166
IX – Multivariate Differential and Integral Calculus
§ 2. Differential Forms of Degree 1
4 – Differential Forms of Degree 1
As seen in Chapter VIII, finding a primitive F of a holomorphic function f
on an open set U ⊂ C = R
2 amounts to constructing a C
1 function in U such
that dF = f (z)dz = f (z)dx + if (z)dy, in other words such that
D 1 F = f , D 2 F = if .
A somewhat more general problem is to write as dF an expression like
ω = p(x, y)dx + q(x, y)dy ,
(4.1)
i.e. a differential form of degree 1 on U , where p and q, its coefficients, are
given functions; the coefficients are assumed to be always continuous and in
order to avoid serious complications, it is better to assume they are of class
C
1 at least on U .
The aim is, therefore, to find C
1 functions F on U satisfying
D 1 F = p , D 2 F = q .
(4.2)
If such a primitive F of ω exists in U , ω is said to be an exact diffferential ;
if U is connected, F is unique up to an additive constant (Chap. III, n
◦ 21,
consequence of the mean value theorem for several variables).
If p and q are C
1 , in which case F is C
2 , the relation D 2 D 1 F = D 1 D 2 F
requires
D 1 q = D 2 p ;
(4.3)
if this necessary but not always sufficient condition is satisfied, pdx + qdy is
said to be a closed differential. This terminology is inherited from algebraic
topology and “ Stokes ” type integration formulas. In the holomorphic case
(p = f, q = if ), (3) is just Cauchy’s holomorphic condition.
Another case: it was observed in Chap. VII, n
◦ 24 that if H is a real
harmonic function on U , finding a holomorphic function with real part H
amounts to finding a primitive of the holomorphic function D 1 H − iD 2 H,
hence of the differential form
(D 1 H − iD 2 H) (dx + idy) = dH − i (D 2 Hdx − D 1 Hdy) ,
(4.4)
hence of the differential form D 2 Hdx − D 1 Hdy ; the latter is closed since, by
assumption, ΔH = 0, where Δ = D 1 D 1 + D 2 D 2 is the de Laplace operator
d
2 /dx
2 + d
2 /dy
2 .
There are similar problems in arbitrary dimension n. A differential form
(of degree 1) on an open subset U of an n-dimensional space E with chosen
basis (a i ) is for the moment a purely symbolic expression
IX – Multivariate Differential and Integral Calculus
§ 2. Differential Forms of Degree 1
4 – Differential Forms of Degree 1
As seen in Chapter VIII, finding a primitive F of a holomorphic function f
on an open set U ⊂ C = R
2 amounts to constructing a C
1 function in U such
that dF = f (z)dz = f (z)dx + if (z)dy, in other words such that
D 1 F = f , D 2 F = if .
A somewhat more general problem is to write as dF an expression like
ω = p(x, y)dx + q(x, y)dy ,
(4.1)
i.e. a differential form of degree 1 on U , where p and q, its coefficients, are
given functions; the coefficients are assumed to be always continuous and in
order to avoid serious complications, it is better to assume they are of class
C
1 at least on U .
The aim is, therefore, to find C
1 functions F on U satisfying
D 1 F = p , D 2 F = q .
(4.2)
If such a primitive F of ω exists in U , ω is said to be an exact diffferential ;
if U is connected, F is unique up to an additive constant (Chap. III, n
◦ 21,
consequence of the mean value theorem for several variables).
If p and q are C
1 , in which case F is C
2 , the relation D 2 D 1 F = D 1 D 2 F
requires
D 1 q = D 2 p ;
(4.3)
if this necessary but not always sufficient condition is satisfied, pdx + qdy is
said to be a closed differential. This terminology is inherited from algebraic
topology and “ Stokes ” type integration formulas. In the holomorphic case
(p = f, q = if ), (3) is just Cauchy’s holomorphic condition.
Another case: it was observed in Chap. VII, n
◦ 24 that if H is a real
harmonic function on U , finding a holomorphic function with real part H
amounts to finding a primitive of the holomorphic function D 1 H − iD 2 H,
hence of the differential form
(D 1 H − iD 2 H) (dx + idy) = dH − i (D 2 Hdx − D 1 Hdy) ,
(4.4)
hence of the differential form D 2 Hdx − D 1 Hdy ; the latter is closed since, by
assumption, ΔH = 0, where Δ = D 1 D 1 + D 2 D 2 is the de Laplace operator
d
2 /dx
2 + d
2 /dy
2 .
There are similar problems in arbitrary dimension n. A differential form
(of degree 1) on an open subset U of an n-dimensional space E with chosen
basis (a i ) is for the moment a purely symbolic expression