274
III - Convergence: Continuous variables
of U into 1R 2 . For (a, b) E U given, the functions x t--t f(x,b) and y t--t f(a,y)
are defined on a neighbourhood of x = a and of y = b respectively; in fact, the
set of x E IR such that (x, b) E U is a open subset of 1R: apply the definitions.
If they have derivatives at a and b we shall denote them by Dd (a, b) and
D2! (a, b) or, exceptionally, f~ (a, b) and f~ ( a, b), and one meets everywhere
the cumbersome Jacobi notation 8 f j 8x. We shall say that f is of class C1
in U if Dd and D2f exist for any (a, b) E U and are continuous on U. If
Dd and D2! are in their turn of class C1, in which case one says that f is
of class C 2 , one can define the second partial derivatives
(19.1)
DU = f;x = 8 2 f/8x 2 ,
f;x = 8 2 f j8x8y,
D2Dd, DU, etc.
A point of view closer to that of n O 14 consists of extending the concept of
the differential or of the linear tangent map to f at (a, b). Here one attempts
to approximate f(a + u, b + v) - f(a, b) by a lR-linear function of (u, v), i.e.
of the form cu + dv with constants c, dEC, i.e. to write
f(a + u,b +v) = f(a,b) + cu + dv+?
where the error indicated by a ? must be "negligible" with respect to the
dominating term cu + dv when u and v tend to o. The solution is that it
must be negligible with respect to the distance of the point (a, b) from the
point (a+u,b+v), i.e. with respect to lui + Ivl::=:: (u 2 +v 2 )! = lu+ivl. The
preceding relation can then be written as
(19.2)
f(a + u,b + v) = f(a,b) + cu + dv + o(lul + Ivl)
and implies
f(a + u, b) = f(a, b) + cu + o(lul),
whence the existence of Dd(a, b) = c and, similarly, of D2!(a, b) = d. When
the condition (2) is satisfied one says that f is differentiable at the point
(a, b); the linear function (u, v) t--t cu+dv is called, depending on the author,
the differential or the linear tangent map or the derivative of f at (a, b), and
denoted by
(19.3)
df(a, b) : (u, v) t--t Dd(a, b)u + D2!(a, b)v,
or D f (a, b), or even ff (a, b). We can now write (2) in a much more concise
way:
(19.2')
f(z + h) = f(z) + Df(z)h + o(h)
on putting z = (a,b) E U, h = (u,v) E lR.2, and agreeing to write Df(z)h for
III - Convergence: Continuous variables
of U into 1R 2 . For (a, b) E U given, the functions x t--t f(x,b) and y t--t f(a,y)
are defined on a neighbourhood of x = a and of y = b respectively; in fact, the
set of x E IR such that (x, b) E U is a open subset of 1R: apply the definitions.
If they have derivatives at a and b we shall denote them by Dd (a, b) and
D2! (a, b) or, exceptionally, f~ (a, b) and f~ ( a, b), and one meets everywhere
the cumbersome Jacobi notation 8 f j 8x. We shall say that f is of class C1
in U if Dd and D2f exist for any (a, b) E U and are continuous on U. If
Dd and D2! are in their turn of class C1, in which case one says that f is
of class C 2 , one can define the second partial derivatives
(19.1)
DU = f;x = 8 2 f/8x 2 ,
f;x = 8 2 f j8x8y,
D2Dd, DU, etc.
A point of view closer to that of n O 14 consists of extending the concept of
the differential or of the linear tangent map to f at (a, b). Here one attempts
to approximate f(a + u, b + v) - f(a, b) by a lR-linear function of (u, v), i.e.
of the form cu + dv with constants c, dEC, i.e. to write
f(a + u,b +v) = f(a,b) + cu + dv+?
where the error indicated by a ? must be "negligible" with respect to the
dominating term cu + dv when u and v tend to o. The solution is that it
must be negligible with respect to the distance of the point (a, b) from the
point (a+u,b+v), i.e. with respect to lui + Ivl::=:: (u 2 +v 2 )! = lu+ivl. The
preceding relation can then be written as
(19.2)
f(a + u,b + v) = f(a,b) + cu + dv + o(lul + Ivl)
and implies
f(a + u, b) = f(a, b) + cu + o(lul),
whence the existence of Dd(a, b) = c and, similarly, of D2!(a, b) = d. When
the condition (2) is satisfied one says that f is differentiable at the point
(a, b); the linear function (u, v) t--t cu+dv is called, depending on the author,
the differential or the linear tangent map or the derivative of f at (a, b), and
denoted by
(19.3)
df(a, b) : (u, v) t--t Dd(a, b)u + D2!(a, b)v,
or D f (a, b), or even ff (a, b). We can now write (2) in a much more concise
way:
(19.2')
f(z + h) = f(z) + Df(z)h + o(h)
on putting z = (a,b) E U, h = (u,v) E lR.2, and agreeing to write Df(z)h for