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

220_ Functions of Several Variables_[C/i. 6
Sec. 12. Taylor's Formula for a Function of Several Variables
Let a function f (x, y) have continuous partial derivatives of all orders
up to the (rc+l)th inclusive in the neighbourhood of a point (a, b). Then
Taylor's formula will hold in the neighbourhood under consideration:
where
or
In other notation,
-j
) + -Jy [/tfX if
2
df (x, y) + rf . . .
/ (x, y) +
-a) + f' y (a, b)(y-b)] +
...+[^^
The particular case of formula (1), when a b Q,
is called Maclaunn's
formula.
Similar formulas hold for functions of three and a larger number of
variables.
Example.' Find the increment obtained by the function f (x, y) = A 3
when passing from the values *= 1, y 1 to the values *,-- 1 -{-//,
Solution. The desired increment may be found by applying formula (2).
First calculate the successive partial derivatives and their values at the
given point (1, 2):
x f /;il,2)=-6.4
/; (1,2) = 3- 1+3.2=9,
+ 3.1 = -21 f
fxx (\, 2) = 6-l=6 f
^(1,2)= -12.2= -24,