158
IX – Multivariate Differential and Integral Calculus
Then the map ϕ is (x, y) → (r, θ) and it is a diffeomorphism from U onto the
open subset of R
2 defined by the inequalities imposed on r and θ ; its inverse
map is f (r, θ) = (r. cos θ, r. sin θ). Then, to say that a function p defined on
an open set G ⊂ U is of class C
1 means that
p(x, y) = P (r, θ)
with a C
1 function P defined on the open subset ϕ(G). The differential of
f is (cos θ.dr − r sin θ.dθ, sin θ.dr + r cos θ.dθ) and, for ξ = (r, θ), the vectors
a i (ξ) are obtained by replacing (dr, dθ) either by (1, 0) or by (0, 1) ; hence
a 1 (ξ) = (cos θ, sin θ) ,
a 2 (ξ) = (−r sin θ, r cos θ) .
Exercise. In R
3
− {0}, use spherical coordinates defined by
x = r cos ϕ. cos θ , y = r sin ϕ. cos θ , z = r sin θ ,
with r > 0, 0 ≤ ϕ < 2π, |θ| ≤ π/2. (It is not exactly a diffeomorphism onto
an open set, but this does not matter). Calculate the a i (ξ).
The basic idea of classical tensor analysis is to use for all calculations at
the point x = f (ξ) what ´
Elie Cartan called a moving frame (a i (ξ)), instead
of a basis for E chosen once for all. This amounts to regarding a function
of x ∈ U as a function of the corresponding point ξ = ϕ(x) of U
and to
calculating in the canonical basis of R
n ; we will see (§ 4) that this is also
what we are forced to do in “ curved spaces ”, i.e. the differential manifolds of
the modern theory, since they are not contained in a Cartesian space whose
basis we could choose.
For example, let us consider a vector h ∈ E and calculate its components
h
i (ξ) with respect to the basis a i (ξ) = f
(ξ)e i attached to the point x = f (ξ).
Set ϕ(x) = ϕ
i (x)e i , so the coordinates of x in the chart (U, f ) are ξ
i = ϕ
i (x) ;
taking (3) and f
(ξ) = ϕ
(x)
−1 into account,
h = f
(ξ)ϕ
(x)h = f
(ξ)dϕ(x; h) = f
(ξ)dϕ
i (x; h)e i = dϕ
i (x; h)a i (ξ) .
This gives the coordinates h
i (ξ) = dϕ
i (x; h) sought and the relation
h = dϕ
i (x; h)a i (ξ)
(3.4)
that ´
Elie Cartan, following Leibniz, used to write as
dx = a i (ξ)dξ
i ,
(3.4’)
the expression dξ
i in these formulas being the differential of the i
th “curvilinear ”
coordinate of x → ξ
i = ϕ
i (x) considered a function of x This relation merely
expresses the fact that the a i (ξ) are the partial derivatives D i f (ξ) with respect to the coordinates ξ
i of the inverse map x = f (ξ) of ϕ.
IX – Multivariate Differential and Integral Calculus
Then the map ϕ is (x, y) → (r, θ) and it is a diffeomorphism from U onto the
open subset of R
2 defined by the inequalities imposed on r and θ ; its inverse
map is f (r, θ) = (r. cos θ, r. sin θ). Then, to say that a function p defined on
an open set G ⊂ U is of class C
1 means that
p(x, y) = P (r, θ)
with a C
1 function P defined on the open subset ϕ(G). The differential of
f is (cos θ.dr − r sin θ.dθ, sin θ.dr + r cos θ.dθ) and, for ξ = (r, θ), the vectors
a i (ξ) are obtained by replacing (dr, dθ) either by (1, 0) or by (0, 1) ; hence
a 1 (ξ) = (cos θ, sin θ) ,
a 2 (ξ) = (−r sin θ, r cos θ) .
Exercise. In R
3
− {0}, use spherical coordinates defined by
x = r cos ϕ. cos θ , y = r sin ϕ. cos θ , z = r sin θ ,
with r > 0, 0 ≤ ϕ < 2π, |θ| ≤ π/2. (It is not exactly a diffeomorphism onto
an open set, but this does not matter). Calculate the a i (ξ).
The basic idea of classical tensor analysis is to use for all calculations at
the point x = f (ξ) what ´
Elie Cartan called a moving frame (a i (ξ)), instead
of a basis for E chosen once for all. This amounts to regarding a function
of x ∈ U as a function of the corresponding point ξ = ϕ(x) of U
and to
calculating in the canonical basis of R
n ; we will see (§ 4) that this is also
what we are forced to do in “ curved spaces ”, i.e. the differential manifolds of
the modern theory, since they are not contained in a Cartesian space whose
basis we could choose.
For example, let us consider a vector h ∈ E and calculate its components
h
i (ξ) with respect to the basis a i (ξ) = f
(ξ)e i attached to the point x = f (ξ).
Set ϕ(x) = ϕ
i (x)e i , so the coordinates of x in the chart (U, f ) are ξ
i = ϕ
i (x) ;
taking (3) and f
(ξ) = ϕ
(x)
−1 into account,
h = f
(ξ)ϕ
(x)h = f
(ξ)dϕ(x; h) = f
(ξ)dϕ
i (x; h)e i = dϕ
i (x; h)a i (ξ) .
This gives the coordinates h
i (ξ) = dϕ
i (x; h) sought and the relation
h = dϕ
i (x; h)a i (ξ)
(3.4)
that ´
Elie Cartan, following Leibniz, used to write as
dx = a i (ξ)dξ
i ,
(3.4’)
the expression dξ
i in these formulas being the differential of the i
th “curvilinear ”
coordinate of x → ξ
i = ϕ
i (x) considered a function of x This relation merely
expresses the fact that the a i (ξ) are the partial derivatives D i f (ξ) with respect to the coordinates ξ
i of the inverse map x = f (ξ) of ϕ.