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

202_Functions of Several Variables_[C/i. 6
1920. Find d'z if
1921. Find d*z if
1922. Find d*z if
z~f(u, v), where u = ax, v = by^
z = f(u, v), where u = xe y , v = ye*.
z = e x cos y.
1923. Find the third differential of the function
z = * cos y + y sin x.
Determine all third partial derivatives.
1924. Find df(l 9 2) and d*f(l, 2) if
f(x, y) = x 2
1925. Find d 2
/(0, 0,0) if
Sec. 8. Integration
f(x 9 y, z) = x z
+ xy + y* 4\nx \Q\ny.
of Total Differentials
t. The condition for a total differential. For an expression P (x, y)dx-}~
+ Q(* y)dy> where the functions P (x, y) and Q (x, y) are continuous in a
simply connected region D together with their first partial derivatives, to be
(in D) the total differential ol some function u (x, y), it is necessary and sufficient
that
aq^ap
dx ~~
dy '
Example t. Make sure that the expression
is a total differential of some function, and find that function.
Solution. In the given case, P = 2x + y t Q x+2y. Therefore, ,5 = -- =
= 1, and, hence,
where u is the desired function.
It is given that - = 2jt + #; therefore,
But on the other hand = x + y' (y) = x + 2y, whence q>' (y) = 2y, (p(f/) =
and
Finally we have