§5. Differentiable functions of several variables
281
(21.4)
d
dtg(x + tu, y + tv) =
= D1g(x + tu, y + tv)u + D2g(X + tu, y + tv)v
or, in condensed notation,
(21.4')
d
dtg(z + th) = Dg(z + th)h.
Suppose that z + th E V for any t E [0,1]; this means that the line segment
joining the points z and z+h lies entirely in V, as is the case if Ihl is sufficiently
small or, as well, if V is convex. Let us apply Theorem 18, the mean value
theorem, and its corollaries to the function p(t) = g(x + tu, y + tv), in the
form p(l) -p(O) = ... ; Theorem 18 shows that if 9 has real values, then there
exists atE [0, 1] such that
(21.5)
g(x + u, y + v) - g(x, y) =
= D1g(x + tu, y + tv)u + D2g(X + tu, y + tv)v,
(21.5')
g(z + h) - g(z) = Dg(z + th)h;
if 9 has complex values one finds only that
(21.6)
Ig(x + u, y + v) - g(x, y)1 ~
or
(21.6')
~ sup ID1g(x+tu,y+tv)u+D2g(x+tu,y+tv)vl,
09:9
Ig(z + h) - g(z)1 ~ sup IDg(z + th)hl·
Now every linear map A ofR? into]R2 (or of]RP into ]Rq) has a norm IIAII,
namely the least number M ~ 0 such that IIAhl1 ~ Mllhll for all vectors h in
the domain space; if one uses the usual Pythagorean norm, and if
(21.7)
A(U) = (au + bv),
v
cu + dv
so that a, b, e, d are the coefficients of the matrix of A, one finds easily, from
the Cauchy-Schwarz inequality in ]R2, that
(21.8)
IIAII = (a 2 + b 2 + e 2 + d 2 )! ;:::: lal + Ibl + lei + Idl·
If A = Dg(z) as above, the (real) coefficients of the matrix of A are the real
and imaginary parts of D1g(z) and D29(Z), whence
(21.9)
In any case, the formulae used to measure the "length" of a vector and thus
the norm of a linear map or matrix are of little importance, so long as they