文档介绍：Caltech – Ph106 – Fall 2001
Math for physicists: differential forms
Disclaimer: this is a first draft, so a few signs might be off.
1 Basic properties
Differential e up in various parts of theoretical physics, including advanced
classical mechanics, ism, thermodynamics, general relativity, and quan-
tum field theory. So they’re well worth knowing about. This is supposed to be a
self-contained exposition for someone who has some knowledge of multivariable calcu-
lus.
Forms are like infinitesimal objects, but this is (or can be made) pletely rigorous
subject. The most basic object is exterior derivative, d. For a function F (x1, . . . , xn),
∂F
dF = dxi (1)
∂xi
by definition. Summation over i from 1 to n is implicit. In some subjects (particularly
relativity), there is a preference to write xi and/or dxi, but I will eschew this here. Eq.
(1) looks like an infinitesimal variation. The special thing about forms is that they
involve an muting wedge product, ∧, defined so that
dxi ∧ dxj = −dxj ∧ dxi . (2)
A general p-form (for p ≤ n) is
1
ω= ω dx ∧ dx ∧. . . ∧ dx . (3)
p! i1...ip i1 i2 ip
Again there is an implicit sum on i1 through ip, each from 1 to n. The coefficient
functions ωi1∧ip may be assumed to be antisymmetric in i1 through ip. So for instance,
if p = 2, ωij = −ωji. The 1/p! is there in (3) because the sum contains p! copies of
each term