two
variables x and y, we need to pay attention to what follows. If V is any
function whatsoever of x and y, and we substitute x + dx for x, then V is
transformed into R.If y + dy is substituted for y, then V is transformed
into S. If simultaneously x+dx and y+dy are substituted for x and y, then
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V is changed into V . Since R comes from V when x + dx is substituted
for x, it is clear that if furthermore y + dy is substituted in R, the result is
I
V . It comes to the same thing as substituting x + dx for x and y + dy for
y immediately. In a similar way, if x + dx is substituted for x in S, since
S has already arisen from V by substituting y + dy for y, once again we
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obtain V , as may be seen more clearly from the following table.

The quantity becomes if for we put

V R x x + dx
V S y y + dy
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V V x x + dx
y y + dy

R V I y y + dy
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S V x x + dx

227. If we differentiate V as if x were the only variable and y is treated as a
constant, since we substitute x+dx for x, the function V becomes R, whose
differential will be equal to R−V . From the form dV = Pdx+Qdy it follows
that the same differential will be equal to Pdx, so that R−V = Pdx.Ifwe
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substitute y + dy for y and treat x as a constant, since R becomes V and
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V becomes S; the quantity R − V becomes V − S. Then the differential of
R − V = Pdx, which arises if only y is considered variable, will be equal
to
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V − R − S + V.

In a similar way, when we substitute y+dy for y, V becomes S, so that S−V
is the differential of V if we let only y be variable, so that Q − V = Qdy.
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Now when we substitute x + dx for x, S becomes V and V becomes R,so
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that the quantity S−V becomes V −R and the differential of S−V = Qdy,
which arises when only x is variable, is equal to
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V − R − S + V,
which is equal to the differential we found previously.