Page 89 - Curvature and Homology

whenever the integral converges as will always be the case in the sequel.
(It is assumed that M is orientable and that an orientation of M has been
chosen). The scalar product evidently has the properties:

where a, a,, a,, P, /I1 and P2 have the same degree.
If (a, P) = 0, a and P are said to be orthogonal.
It should be remarked that the r operation is an isomorphism between
the spaces AP (Tg) and A"-P(T,*) at each point P of M.

2.8. Harmonic forms. The operators 6 and A
There are several well-known examples from classical physics
(potential theory) where relations analogous to Laplace's equation hold.
The electrical potential due to a system of charges or the vector potential
due to a system of currents is not uniquely determined. To the former
an arbitrary constant may be added and to the latter an arbitrary vector
with vanishing curl. In defining electrical potential we begin with a
vector field E representing the electrical intensity which satisfies the
equation curl E = 0. This is the condition that the electric field be
conservative. A function f is then defined as follows:

where r denotes the position vector of a point in E3 and the . denotes
the inner product of vectors in ES. It follows that E = grad f and f
is determined to within an additive constant.
In defining the vector potential, on the other hand, we begin with
the magnetic induction B which satisfies the equation div B = 0. As it
turns out, this is a sufficient condition for the existence of a vector
field A (unique up to a vector field whose curl vanishes) satisfying
B = curl A.
We now re-write the above equations as tensor equations in E3. We
may distinguish between covariant and contravariant tensor fields
provided the coordinate system is not Euclidean. Let Ei denote the
components of the covariant vector field E and Bi the components of
the contravariant vector field B. Then,
D, E, - D, E, = 0 (2.8.2)