234
IX – Multivariate Differential and Integral Calculus
associating to each h ∈ X
(x) its “ initial point ” x, we get a canonical map
p : T (X) −→ X. The set T (X) can easily be made into a manifold. First, if
(U, ϕ) is a chart of X, the inverse image p
−1 (U ) is the set T (U ) of tangent
vectors to the manifold U ; if n = dim(X), the map ϕ
: (x, h) → (ϕ(x), h(ϕ))
from T (U ) to the Cartesian product ϕ(U ) × R
n is bijective. Formulas for
chart change of section (i) of the present n
◦ show that, if (V, ψ) is another
chart of X and if the chart change is of class C
r on U ∩ V , then the image of
(ϕ(x), h(ϕ)) is obviously (ψ(x), h(ψ)) under a C
r−1 map. This gives a C
r−1
structure on T (X) = X
: the open subsets Ω of T (X) are defined by the
condition that, for any chart (U, ϕ) of X, the image of Ω ∩ T (U ) under ϕ
is
an open subset of ϕ(U ) × R
n , so that the couples (T (U ), ϕ
) become charts
of class C
0 for T (X) ; as they constitute an atlas of class C
r−1 for T (X), the
definition of the manifold T (X) follows.
To any homomorphism f : X −→ Y of manifolds can be associated a
homomorphism f
: X
−→ Y
, namely
f
: (x, h) −→ (f (x), f
(x)h) .
(12.16)
If there is another homomorphism g : Y −→ Z, then setting p = gf , the
multivariate chain rule shows that
p
= g
◦ f
.
(12.17)
Indeed, f
maps (x, h) to (f (x), f(x)h), which is then mapped by g
to
(g(f (x)), g
(f (x))f
(x)h) . It, therefore, remains to check that p(x) = g(f (x)),
which is trivial, and that p
(x) = g
(f (x)) ◦ f
(x).
For a product manifold Z = X × Y , there is a canonical isomorphism
Z
= X
× Y
since, for x ∈ X and y ∈ Y , the tangent space T (x,y) (Z) was
identified with T x (X) × T y (Y ).
All this is too easy though sometimes convenient, especially in Lie
group theory. Explaining the structure of the manifolds T (T (X)) = T
2 (X),
T (T (T (X))) = T
3 (X), etc. is, however, far less simple. Then, any homomorphism f : X −→ Y has “ extensions ” f
(r) : T
r (X) −→ T
r (Y ) that are
homomorphisms and for which formula (17) becomes
p
(r) = g
(r)
◦ f
(r) .
Interpreted in classical terms this order r multivariate chain rule is also not
obvious. For a start, try to understand the case r = 2.
Exercise. Call an arbitrary basis of a space T x (X) a frame. Construct a
natural manifold structure on the set of frames of X.
IX – Multivariate Differential and Integral Calculus
associating to each h ∈ X
(x) its “ initial point ” x, we get a canonical map
p : T (X) −→ X. The set T (X) can easily be made into a manifold. First, if
(U, ϕ) is a chart of X, the inverse image p
−1 (U ) is the set T (U ) of tangent
vectors to the manifold U ; if n = dim(X), the map ϕ
: (x, h) → (ϕ(x), h(ϕ))
from T (U ) to the Cartesian product ϕ(U ) × R
n is bijective. Formulas for
chart change of section (i) of the present n
◦ show that, if (V, ψ) is another
chart of X and if the chart change is of class C
r on U ∩ V , then the image of
(ϕ(x), h(ϕ)) is obviously (ψ(x), h(ψ)) under a C
r−1 map. This gives a C
r−1
structure on T (X) = X
: the open subsets Ω of T (X) are defined by the
condition that, for any chart (U, ϕ) of X, the image of Ω ∩ T (U ) under ϕ
is
an open subset of ϕ(U ) × R
n , so that the couples (T (U ), ϕ
) become charts
of class C
0 for T (X) ; as they constitute an atlas of class C
r−1 for T (X), the
definition of the manifold T (X) follows.
To any homomorphism f : X −→ Y of manifolds can be associated a
homomorphism f
: X
−→ Y
, namely
f
: (x, h) −→ (f (x), f
(x)h) .
(12.16)
If there is another homomorphism g : Y −→ Z, then setting p = gf , the
multivariate chain rule shows that
p
= g
◦ f
.
(12.17)
Indeed, f
maps (x, h) to (f (x), f(x)h), which is then mapped by g
to
(g(f (x)), g
(f (x))f
(x)h) . It, therefore, remains to check that p(x) = g(f (x)),
which is trivial, and that p
(x) = g
(f (x)) ◦ f
(x).
For a product manifold Z = X × Y , there is a canonical isomorphism
Z
= X
× Y
since, for x ∈ X and y ∈ Y , the tangent space T (x,y) (Z) was
identified with T x (X) × T y (Y ).
All this is too easy though sometimes convenient, especially in Lie
group theory. Explaining the structure of the manifolds T (T (X)) = T
2 (X),
T (T (T (X))) = T
3 (X), etc. is, however, far less simple. Then, any homomorphism f : X −→ Y has “ extensions ” f
(r) : T
r (X) −→ T
r (Y ) that are
homomorphisms and for which formula (17) becomes
p
(r) = g
(r)
◦ f
(r) .
Interpreted in classical terms this order r multivariate chain rule is also not
obvious. For a start, try to understand the case r = 2.
Exercise. Call an arbitrary basis of a space T x (X) a frame. Construct a
natural manifold structure on the set of frames of X.