90
6 Tensor Analysis
t =
d
dλ
=
dx
i
dλ
∂
∂ x i versus
t =
dx
i
ds
e i , so
∂
∂ x i ↔ ↔
e i .
(6.37)
That is the curve derivative operators along coordinate lines act just like the coordinate
basis vectors, and the curve derivative itself acts like a vector.
Appendix 2: p-Forms and Exterior Derivatives
The reader may encounter objects called p-forms in the literature. Here we will only
mention the basic concept and the nomenclature (Misner 1973; Ohanian 1994).
The 1-forms we have used naturally generalize to these larger p-forms, which
are useful in some physics applications. A 2-form is defined as the antisymmetrized
exterior product of 1-forms, say ˜
α and ˜
β. Specifically
˜
α ∧ ˜
β ≡ ˜
α ⊗ ˜
β − ˜
β ⊗ ˜
α.
(6.38)
If the 1-forms are expanded in terms of a coordinate basis then the 2-form may be
written
˜
α = α μ ˜
dx
μ
, ˜
β = β σ ˜
dx
σ
, ˜
α ∧ ˜
β =
1
2
α μ β σ − β μ α σ
˜
dx
μ
∧ ˜
dx
σ
. (6.39)
The general p-form is defined as the anti-symmetric product of p such factors,
˜
α ≡
1
p!
α μσ ...λ
˜
dx
μ
∧ ˜
dx
σ
. . . ∧ ˜
dx
λ
,
(6.40)
where the coefficient set is anti-symmetric.
The exterior derivative is defined as the only p + 1 form that can be built from a
p-form by differentiation; it is explicitly
d ˜
α ≡
1
p!
α μσ ...λ,β
( ˜
dx
β
∧ ˜
dx
μ
∧ ˜
dx
σ
. . . ∧ ˜
dx
λ
).
(6.41)
If we differentiate the exterior derivative again we obviously get zero because
the coefficient set becomes symmetric in two indices, and the basis form is
anti-symmetric. We may abbreviate this statement as d(d ˜
α) = 0.
Such p-forms are useful in physics where antisymmetric tensors occur; in electromagnetism they allow an elegant treatment of Maxwell’s equations. A few brief
comments on p-forms in four dimensions are thus in order: A 0-form is simply defined
as a scalar and has 1 independent component. The 1-forms we have been using have
four independent components. The 2-forms have the same number of components
as an anti-symmetric 4 by 4 matrix, which is 6; this is the number of components
in the electromagnetic Maxwell tensor. The 3-forms can be labeled by the missing
6 Tensor Analysis
t =
d
dλ
=
dx
i
dλ
∂
∂ x i versus
t =
dx
i
ds
e i , so
∂
∂ x i ↔ ↔
e i .
(6.37)
That is the curve derivative operators along coordinate lines act just like the coordinate
basis vectors, and the curve derivative itself acts like a vector.
Appendix 2: p-Forms and Exterior Derivatives
The reader may encounter objects called p-forms in the literature. Here we will only
mention the basic concept and the nomenclature (Misner 1973; Ohanian 1994).
The 1-forms we have used naturally generalize to these larger p-forms, which
are useful in some physics applications. A 2-form is defined as the antisymmetrized
exterior product of 1-forms, say ˜
α and ˜
β. Specifically
˜
α ∧ ˜
β ≡ ˜
α ⊗ ˜
β − ˜
β ⊗ ˜
α.
(6.38)
If the 1-forms are expanded in terms of a coordinate basis then the 2-form may be
written
˜
α = α μ ˜
dx
μ
, ˜
β = β σ ˜
dx
σ
, ˜
α ∧ ˜
β =
1
2
α μ β σ − β μ α σ
˜
dx
μ
∧ ˜
dx
σ
. (6.39)
The general p-form is defined as the anti-symmetric product of p such factors,
˜
α ≡
1
p!
α μσ ...λ
˜
dx
μ
∧ ˜
dx
σ
. . . ∧ ˜
dx
λ
,
(6.40)
where the coefficient set is anti-symmetric.
The exterior derivative is defined as the only p + 1 form that can be built from a
p-form by differentiation; it is explicitly
d ˜
α ≡
1
p!
α μσ ...λ,β
( ˜
dx
β
∧ ˜
dx
μ
∧ ˜
dx
σ
. . . ∧ ˜
dx
λ
).
(6.41)
If we differentiate the exterior derivative again we obviously get zero because
the coefficient set becomes symmetric in two indices, and the basis form is
anti-symmetric. We may abbreviate this statement as d(d ˜
α) = 0.
Such p-forms are useful in physics where antisymmetric tensors occur; in electromagnetism they allow an elegant treatment of Maxwell’s equations. A few brief
comments on p-forms in four dimensions are thus in order: A 0-form is simply defined
as a scalar and has 1 independent component. The 1-forms we have been using have
four independent components. The 2-forms have the same number of components
as an anti-symmetric 4 by 4 matrix, which is 6; this is the number of components
in the electromagnetic Maxwell tensor. The 3-forms can be labeled by the missing