4
VIII – Cauchy Theory
(ii) Differential calculus in R
2 . Let U be an open subset of R
2 and f a
map from U to R or R
2 . It is said to be differentiable at a point c ∈ U if,
h ∈ R
2 being a varying vector, f (c + h) − f (c) is “ approximately linear ” in
h for sufficiently small h; to be precise, there needs to be a linear map from
R
2 to R or R
2 , the tangent map to f (or derivative, or differential of f ) of f
at c, written f
(c), such that
f (c + h) = f (c) + f
(c)h + o(h)
as the length |h| of the vector h tends to 0; hence
f
(c)h =
d
dt
f (c + th)
fort = 0 .
(1.1)
If c = (a, b), then
f (a + u, b + v) = f (a, b) + pu + qv + o(|u| + |v|) ,
where the coefficients p, q, elements of R or R
2 as the case may be, do not
depend on u, v. These are the partial derivatives
3
p = D 1 f (c) = lim
u=0
f (a + u, b) − f (a, b)
u
,
q = D 2 f (c) = lim
v=0
f (a, b + v) − f (a, b)
v
of f at c. Hence, if h = (u, v) ∈ R
2 , then
f
(c)h = D 1 f (c)u + D 2 f (c)v .
(1.2)
In the case of a function with values in R
2
, if we set f (x, y) = (f 1 (x, y), f 2 (x,
y)), then for c = (a, b), the function f
(c), therefore, maps h = (u, v) to the vector
f
(c)h = D 1 f (c)u + D 2 f (c)v =
(1.3)
= (D 1 f 1 (c), D 1 f 2 (c)) u + (D 2 f 1 (c) , D 2 f 2 (c)) v =
= (D 1 f 1 (c)u + D 2 f 1 (c)v, D 1 f 2 (c)u + D 2 f 2 (c)v) .
Conversely, if the partial derivatives exist for all c ∈ U and are continuous on
U , in which case f is said to be of class C
1 in U , then f is differentiable at
all points of U . If D 1 f and D 2 f are also of class C
1 , f is said to be of class
C
2 , and so on. We then have
D 1 D 2 f = D 2 D 1 f .
(1.4)
3 A notation such as D1f (c) will always denote the value of the function D1f at
c.
VIII – Cauchy Theory
(ii) Differential calculus in R
2 . Let U be an open subset of R
2 and f a
map from U to R or R
2 . It is said to be differentiable at a point c ∈ U if,
h ∈ R
2 being a varying vector, f (c + h) − f (c) is “ approximately linear ” in
h for sufficiently small h; to be precise, there needs to be a linear map from
R
2 to R or R
2 , the tangent map to f (or derivative, or differential of f ) of f
at c, written f
(c), such that
f (c + h) = f (c) + f
(c)h + o(h)
as the length |h| of the vector h tends to 0; hence
f
(c)h =
d
dt
f (c + th)
fort = 0 .
(1.1)
If c = (a, b), then
f (a + u, b + v) = f (a, b) + pu + qv + o(|u| + |v|) ,
where the coefficients p, q, elements of R or R
2 as the case may be, do not
depend on u, v. These are the partial derivatives
3
p = D 1 f (c) = lim
u=0
f (a + u, b) − f (a, b)
u
,
q = D 2 f (c) = lim
v=0
f (a, b + v) − f (a, b)
v
of f at c. Hence, if h = (u, v) ∈ R
2 , then
f
(c)h = D 1 f (c)u + D 2 f (c)v .
(1.2)
In the case of a function with values in R
2
, if we set f (x, y) = (f 1 (x, y), f 2 (x,
y)), then for c = (a, b), the function f
(c), therefore, maps h = (u, v) to the vector
f
(c)h = D 1 f (c)u + D 2 f (c)v =
(1.3)
= (D 1 f 1 (c), D 1 f 2 (c)) u + (D 2 f 1 (c) , D 2 f 2 (c)) v =
= (D 1 f 1 (c)u + D 2 f 1 (c)v, D 1 f 2 (c)u + D 2 f 2 (c)v) .
Conversely, if the partial derivatives exist for all c ∈ U and are continuous on
U , in which case f is said to be of class C
1 in U , then f is differentiable at
all points of U . If D 1 f and D 2 f are also of class C
1 , f is said to be of class
C
2 , and so on. We then have
D 1 D 2 f = D 2 D 1 f .
(1.4)
3 A notation such as D1f (c) will always denote the value of the function D1f at
c.