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

206_Functions of Several Variables_[Ch. 6
Example 2. Find -T- and j- if
Solution. First method. Denoting the left side of this equation by F (x, //, z),
we find the partial derivatives
F' x (x, y, z) = 2x, F' y (x, y, z) = 40-z+l, F z (x, y, 2) = 6z-0.
Applying formulas (2), we get
dz _ F'x(x> y. *) _ 2x
~~
d*~ F' z (x t y, z)
6* y '
dz^ Jy(*. !/, 2) ^ 1 ~"
dy~ ^ (x> ^ z)
Second method. Differentiating the given equation, we obtain
2x dx 4f/ dy + 6zdz y dz zdy + dy = 0.
4// z
62 /
Whence we determine dz t that is, the total differential of the implicit function:
\ 4// z}dy
see that
dz 2x dz \4yz
dx y 6z '
dy y 62
Comparing with the formula dz = dx ~ -\- dy we ,
-Q-
3. A system of implicit functions. If a system of two equations
f F(x, y, u = t i>) 0,
\ G(x, y, u, o) =
defines u and v as functions of the variables x and y
D(F, G)
D(u, v)'
dF_ dF_
dudv
dGdG
du dv
and the Jacobian
then the differentials of these functions (and hence their partial derivatives
as well) may be found from the following set of equations
Example 3. The equations
'dF dF . dF . dF . .
,, t ,
,
^- dx + -z- dy + -^~ du + ^- dv =0,
,
dx ^ dy y ^du ^ dv /Q .
dG
du
3- -r~ l y *
dx
-^---' 3-
dy dv
define u and v as functions of x and w; find -,,- and rr? .
^
dx dy dx dy