(^638) Series
(7) can be put in the form
(9) fck)(t) =[(x- xo)a + (Y- yo) ay] G
when k = 2. It turns outthatTaylor formulas with remainders are obtained
by substituting (9) into (1), the partial derivatives being evaluated at (xo,yo)
when k 5 n and at (x,y) when k = n + 1.
7 LetG be a function having continuous partial derivatives of first and second
orders over a neighborhood of (xo,yo) in which (x,y) is supposed to lie. The
Taylor formula of Problem 6 which terminates with second derivatives then takes
the form
(1) G(x,Y) = G(xo,Yo) + G.(xo,Yo)(x - xo) + G.(xo,yo)(Y - Yo)
-1- '(G:=(x,Y)(x - xo)2 + 2Gzv(x,Y)(x - xo)(y - yo) +GYU(x,Y)(Y - Yo)2I,
where x lies between xo and x and y lies between yo and y except that x = xo
when x = xo and y = yo when y = yo. This formula is useful. For example,
it provides an easy way of estimating the difference between G(x,y) and G(xo,yo)
that is especially useful when Ix - xol and Iy - yol are so small that the last term
is negligible in comparison to the two preceding terms. In particular, (1) gives
us a chance to estimate the magnitude of the error involved when the number
dz defined by
(2) dz = G.(xo,Yo) (x - xo) + G, (xo,yo) (y - Yo)
is taken as an approximation to the number Az defined by
(3) As = G(x,y) - G(xo,yo)
It is quite possible to spend a few days solving problems involving these ideas,
and the investment of time might even be a reasonably good one. We invest a
few minutes to study extrema (local and global minima and maxima) of G. If
G(x,y) has an extremum at (xo,yo), then G(x,yo) must have an extremum at xo
and hence G,(xo,yo) = 0. Similarly, if G(x,y) has an extremum at (xo,yo), then
G(xo,y) must have an extremum at yo and hence G cannot
have an extremum at (xo,yo) unless
(4) G.(xo,yo) = G5(xo,Yo) = 0.
To investigate the question whether G has an extremum at a point (xo,yo) for
which (4) holds, we put (1) in the form
(5) G(x,y) - G(xo,yo) _ [(1 + El)h2 + 2(B + e2)hk + (C + Es)k2],
where
(6) -4 = G::(xo,yo), B = G:v(xo,Yo), C = Gvv(xo,Yo),
h = x - xo, k = y - yo, and the numbers el, C2i es dependupon h and k and are
small when jhI and Iki are small. In case .4 0 0 we can, when Jhj and Jkl are
small enough to make .4 + El s 0, put (5) in the form
(7) G(x,y) - G(xo,yo) = 2(1111-
El)] ((A + El)h + (B +
E2)k]2
- [(A' + E1)(C + ES) - (B + 2)2]k2}.^1