Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

Sec. ti\_Integration of Total Differentials_203
2. The case of three variables. Similarly, the expression
P(x, y, z)dx + Q(x, y, z)dy + R(x t y, z)dz,
where P (x, y, z), Q(x, y, z), R(x, y, z) are, together with their first partial
derivatives, continuous functions of the variables x, y and 2, is the total
differential of some function u (x t y, z) if and only if the following conditions
are fulfilled:
dQ^W dR^dQ dP^dR
dx dy '
dy dz '
Example 2. Be sure that the expression
dz dx
is the total differential of some function, and find that function.
Solution. Here, p = 3jc f + 30 1, Q=z 2 + 3x, R = 2yz+\. We establish
the fact thai
and, hence,
= = O . =
where u is the sought-for function.
We have
hence,
On the other hand,
-r = c . rr = r = V. /
dQ dP dR dQ dP OR
dx dy dy dz dz dx
u= (3x 2 + 3y \)dx = x* + 3xy x + (#, 2) whose partial derivatives are known and the condition
for total differential is fulfilled.
We find q>:
that is, y(y, e) = ^2 2 + 2 + C, And finally,
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