§ 1. Classical Differential Calculus
151
with numerical valued functions f
j (x), g
k (y) and p
k (x), we get
p
k (x) = g
k [f (x)] .
(2.16)
On the other hand, relations (6) and (8) show that, for h ∈ E,
dp(x; h) = dg
y; df (x; h)
j b j
= dg (y; b j ) df (x; h)
j =
(2.17)
= D j g
k (y)c k .D i f
j (x)h
i .
Since by (8) applied to p, dp(x; h) = D i p
k (x)h
i c k ,
D i
g
k [f (x)]
= D i p
k (x) = D i f
j (x).D j g
k (y) where y = f (x)
(2.18)
finally follows. This expresses (13) in terms of matrices : the Jacobian matrix
of p at x is the product of the Jacobian matrix of f at x and that of g
at y = f (x). The left hand side of (18) denotes the effect of the operator
D i = d/dx
i on the function x → g
k [f (x)], not to be confused with D i g
k [f (x)],
which is the value of the function D i g
k at f (x) ; this value is not usually
well-defined since the function g
k depends on y and not on x. For example,
the second expression, D i p
k (x), is the value of the function D i p
k at x, and
D j g
k (y) the value of the la fonction
17 D j g
k at y, where D j = d/dy
j . The
presence of a punctuation point in the expression D i f
j (x).D j g
k (y) indicates
that the operator D i applies only to f
j (x) and not to f
j (x)D j g
k (y). These
conventions will be systematically used in order to avoid confusion.
These formulas can be simplified if the function g is real valued; this is
then also the case of p and we get
D i {g [f (x)]} = D j g [f (x)] .D i f
j (x) .
(2.19)
In particular, if (case E = R) there is map, written t → μ(t) rather than f ,
from an interval of R to the domain of definition V of g. So setting D = d/dt,
D {g [μ(t)]} = dg [μ(t); μ
(t)] = D j g [μ(t)] .Dμ
j (t)
(2.20)
at each point t where Dμ(t) = μ
(t) ∈ F exists.
(iii) Partial differentials. Seemingly more complicated composite functions often need to be considered. For example,
p(u) = g [x(u), y(u), z(u)] ,
(2.21)
where u varies in an open subset Ω of a Cartesian space, where the functions
x, y, z map Ω to open subsets U, V, W of three Cartesian spaces E, F, G and
where g is defined on the open subset U × V × W of the Cartesian space
17 Generally, the symbol Dif always denotes the partial derivative of the function f
with respect to the i
th variable on which it depends, irrespective of the letters
used to denote them.
151
with numerical valued functions f
j (x), g
k (y) and p
k (x), we get
p
k (x) = g
k [f (x)] .
(2.16)
On the other hand, relations (6) and (8) show that, for h ∈ E,
dp(x; h) = dg
y; df (x; h)
j b j
= dg (y; b j ) df (x; h)
j =
(2.17)
= D j g
k (y)c k .D i f
j (x)h
i .
Since by (8) applied to p, dp(x; h) = D i p
k (x)h
i c k ,
D i
g
k [f (x)]
= D i p
k (x) = D i f
j (x).D j g
k (y) where y = f (x)
(2.18)
finally follows. This expresses (13) in terms of matrices : the Jacobian matrix
of p at x is the product of the Jacobian matrix of f at x and that of g
at y = f (x). The left hand side of (18) denotes the effect of the operator
D i = d/dx
i on the function x → g
k [f (x)], not to be confused with D i g
k [f (x)],
which is the value of the function D i g
k at f (x) ; this value is not usually
well-defined since the function g
k depends on y and not on x. For example,
the second expression, D i p
k (x), is the value of the function D i p
k at x, and
D j g
k (y) the value of the la fonction
17 D j g
k at y, where D j = d/dy
j . The
presence of a punctuation point in the expression D i f
j (x).D j g
k (y) indicates
that the operator D i applies only to f
j (x) and not to f
j (x)D j g
k (y). These
conventions will be systematically used in order to avoid confusion.
These formulas can be simplified if the function g is real valued; this is
then also the case of p and we get
D i {g [f (x)]} = D j g [f (x)] .D i f
j (x) .
(2.19)
In particular, if (case E = R) there is map, written t → μ(t) rather than f ,
from an interval of R to the domain of definition V of g. So setting D = d/dt,
D {g [μ(t)]} = dg [μ(t); μ
(t)] = D j g [μ(t)] .Dμ
j (t)
(2.20)
at each point t where Dμ(t) = μ
(t) ∈ F exists.
(iii) Partial differentials. Seemingly more complicated composite functions often need to be considered. For example,
p(u) = g [x(u), y(u), z(u)] ,
(2.21)
where u varies in an open subset Ω of a Cartesian space, where the functions
x, y, z map Ω to open subsets U, V, W of three Cartesian spaces E, F, G and
where g is defined on the open subset U × V × W of the Cartesian space
17 Generally, the symbol Dif always denotes the partial derivative of the function f
with respect to the i
th variable on which it depends, irrespective of the letters
used to denote them.