ABSTRACT
R n with coordinates {xi}, where i = 1, ..., n. (Any surface M in Rn can
be used instead.) Now construct all linear combinations of the differentials
{dxi}, that is, consider the space V defined by
V = 〈{dxi}〉 = {ai dxi} (13.1)
where we have introduced the Einstein summation convention, under which
repeated indices must be summed over. If the coefficients ai are numbers,
then V is an n-dimensional vector space with basis {dxi}. We will, however, instead allow the coefficients to be functions on Rn, which turns V into
a module over the ring of functions. (This is entirely analogous to the
transition from vectors to vector fields.)
What are the elements of V ? Any differential df can be expanded in
terms of our basis using calculus, namely
df = ∂f
∂xi dxi (13.2)
so that df ∈ V . What integrand does df correspond to? The fundamental theorem for the gradient says that∫
∇f · dr = f ∣∣B A
(13.3)
for any curve C from point A to point B. We rewrite this relationship in
terms of integrands as
df = ∇f · dr, (13.4) which we refer to as the Master Formula because of its importance in
vector calculus. Thus, the differential form df represents the integrand
corresponding to the vector field ∇f ; we will have more to say about this
A special case of (13.2) occurs when f = xj is a coordinate function, in
which case
d(xj) = ∂xj
∂xi dxi = dxj , (13.5)
which shows that our basis really is what we thought. And we can use
calculus to check that this construction does not depend on the choice of
coordinates, that is, that df is the same when expanded in terms of any set
of coordinates on Rn. Schematically, this argument goes as follows:
df = ∂f
∂u du +
∂f
∂v dv + ...
