174
IX – Multivariate Differential and Integral Calculus
differential form ω on U can be deduced from and f , the inverse image
26
of under f , which could also be called the composition of ω and f ; the
nature of this operation is imposed by the necessity of making it well-defined.
Some authors denote it by f
∗ () or
t f () ; this barbarian notation presents
occasional advantages, but no one has ever denoted the composite q ◦ f of
two functions by f
∗ q or
t f (q). So we will denote the inverse image by the
notation
◦ f : (x; h) −→ [f (x); f
(x)h]
(6.5)
which, in Leibniz style, can be written
◦ f (x; dx) = [f (x); df (x)] = [f (x); f
(x)dx]
(6.6)
in a similar way as the differentiation theorem of composite functions; conversely, it can now be written as
dg ◦ f = d(g ◦ f ) .
(6.7)
A curvilinear integral can then be interpreted in the following manner.
Thanks to the path μ : I −→ E, ω is transformed into a differential form
ω ◦ μ on I, given more explicitly by
ω ◦ μ(t; dt) = ω [μ(t); μ
(t)] dt .
This is precisely the expression integrated over I in order to integrate ω
along μ. In dimension one, any differential form can be written p(t)dt with
a function p(t), and the extended integral of p over I is just the integral of
the form p(t)dt along the rather commonplace path t → t. Hence, in Leibniz
style, we get the formula
μ
ω(x; dx) =
I
ω ◦ μ(t; dt) .
(6.8)
Now consider three open subsets U , V and W in the Cartesian spaces
E, F and G and two maps f : U −→ V and g : V −→ W . So there is a
composite map g ◦ f : U −→ W . Given a form ω on W , a form ω ◦ g on V ,
then a form (ω ◦ g) ◦ f on U can be successively defined or, a form ω ◦ (g ◦ f )
on U can be directly defined. Like in set theory, in this case,
(h ◦ g) ◦ f = h ◦ (g ◦ f )
trivially holds. We have here an associativity formula
(ω ◦ g) ◦ f = ω ◦ (g ◦ f ) .
(6.9)
26 The “ direct ” image would consist in deducing from f and from a form ω on U
a form on V . This is impossible to do if f is not a diffeomorphism.
IX – Multivariate Differential and Integral Calculus
differential form ω on U can be deduced from and f , the inverse image
26
of under f , which could also be called the composition of ω and f ; the
nature of this operation is imposed by the necessity of making it well-defined.
Some authors denote it by f
∗ () or
t f () ; this barbarian notation presents
occasional advantages, but no one has ever denoted the composite q ◦ f of
two functions by f
∗ q or
t f (q). So we will denote the inverse image by the
notation
◦ f : (x; h) −→ [f (x); f
(x)h]
(6.5)
which, in Leibniz style, can be written
◦ f (x; dx) = [f (x); df (x)] = [f (x); f
(x)dx]
(6.6)
in a similar way as the differentiation theorem of composite functions; conversely, it can now be written as
dg ◦ f = d(g ◦ f ) .
(6.7)
A curvilinear integral can then be interpreted in the following manner.
Thanks to the path μ : I −→ E, ω is transformed into a differential form
ω ◦ μ on I, given more explicitly by
ω ◦ μ(t; dt) = ω [μ(t); μ
(t)] dt .
This is precisely the expression integrated over I in order to integrate ω
along μ. In dimension one, any differential form can be written p(t)dt with
a function p(t), and the extended integral of p over I is just the integral of
the form p(t)dt along the rather commonplace path t → t. Hence, in Leibniz
style, we get the formula
μ
ω(x; dx) =
I
ω ◦ μ(t; dt) .
(6.8)
Now consider three open subsets U , V and W in the Cartesian spaces
E, F and G and two maps f : U −→ V and g : V −→ W . So there is a
composite map g ◦ f : U −→ W . Given a form ω on W , a form ω ◦ g on V ,
then a form (ω ◦ g) ◦ f on U can be successively defined or, a form ω ◦ (g ◦ f )
on U can be directly defined. Like in set theory, in this case,
(h ◦ g) ◦ f = h ◦ (g ◦ f )
trivially holds. We have here an associativity formula
(ω ◦ g) ◦ f = ω ◦ (g ◦ f ) .
(6.9)
26 The “ direct ” image would consist in deducing from f and from a form ω on U
a form on V . This is impossible to do if f is not a diffeomorphism.