§5. Differentiable functions of several variables
273
§5. Differentiable functions of several variables
To conclude this chapter we shall generalise the results obtained in the preceding nO to functions of several real variables. We shall need them occasionally,
mainly d propos holomorphic functions, and almost always for functions of
two variables, i.e. defined on an open subset U of JR.2 or of C. The proofs will
be presented so that they be extended immediately to the general case (see
Vol. III, Chap. IX, §1). Apart from the particular case of holomorphic functions, we shall state few theorems, for the reason that everything contained in
this § is fundamental and in general easy to remember, since they are direct
generalisations of the results for a single variable.
In all that follows, when it is a question of the values taken by the variables
or the functions considered, we shall make no distinction between the complex
number z = x + iy and the vector (x, y) of JR.2; anyhow, this is how complex
numbers are defined. Note that whenever we speak of linear functions or
maps in C, this will almost always be in the real sense, as in any vector space
over JR.; such a map is necessarily of the form
(u, v) I-----> cu + dv
with real variables u and v and constant coefficients e, dEC = JR. 2 . This
point of view generalises to any number of real variables subject to taking
the coefficients e and d as vectors having real coordinates; to exhibit this fact
we must separate the real and imaginary parts of e and d in the preceding
formula, whence a formula which we write in terms of real matrices as
( u) I-----> (a /3) (u) = (au + /3 v ) .
v
,8
v
,u+ 8v
In C, the linear functions in the complex sense are the functions z ~ ez, with
e E C, since C is a vector space of dimension lover the field C; in other
words, these are the maps of the form (*) satisfying the condition
d = ie,
which allows us to exhibit u + iv as a factor; in the form (**) this means that
6 = a, , = -/3. These C-linear functions are also JR.-linear, but too specialised
to feature outside the theory of analytic or holomorphic functions.
19 - Partial derivatives and differentials
Let f = It + ih be a complex-valued function defined and continuous on an
open subset U of JR.2; f is also a map
(x,y) I-----> (It(x,y),h(x,y))
273
§5. Differentiable functions of several variables
To conclude this chapter we shall generalise the results obtained in the preceding nO to functions of several real variables. We shall need them occasionally,
mainly d propos holomorphic functions, and almost always for functions of
two variables, i.e. defined on an open subset U of JR.2 or of C. The proofs will
be presented so that they be extended immediately to the general case (see
Vol. III, Chap. IX, §1). Apart from the particular case of holomorphic functions, we shall state few theorems, for the reason that everything contained in
this § is fundamental and in general easy to remember, since they are direct
generalisations of the results for a single variable.
In all that follows, when it is a question of the values taken by the variables
or the functions considered, we shall make no distinction between the complex
number z = x + iy and the vector (x, y) of JR.2; anyhow, this is how complex
numbers are defined. Note that whenever we speak of linear functions or
maps in C, this will almost always be in the real sense, as in any vector space
over JR.; such a map is necessarily of the form
(u, v) I-----> cu + dv
with real variables u and v and constant coefficients e, dEC = JR. 2 . This
point of view generalises to any number of real variables subject to taking
the coefficients e and d as vectors having real coordinates; to exhibit this fact
we must separate the real and imaginary parts of e and d in the preceding
formula, whence a formula which we write in terms of real matrices as
( u) I-----> (a /3) (u) = (au + /3 v ) .
v
,8
v
,u+ 8v
In C, the linear functions in the complex sense are the functions z ~ ez, with
e E C, since C is a vector space of dimension lover the field C; in other
words, these are the maps of the form (*) satisfying the condition
d = ie,
which allows us to exhibit u + iv as a factor; in the form (**) this means that
6 = a, , = -/3. These C-linear functions are also JR.-linear, but too specialised
to feature outside the theory of analytic or holomorphic functions.
19 - Partial derivatives and differentials
Let f = It + ih be a complex-valued function defined and continuous on an
open subset U of JR.2; f is also a map
(x,y) I-----> (It(x,y),h(x,y))