4.3 Vectors and 1-Forms, Abstract View
43
orthogonality relation inferred by (4.28),
˜
p
V
=
p m ω
m
e j V
j
= p m V
j
ω
m
e j
= p j V
j so ω
m
e j
= δ
m
j .
(4.30)
That is, we have set up the basis forms to obey the orthogonality relation in (4.30).
One special 1-form is of particular interest and leads to a curious notation. In
the context of forms the “gradient” of a function φ is defined as a form having
components which are the usual partial derivatives φ ,k ; the gradient is thus
dφ ≡ φ ,k ω
k
.
(4.31)
The gradient 1-form of the coordinate x
i is then given by
dx
i
= x
i
,k ω
k
=
∂ x
i
∂ x k ω
k
= δ
i
k ω
k
= ω
i
, dx
i
= ω
i
,
(4.32)
Thus the basis may be expressed as the set of coordinate gradients. This implies that
we may rewrite (4.31) as
dφ = φ ,k dx
k
(4.33)
which looks like the analogous elementary calculus expression, but is a relation
between 1-forms. It is clear from (4.27) and (4.33) that the 1-forms should transform
covariantly.
Example 4.3 Let us return to the polar coordinate system in Example 4.1
and obtain the relevant vectors and forms. From Fig. 4.2 it is clear that the
coordinate basis vectors in the radial and angle directions must be
e ρ = a
cos ϕ e x + sin ϕ e y
,
e θ = b(− sin ϕ e x + cos ϕ e y ).
(4.34)
where a and b are some constants. The basis vectors can have any length we
choose.
According to (4.19) the metric is then,
g μν =
a
2 0
0 b
2
.
(4.35)
If we wish this to be the metric for flat Euclidean 2-space as in (4.18) we choose
a = 1 and b = ρ and have
e ρ = cos ϕ e x + sin ϕ e y ,
e θ = ρ(− sin ϕ e x + cos ϕ e y ).
(4.36)
43
orthogonality relation inferred by (4.28),
˜
p
V
=
p m ω
m
e j V
j
= p m V
j
ω
m
e j
= p j V
j so ω
m
e j
= δ
m
j .
(4.30)
That is, we have set up the basis forms to obey the orthogonality relation in (4.30).
One special 1-form is of particular interest and leads to a curious notation. In
the context of forms the “gradient” of a function φ is defined as a form having
components which are the usual partial derivatives φ ,k ; the gradient is thus
dφ ≡ φ ,k ω
k
.
(4.31)
The gradient 1-form of the coordinate x
i is then given by
dx
i
= x
i
,k ω
k
=
∂ x
i
∂ x k ω
k
= δ
i
k ω
k
= ω
i
, dx
i
= ω
i
,
(4.32)
Thus the basis may be expressed as the set of coordinate gradients. This implies that
we may rewrite (4.31) as
dφ = φ ,k dx
k
(4.33)
which looks like the analogous elementary calculus expression, but is a relation
between 1-forms. It is clear from (4.27) and (4.33) that the 1-forms should transform
covariantly.
Example 4.3 Let us return to the polar coordinate system in Example 4.1
and obtain the relevant vectors and forms. From Fig. 4.2 it is clear that the
coordinate basis vectors in the radial and angle directions must be
e ρ = a
cos ϕ e x + sin ϕ e y
,
e θ = b(− sin ϕ e x + cos ϕ e y ).
(4.34)
where a and b are some constants. The basis vectors can have any length we
choose.
According to (4.19) the metric is then,
g μν =
a
2 0
0 b
2
.
(4.35)
If we wish this to be the metric for flat Euclidean 2-space as in (4.18) we choose
a = 1 and b = ρ and have
e ρ = cos ϕ e x + sin ϕ e y ,
e θ = ρ(− sin ϕ e x + cos ϕ e y ).
(4.36)