Mukhi S., Mukunda N. Lectures on Advanced Mathematical Methods for Physicists
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52
Chapter
3. Differentiable Manifolds I
Tangent
vectors
at
a
point
form a
vector
space.
If
we have two curves
p(t), q(t)
C
M
passing
through
the
same
point
at
to,
then
we
can
take
the
sum
of
the
tangent
vectors
to
each curve
at
to
to
get
a new vector in
IR
n
,
denoted
. [dxi
dxi]
a'(xo)
=
ill
(p(t))
+
ill
(q(t))
t=to
(3.23)
This
defines a
straight
line in
IR
n
:
Xi(t)
=
Xo
+
ai(xo)t
(3.24)
where
Xo
is
the
image of
the
point
p(O)
on
the-manifold.
This
straight
line
can
be
mapped
back
to
M,
to
give a new curve
on
M
around
the
point
where
p(t), q(t)
intersect, using
¢;;1.
Thus,
the
sum
of
tangent
vectors
to
two curves passing
through
a
point
can
be
regarded
as a
tangent
vector
to
a
third
curve passing
through
the
same
point.
Definition:
The
vector space
of
tangent
vectors
at
a
point
p
E
!'vI
is called
the
tangent
space
Tp(M)
at
p.
Note
that
any
collection
of
numbers
a
i
and
a
coordinate
system
Xi
define
a
tangent
vector. Here is
the
construction:
(i) Pick a
point
xb
E
IR
n
and
define
any
curve
xi(t)
in
IR
n
such
that
d:fti
It=to
a'
.
(ii) "Lift"
the
curve back
to
Musing
¢;;1
for
that
coordinate
system.
(iii)
In
any
other
coordinate
system
yi(X),
define
a,i
=
~I
a
j
to
be
the
same
Xo
object
in
the
new coordinates.
Let
us now combine
the
two concepts we have
just
discussed: differentiation
of functions
on
M,
and
tangent
vectors
to
curves
on
M.
In
this
way we will
be
able
to
provide a
coordinate invariant
definition
of
a
tangent
vector.
Recall how we differentiated a
C=
function
on
M
along a given curve.
If
the
function
on
M
is
f,
then
along
the
curve
it
is
f(p(t)),
and
its derivative
along
the
curve is defined
to
be
'!it
(p(
t)).
Clearly
this
defines a
new
function
on
the
same
curve.
U sing
the
coordinate
map
we
can
express
this
as
df
df
dx
i
dt
=
dXi
ill
(p(t))
(3.25)
Thus,
the
derivative
of
f
along a curve,
at
some point, is expressed in
terms
of
its
gradient
of
/
ax
i
in
the
chosen
coordinate
system,
contracted
with
the
tangent
vector
at
that
point.
But
while
both
of
/
ax
i
and
dx
i
/
dt
are
coordinate-
dependent
objects,
when
contracted
together
they
give
df
/dt
which is manifestly
coordinate
independent.
Writing
df
dt
[
dxi
a ]
ill
(p(t))
ax
i
f
(3.26)
3.4. Calculus
on
Manifolds: Vector
and
Tensor Fields
53
we
see
that
d:ft'
(p(t))
&~i
is a coordinate-invariant first
order
differential
operator
which
maps
any
Coo
function
on
a curve
in
a manifold into
another
such function
on
this
curve.
At
a
point
Xo
=
Xi(tO),
the
operator
is
d:fti
It=to
d~i.
This
carries
precisely
the
same
information as
the
tangent
vector
d:fti
Ito.
So we
can
as well
call
this
differential
operator
a
tangent
vector
at
To.
All we have done is
to
take
our
previous definition
of
tangent
vector
and
contract
it
with
8/
8x
i
to
make
it
a
differential operator
on
functions.
By
this
process it also becomes manifestly
coordinate
invariant.
So far we have confined our
attention
to
tangent
vectors defined
at
a fixed
point
p
E
M.
These
can
be
written
a
i
8/
8x
i
for some
constants
a
i
.
But
now
consider
the
differential
operator
X
=
ai(x)
8/8xi,
where
ai(x)
are
Coo
func-
tions.
This
operator
maps
Coo
functions over a whole
coordinate
patch
into
new
Coo
functions
on
the
same
patch:
.
8f
X:
f
--+
Xf
==
a'(x)-8
.
x'
(3.27)
We call X a
vector field.
Given two functions
f
and
9
on
M,
we have
X(fg)
=
(Xf)g
+
f(Xg).
In
fact, this
property
characterizes vector fields,
and
we
can
as well define a vector
field
this
way.
Definition:
A
vector field
is a linear
map
from
Coo
functions
to
Coo
functions
on
a manifold
M,
satisfying
X(fg)
=
(Xf)g
+
f(Xg).
Let
us
now generalise
this
concept
to
tensor
fields.
Just
as vector fields
are
linear maps,
tensor
fields will
be
defined as multilinear
maps
from sets of
functions
to
sets
offunctions.
Consider
an
ordered
set
of
n
functions
(h,
...
,
f
n)
on
a manifold
M.
This
set
forms a vector space
under
addition
in
the
obvious
way. Moreover, we
can
multiply
two elements
of
this
space as follows:
Now a
map
X :
(h,
...
,
fn)
--+
X(h,.··,
fn),
taking
each
set
of functions in
this
space
to
another
set, is called
multilinear
if
(3.28)
Definition:
A
tensor
field
of
rank n
is a multilinear
map
(3.29)
satisfying
X((h,···,fn)*(gl,
...
,gn))
=
(X(h,···,
fn))
* (gl,
...
, gn)
+
(h,···,
fn)
*
(X(gl,
...
,gn))
(3.30)
54
Chapter
3. Differentiable Manifolds I
The
nth
rank
tensor
fields form a vector
space
which we
denote
Tp(M)
(>9
Tp(M)
(>9
•..
(>9
Tp(M),
or
simply
(>9nTp(M).
This
is
the
n-fold
tensor
product
of
the
tangent
space
with
itself. Since
fJ
/
fJxi
form a basis for
Tp
(M),
a basis
for
the
n-fold
tensor
product
space is given
by
fJ
/ fJxil
(>9
fJ
/
fJxi2
(>9
.•.
(>9
fJ
/ fJxin.
Thus
any
A
E
(>9nTp(M)
can
be
written
...
fJ fJ fJ
A
=
A21t2
...
2n(x)_.
(>9
-.
(>9
...
(>9
-.
(3.31)
fJX"'
fJxt2
fJxtn
Clearly
the
coefficients
Ail
i2
...
in
(x)
transform
like
nth
rank
contravariant
ten-
sors,
in
"physics" terminology,
and
we
call
A
E
(Tp(M))n
an
nth
rank
tensor
field.
Now
suppose
we have two manifolds M
and
N
and
a
COC!
map
(3.32)
This
map
induces
another
map,
called
the
differential map
(3.33)
which
maps
the
tangent
space
to
M
at
p,
to
the
tangent
space
to
N
at
rP(p).
To define
this
map,
note
the
following: if
f :
N
---7
lR is a
COC!
function
on
N,
then
f
0
rP
:
M
---7
lR is a
COC!
function
on
M (Fig. 3.9).
M
N
p
<p(p)
)
R
Figure
3.9: How
to
define
the
differential
map
rP*.
A
tangent
vector
acts
on
functions,
as
we have discussed above. Given
a vector
X
E
Tp(M),
we define
rP*X
E
T¢(p)(N)
to
have
the
same
action
on
f :
N
---7
lR as
X
E
Tp(M)
would have
on
f
0
rP
: M
---7
lR.
(3.34)
Notice
that
for given
X
and
f,
both
sides
of
this
equation
are
just
real
numbers.
The
above
equation
defines a
map
rP*
:
Tp(M)
---7
T¢(p)(N)
by
rP*
:
X
---7
rP*X.
3.5.
Calculus
on
Manifolds: Differential
Forms
55
3.5
Calculus
on
Manifolds:
Differential
Forms
Having defined
vector
and
tensor
fields
on
a manifold, we now
turn
to
the
definition
of
their
"dual" objects, differential forms.
Both
tensor
fields
and
differential forms
playa
crucial role in physics, where
we
tend
to
think
of
their
roles
as
rather
similar. However in
the
context
of
differentiable manifolds
the
two
have very different meanings, as we will
try
to
bring
out
in
what
follows.
Given a vector space
V
with
basis
ei,
i
=
1""
,n,
the
dual
vector
space
V*
is defined
to
be
the
vector
space
generated
by
the
dual
basis
e*i
given
by
an
inner
product
(3.35)
For finite-dimensional vector spaces,
the
dual
space
is isomorphic
to
the
original one. However,
the
pairing
between
a vector
space
and
its
dual
can
lead
to
different
properties
for
the
elements
of
the
two spaces.
These
properties
are
induced
by
a
space
on
its
dual
if we require
that
the
inner
product
remain
invariant
under
certain
transformations.
In
the
present
case we
want
to
study
the
dual
of
the
tangent
space,
and
we will derive
the
transformation
laws across
coordinate
patches
for elements
of
this
space
by
requiring
that
the
inner
product
be
coordinate
invariant.
In
general, elements
of
V
and
V*
can
be
expressed
in
terms
of
their
respec-
tive bases:
a
E
V,
a'
E
V*,
a=I:jajej
a'
=
"'.
a'e*i
D,
,
(3.36)
Then,
(a',a)
=
a~aj
(e*i,ej)
=
a~ai
(summation
implied).
Thus
the
pair
a'
E
V*, a
E
V
gets
mapped
onto
a
real
number,
or
in
other
words, ( ) is a bilinear
map:
V*
®
V
---+
JR.
There
is
an
alternative
way
to
look
at
this. For a given
a'
E
V*,
we
have a
linear map
(a',
):
V
---+
JR
(3.37)
defined
by
(a',
):
a
E
V
---+
(a',
a)
E
JR
(3.38)
Thus
the
dual
vector
space
to
V
is
the
space
of
linear Junctionals
on
V.
Let
us clarify
what
we
mean
by
linear functionals. A functional is a
map
which
takes
an
element
of
vector
space
to
a
number.
In
our
case, clearly
(a,
)
is a functional,
and
it
has
the
additional
property
that:
(a', ) :
'xa
+
J.1.b
---+
'x(a', a)
+
{l(a',
b)
(3.39)
which is
just
what
we
mean
by
linearity.
An
important
property
of
dual
spaces is
that
if
we
have a
map
of
vector
spaces
V,
W:
J:V---+W
56
Chapter
3.
Differentiable Manifolds I
then
this
induces a
dual map
in
the
opposite
direction:
f*
:
W*
---7
V*
Let
us show this. Given
b'
E
W*,
we
want
to
define
f*(b')
E
V*.
We
can
define
it
as
a linear functional
on
V:
(f*(b'),
):
(f*(b'),
a)
=
(b',
f(a))
(3.40)
The
inner
product
on
the
left
of
the
equality
is clearly
between
elements
of
V
and
V*,
while
the
one
on
the
right
is
between
elements
of
Wand
W*.
Since
the
map
f : V
---7
W
is given,
the
above
equation
defines
the
dual
map
f*
:
W*
---7
V*.
In
the
study
of
differentiable manifolds, we defined
the
tangent
space
Tp(M)
at
a
point
p,
with
basis
a / axi.
Now we will define
its
dual.
Definition:
The
space
dual
to
Tp(M)
is called
the
cotangent space
T;(M).
Denote
the
dual
basis of
T;(M)
by
dXi.
Note
that
this
is a formal
symbol
for
the
moment
and
does
not
represent
an
infinitesimal
amount
of
anything!
Thus
there
is
an
inner
product,
(3.41)
This
is manifestly
invariant
under
local changes
of
coordinates, if we assign
to
dxi
the
transformation
law
. . ayi .
dx'
---7
dy'
=
--dx)
ax)
(3.42)
The
notation
dx
i
serves
to
remind
us
that
this
object
transforms like an
infinitesimal distance.
As we have
indicated
above,
this
is only
notation
-
but
now we
understand
what
motivates
it.
Definition:
An
element
of
the
cotangent
space is
written
and
is called a
I-form
at
p.
The
coefficient
Wi
transforms
like a "covariant vector" ,
in
"physics" terminology.
As we
did
in
going from vectors
to
vector fields, we
can
generalize a form
at
a single
point
to
a field defined all over
the
coordinate
patch:
W(x)
=
Wi(X)
dx
i
with
Wi(X)
a
set
of
functions
on
IRn.
Such
an
object,
with
the
coefficients
Wi(X)
being differentiable functions, is called a
differential
i-form.
The
inner
product
between
basis vectors
of
the
tangent
space
Tp(M)
and
the
cotangent
space
T;(M)
gives a
product
between
arbitrary
elements
of
these
3.5
. Calculus on Manifolds: Differential Forms
57
two spaces.
If
X
=
a
j
(x)8jax
j
is a vector field
and
w(x)
=
Wi
(x)dx
i
is
a I-form,
then
(3.43)
This
is coordinate-independe
nt,
as it should be.
The
left
hand
side is a pairing
between a ve
ctor
field
and
a form, each of which
is
coordinate
independent,
while on
the
right
hand
side we find
the
coefficients (or "components") of these
objects
, which
are
coordinate
dependent
but
transform
oppositely across patches
precisely in such a way
that
the
product
is invariant.
We now define
the
exterior derivative on functions
on
the
manifold as a
map
d
defined by:
af
.
d : f
->
df
=
-a
.
dx'
x'
(3.44)
This
associa
tes
a I-form
to
any
function
f
on
a manifold.
The
components of
this
I-form
ar
e
just
the
derivatives of
f
along each of
the
coordinates.
Later
we
will see
that
the
exterior derivative
can
be
defined
to
act
on differential forms
and
not
just
functions. We will provide
the
complete definition
at
that
stage.
The
exterior derivative plays a
fundam
e
ntal
role in
the
study
of differential
forms.
If
X
is
an
arbitrary
vector field,
then
(as we discussed above) it
can
be used
to
map
a function
to
another
function by
f
->
X f.
Now we have
the
equality
Xf
=
(
df,X)
(3.45)
which relates
the
action
of
X
on
f
with
the
inner
product
between X
and
the
exterior derivative of
f.
Exercise:
Check
the
above equality.
Just
as for
the
tangent
space, we
can
study
tensor
products
of
th
e cotangent
space:
(T;(M))m
whose basis
is
dx
i
,
0
...
dx
i
",.
More generally, we
can
consider
elements of
the
mixed
product
(Tp(M))n
0
(T;(M))m,
which look like
. . a
a·
.
A
=
A"""
n . .
--
.
0
...
0
--.
0
dx
J
'
0 . . . 0
dx
J
",
•
J,
...
J",
ax"
ax'n
(3.46)
For a physicist,
the
components
can
be
thought
of as forming a "mixed tensor"
which is
nth
rank
contravariant,
mth
rank
covariant.
Now let us
return
to
the
exterior derivative
d.
We
can
think
of functions
as
"O-forms
".
Then
the
operator
d
acting
on
a O-form produces a I-form. How
about
the
action
of
d
on I-forms?
That
is something we have
yet
to
define.
We could
perhaps
try
something like
d:
d
i
?
aWi
did]
W
=
4-'
i
X
->
-a
.
x
0
X
x
J
But
this
does
not
work!
If
Wi
tranforms
like
the
components of a I-form,
aw
i
j
ax]
58
Chapter
3.
Differentiable Manifolds I
does
not
transform
in a covariant way.
Exercise:
Check
the
above
statement.
Physicists
are
familiar
with
one way
to
remedy
this, which is
to
introduce
an
"affine connection"
and
a "covariant derivative".
But
that
relies
on
the
existence
of
a
Riemannian
metric
on
the
manifold,
something
we have
not
yet
defined.
For
now
our
manifolds
are
equipped
only
with
a differentiable
structure.
So if we
want
to
define generalizations
of
I-forms
which
transform
nicely
under
coordinate
changes, we have
to
do
something
different.
Indeed
our
problem
will
be
solved
if
we
antisymmetrise
the
derivative. We
aWi
aWj
can
show
that
if
Wi
transforms
like a covariant vector,
then
-a
. -
-a
.
does
Xl
X'
transform
like a covariant
2nd-rank
tensor:
a:wj-ajw~=
(~~:
a~k)
(~:;Wl)
-(i~j)
(
ax
k
axl
axk
aym
a
2
xl ) . .
-a
.
-a
.
akWI
+
-a
.
-a
k a
a'
WI
-
(z
~
J)
y'
yl
y'
X
ym
yl
(3.47)
On
antisymmetrising,
the
second
term
drops
out
and
we
get
the
desired result.
The
presence
of
the
second
term
before
antisymmetrising
shows explicitly
that
the
result
would
not
transform
like a
tensor
if we
did
not
antisymmetrise.
To keep
track
of
the
fact
that
aiWj
must
be
antisymmetrised,
we
write
dw
=
aiWj
dx
i
A
dx
j
where
dx
i
Adxj
is defined
to
be
~(dXi
®dx
j
-dx
j
®dXi).
The
object
so
obtained
is called
an
antisymmetric 2-form,
or
simply a
2-form.
This
motivates
us
to
define
the
following
structure:
Definition:
The
wedge product
of
two
copies
of
T;(M)
is
the
vector space
spanned
by
the
basis
dx
i
A
dx
j
.
In
general,
the
basis
dX
i1
A
dXi2
A
...
A
dx
in
spans
the
space
of
totally
antisymmetric
covariant
tensor
fields.
This
is
denoted
An
(M)
and
its elements
are
called
n-forms.
The
union
of
the
spaces of all forms,
A
(M)
==
Un An
(M),
is called
the
exterior algebra
of
the
manifold
M.
We
can
now
complete
our
definition of
the
exterior
derivative by specifying
how
it
acts
on
any
differential
n-form,
mapping
it
to
an
n
+
I-form.
Definition:
The
exterior derivative on
n-forms
is a
map
d:
a
E
An(M)
-7
da
E
An+1(M)
defined by:
(
a).
. .
da
=
--.
-ai
...
i
(x)
dX'n+l
A
dX"
A
...
A
dx'n
aX'n+l
1
n
(3.48)
(3.49)
3.6. Properties of Differential Forms
59
The
components
of
da
are
totally
antisymmetrised, as is manifest in
the
nota-
tion.
Thus
we have
built
up
a
structure
of
forms
on
the
manifold, which have
rank
0,
1,
..
.
d
where
d
is
the
dimension
of
the
manifold M (clearly a
totally
antisymmetrised
object
with
rank
greater
than
d
must
vanish). Along
with
this,
we have a
map
d
which
takes
us from n-forms
to
n+
I-forms by
antisymmetrised
differentiation.
3.6
Properties
of
Differential
Forms
We list a few
properties
satisfied by differential forms.
They
are
rather
easy
to
prove from
the
definition
and
the
reader
is encouraged
to
work
out
all
the
proofs.
(i)
If
am
E
/\m(M),
b
n
E
/\n(M),
then
am
/\
b
n
=
(_I)mn
b
n
/\
am.
This
means
that
two different forms
anticommute
if
th
ey
are
both
odd,
otherwise
they
com-
mute.
(ii)
d(a
m
/\ b
n
)
=
dam /\ b
n
+
(_1)m
am
/\
db
n
.
Thus,
upto
possible signs,
the
d
operator
distributes
over
products
of
forms
just
like
an
ordinary
derivative.
(iii)
d
2
=
O.
In
other
words,
acting
twice
with
the
d
operator
on
the
same
object
gives
O.
We
say
that
d
is a
nilpotent
operator.
This
will
turn
out
to
be
of
fundamental
importance.
The
proof
of
nilpotence goes as follows:
2
(a
. . . )
d
a=d
-a-.-a
i
,
...
i
(X)dX2n+l
/\d
X
2
1
/\···/\dx
2n
x'l.n+l
n
=0
(3.50)
since
the
two derivatives commute, while
the
wedge
product
which
contracts
with
them
anticommutes.
(iv)
The
dimension
of
/\
n
(M)
for a d-dimensional manifold is given
by
the
binomial coefficient
(d)
=
I(
d!
).
This
is due
to
the
antisymmetry,
and
n n. d - n
!
is easy
to
check. Also,
the
dimension vanishes for
n
>
d
as one would
expect
because
there
are
not
enough indices
to
antisymmetrise
.
While
studying
the
tangent
space, we saw
that
¢:
M
--+
N
induces a
map
¢*
:
Tp(M)
--+
T¢(p)(N),
the
differential
map
between
the
tangent
spaces.
It
turns
out
that
this
also induces a
map
between
the
cotangent
spaces,
but
in
the
reverse
direction.
This
follows from
the
dual
pairing between
tangent
and
cotangent
spaces.
The
reverse
map
is
denoted
¢*
:
T¢(p)
(N)
--+
T;
(M)
and
is defined as
60
Chapter
3. Differentiable Manifolds I
follows.
If
w
E
T¢(p/N),
define
¢*w
E
T;(M)
by
(3.51 )
Differential forms
are
useful in
studying
the
topology of general differen-
tiable
manifolds.
Much
of
this
study
can
be
performed
without
introducing
metric
properties.
It
is
important
to
keep
in
mind
that,
although
a
metric
on
a
manifold is
an
essential
construction
in
physics
(particularly
general
relativity),
it
is
an
additional
structure
on
a manifold.
It
is
worth
knowing
what
are
all
the
operations
one
can
perform
without
having
to
define a metric, since
the
consequences
of
these
must
clearly
turn
out
to
be
metric-independent.
In
the
special
theory
of
relativity,
the
manifold
with
which we deal is fiat
Minkowski
spacetime
(or
Euclidean
space
after
Wick
rotation).
Here differential
forms
playa
relatively
trivial
role,
as
the
topology
of
Euclidean
space
is trivial.
But
they
do
serve
as
useful
notation,
and
moreover
many
equations
of
physics
generalise
immediately
to
manifolds which
are
different from
Euclidean
space,
and
whose topology
may
be
nontrivial.
Example:
Free electrodynamics in 4
spacetime
dimensions.
The
electromag-
netic field is
represented
by
a
I-form
A(x)
=
A/L(x)
dx/L.
The
field
strength
is
simply
the
2-form
F(x)
==
dA(x)
=
~F/Lv(x)dx/L
A
dx
v
where
(3.52)
We know
that
d
2
=
O.
Since
F
=
dA,
it
follows
that
dF
=
d
2
A
=
O.
What
does
this
condition
on
F
physically
mean?
Writing
it
out
in
components,
we
find
dF
=
F/LV,A
dx
A
A
dx/L
A
dx
v
=
0
(3.53)
By
total
antisymmetry
of
the
threefold wedge
product,
we
get
(3.54)
This
is called
the
Bianchi identity.
If
we
temporarily
get
ahead
of ourselves
and
introduce
the
Minkowski
met-
ric
of
spacetime,
the
above
equations
can
be
written
in
a
more
familiar form.
First
of
all we find:
Fij,k
+
cyclic
=
0
FOi,j
+
Fij,o
+
FjO,i
=
0
i,j,k
E
{1,2,3}
Next,
defining
the
magnetic
and
electric fields
by
the
Bianchi
identity
becomes
ai
Bi
=
V . B
=
0
!:>·E·
-
!:>·E·
-
E"k
aBk
,f7
x
E~
_
BB
U,
J
U
J
,
- -
'J
at
~
v - -
at
(3.55)
(3.56)
(3.57)
3.6.
Properties
of
Differential
Forms
61
So
the
Bianchi identity,
d
2
A
=
dF
=
0, is
just
equivalent
to
two
of
Maxwell's
equations
in
empty
space.
It
is
tempting
at
this
point
to
display
the
two
remaining
Maxwell
equations
in
the
same
notation.
For
this, we
must
first define
the
constant
antisymmetric
tensor
(under
SO(3,
1)
Lorentz
transformations)
on
JR4:
it
is
denoted
E,WAp,
and
takes
the
value
+
1 if
{p"
v,
A,
p}
is
an
even
permutation
of
{O,
1,2,
3}
and
-1
if
it
is
an
odd
permutation.
It
is
equal
to
°
if
any
two
of
the
indices coincide. We
can
think
of
this
tensor
as
making
up
the
components
of
the
4-form
A
=
~!
E/-LVApdx!"
1\
dx
v
1\
dx
A
1\
dx
P
(3.58)
We
can
use
E/-LVAp
to
define
what
we call
the
dual
of
any
arbitrary
form.
(We will discuss
this
concept
in
more
generality
in
subsequent
chapters).
For
a
I-form
a
=
a/-L
dx/-L,
we define its
dual 3-form
1
*a
=
-*a
dx/-L
1\
dx
v
1\
dx
A
3!
!"VA
(3.59)
by
specifying
the
components
as
a!"VA
=
E/-LVAp
a
P
.
Similarly,
the
dual
of
a 2-form
b
=
~b!"v
dx!"
1\
dx
v
~*b/-Lv
dx/-L
1\
dx
v
with
components
is
the
2-
form
*
b
(3.60)
It
is easy
to
see similarly
that
in general,
the
dual
of
an
n-form
is a (4 -
n
)-form,
and
that
taking
the
dual
of
any
form twice gives
the
original form
back
again:
*(*a)
=
a
for
any
n-form
a.
Now consider
*F
where
F
is
the
Maxwell field
strength
2-form defined
above. Clearly,
in
general we
do
not
have
d*
F
=
0, since
*
F
(unlike
F
itself)
is
not
d
of
anything.
Nevertheless, let us
examine
the
content
of
the
equation
d(*
F)
=
0, which could
be
imposed
by
physical
rather
than
mathematical
re-
quirements.
The
resulting
equations,
in
component
form, will
turn
out
to
be
the
other
two Maxwell equations.
Thus,
these
equations
will
be
seen
to
have
a
dynamical
content,
and
they
have
to
be
imposed
by
hand,
unlike
the
two we
already
obtained,
which
are
just
mathematical
identities.
Working
this
out
in
components
we have:
d(*
F)
=
~E!"VAP
oaFAP
dx
C7
1\
dx/-L
1\
dx
v
Setting
this
to
zero,
and
taking
the
dual
of
the
equation
we find:
which
after
contracting
the
indices
of
the
two E-symbols gives
(3.61 )
(3.62)
(3.63)