Math 53M, Fall 2003 Professor Mariusz Wodzicki

Introduction to differential 3-forms

January 7, 2004

These notes should be studied in conjunction with lectures.1

v3
v2

x v1

Figure 1: The parallelepiped spanned by column-vectors v1 , v2 and v3 an-

chored at a point x ∈ Rm .

1 Orienting a parallelepiped Two ways of ordering the vectors v1 , v2 and v3 up to a

cyclic permutation correspond to two ways of orienting the parallelepiped they span, see
Figure 1. Each of the three orderings: v1 v2 v3 , v3 v1 v2 , and v2 v3 v1 , determines one and the
same orientation, while any of the remaining three: v1 v3 v2 , v3 v2 v1 , or v2 v1 v3 , corresponds
to the other orientation.
In general, there is no preferred orientation. The situation is, however, different when
     
v11 v12 v13
v1 =  v21  , v2 =  v22  , and v3 =  v23  (1)
v31 v32 v33
1
Abbreviations DCVF, LI and 2F stand for Differential Calculus of Vector Functions, Line Integrals, and
Introduction to differential 2-forms, respectively.

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 Math 53M, Fall 2003 Professor Mariusz Wodzicki

are column-vectors in R3 . If the determinant of the 3×3 matrix formed by column-vectors

(1),
 
v11 v12 v13
ω(x; v1 , v2 , v3 ) ˜ det  v21 v22 v23 
v31 v32 v33

˜ v11 v22 v33 + v12 v23 v31 + v13 v21 v32 − (v11 v23 v32 + v12 v21 v33 + v13 v22 v31 ), (2)

is positive then the orientation corresponding to orderings: v1 v2 v3 , v3 v1 v2 , and v2 v3 v1 , is

said to be positive, while the orientation corresponding to orderings: v1 v3 v2 , v3 v2 v1 , and
v2 v1 v3 , is said to be negative. We reverse this terminology if determinant (2) is negative: the
former orientation is then said to be negative and the latter—to be positive.

2 Oriented volume Let us denote by

♦x (v1 , v2 , v3 ) (3)

the parallelepiped spanned by column-vectors v1 , v2 and v3 anchored at point x ∈ R3 . The

absolute value of determinant (2) is equal to the volume of ♦x (v1 , v2 , v3 ). It is therefore
legitimate to call number ω(x; v1 , v2 , v3 ) in (2) the oriented volume of ♦x (v1 , v2 , v3 ).

W Exercise 1 Verify that

 
v11 v12 v13
v1 ¨ (v2 × v3 ) = det  v21 v22 v23  = (v1 × v2 ) ¨ v3 . (4)
v31 v32 v33

Observations on formula (2):

3×3 determinant (2) is the sum of terms v1i v2j v3k with + sign when ijk is
one of the three cycles: 123, 231 or 312, and − sign when ijk is one of the
three transpositions: 132, 321 or 213. 2
Note the following properties of ω:
Note that determinant (2) is also the sum of terms vi1 vj2 vk3 with + sign when ijk is a cycle and −
2

sign when ijk is a transposition.

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Math 53M, Fall 2003 Professor Mariusz Wodzicki

(a) Linearity in each of its three column-vector variables:

ω(x; at + bu, v, w) = aω(x; t, v, w) + bω(x; t, u, v, w) (5)

ω(x; t, au + bv, w) = aω(x; t, u, w) + bω(x; t, v, w) (6)
ω(x; t, u, av + bw) = aω(x; t, u, v) + bω(x; t, u, w) (7)

(b) Antisymmetry: ω changes sign whenever any two of its column-vector ar-
guments are transposed, thus

ω(x; u, w, v) = −ω(x; u, v, w) , (8)

ω(x; w, v, u) = −ω(x; u, v, w) , (9)
ω(x; v, u, w) = −ω(x; u, v, w) (10)

(t, u, v and w being column-vectors and a and b being scalars).

3 Differential 3-forms Any function

ω : D × Rm × Rm × R m → R

satisfying the above two conditions will be called a differential 3-form on a set D ⊆ Rm .
Remark:
We have seen so far differential 0-forms (i.e., functions D → R), 1-forms, 2-
forms and 3-forms. A picture that emerges is that differential q-forms are func-
tions of q column-vectors v1 , . . . , vq anchored at a point x ∈ D, which
behave like the oriented volume of the corresponding q-dimensional “paral-
lelepiped” spanned by these q vectors.
Thus, 1-forms are modelled on the oriented length of a line segment, 2-forms
are modelled on the oriented area of a parallelogram, and finally 3-forms are
modelled on the oriented volume of a parallelepiped.

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 Math 53M, Fall 2003 Professor Mariusz Wodzicki

4 Exterior product of three 1-forms Given three differential 1-forms ϕ1 , ϕ2 and ϕ3 on

D, the formula
 
ϕ1 (x; v1 ) ϕ1 (x; v2 ) ϕ1 (x; v3 )
ω(x; v1 , v2 , v3 ) ˜ det 
 
 ϕ 2 (x; v1 ) ϕ2 (x; v2 ) ϕ2 (x; v3 ) 
 (11)
ϕ3 (x; v1 ) ϕ3 (x; v2 ) ϕ3 (x; v3 )

gives us a differential 3-form. We denote it ϕ1 ∧ ϕ2 ∧ ϕ3 and call it the exterior product

of 1-forms ϕ1 , ϕ2 and ϕ2 .
Note that

ϕi ∧ ϕj ∧ ϕk = ϕ1 ∧ ϕ2 ∧ ϕ3 (if ijk is a cycle) (12)

= −ϕ1 ∧ ϕ2 ∧ ϕ3 (if ijk is a transposition) . (13)

This follows from the fact that transposing any two columns in a matrix changes the sign
of its determinant.

W Exercise 2 Verify that for any differential 1-forms ϕ, χ, υ, ϑ and scalars a and b, one
has:
(a 1 ) (aϕ + bχ) ∧ υ ∧ ϑ = a ϕ ∧ υ ∧ ϑ + b χ ∧ υ ∧ ϑ ;
(a 2 ) ϕ ∧ (aχ + bυ) ∧ ϑ = a ϕ ∧ χ ∧ ϑ + b ϕ ∧ υ ∧ ϑ ;
(a 3 ) ϕ ∧ χ ∧ (aυ + bϑ) = a ϕ ∧ χ ∧ υ + b ϕ ∧ χ ∧ ϑ .

5 Exterior product of 1-forms and 2-forms Recall that any 1-form ϕ is uniquely repre-
sented as X
fi dxi
i

and that any 2-form ψ is uniquely represented as

X
gjk dxj ∧ dxk . (14)
j,k

We can define exterior products ϕ ∧ ψ and ψ ∧ ϕ as:

X
ϕ∧ψ˜ fi gjk dxi ∧ dxj ∧ dxk (15)
i,j,k
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 Math 53M, Fall 2003 Professor Mariusz Wodzicki

and
X
ψ∧ϕ˜ gjk fi dxj ∧ dxk ∧ dxi , (16)
i,j,k

respectively. It follows immedietely from definition (11) that

dxj ∧ dxk ∧ dxi = dxi ∧ dxj ∧ dxk ; (17)

hence,

ψ∧ϕ = ϕ∧ψ (18)

for any 2-form ψ.

W Exercise 3 Verify that for any differential 1-forms ϕ, χ, differential 2-forms ψ, ξ and
scalars a and b, one has:
(b 1 ) (aϕ + bχ) ∧ ψ = a ϕ ∧ ψ + b χ ∧ ψ ;
(b 2 ) ϕ ∧ (aψ + b ξ) = a ϕ ∧ ψ + b ϕ ∧ ξ .

6 dx ∧ dy ∧ dz Note that
 
v11 v12 v13
dx ∧ dy ∧ dz (x; v1 , v2 , v3 ) = det  v21 v22 v23  (19)
v31 v32 v33

which is the right-hand-side of (2) and, up to a sign, the volume of parallelepiped formed
by column-vectors v1 , v2 and v3 at point x ∈ R3 . We call the differential 3-form on R3 ,
dx ∧ dy ∧ dz, the oriented volume-element.

7 Differential 3-forms on R3 For any differential 3-forms ω on a subset D of R3 , and

column-vectors  
v1i
vi =  v2i  ; (i = 1, 2, 3) , (20)
v3i

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Math 53M, Fall 2003 Professor Mariusz Wodzicki

plugging (20) into ω(x; v1 , v2 , v3 ) and using properties (5)–(10), yields the following simple
formula

ω = f dx ∧ dy ∧ dz where f(x) ˜ ω(x; i, j, k) . (21)

In particular, every differential 3-form on a set D ⊆ R3 , is a multiple, with a function-

coefficient, of the oriented-volume element, dx ∧ dy ∧ dz. Compare this with the situation
regarding 1-forms in R1 , and regarding 2-forms in R2 (see Section 7 of 2F).
The function-coefficient f in (21) is, for obvious reasons, denoted
ω
(22)
dx ∧ dy ∧ dz

(compare this with formula (19) in 2F).

Any differential 3-form on a set in one- or two-dimensional Euclidean space is identically
zero.

8 Example In Section 4 of 2F we calculated df1 ∧ df2 for two functions in R2 . Similarly,

one can calculate df1 ∧ df2 ∧ df3 for three functions in R3 . The formula we obtain is
remarkably similar to formula (8) of 2F:

df1 ∧ df2 ∧ df3 = (det Jf (x)) dx1 ∧ dx2 ∧ dx3 , (23)

 
f1
where f ˜  f2  denotes the vector function D → R3 having f1 , f2 and f3 as its compo-
f3
nents.
Let us collect various formulae for the determinant of the Jacobi matrix of a vector function
f : D → Rd , whose domain D is a subset of Rm ,3 for three smallest values of dimension
3
Recall that such functions are called in College textbooks of Multivariable Calculus vector fields (on a set
D); cf. Section 13 of 2F.

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Math 53M, Fall 2003 Professor Mariusz Wodzicki

m = 1, 2 and 3:
df
det Jf (x) = (for m = 1) (24)
dx
df1 ∧ df2
= (for m = 2) (25)
dx1 ∧ dx2
df1 ∧ df2 ∧ df3
= (for m = 3) . (26)
dx1 ∧ dx2 ∧ dx3

The determinant of the Jacobi matrix of f is often referred to as the Jacobian of f . L

9 The differential of a 2-form We already know that differential df of a function (i.e.,

of a 0-form) is a 1-form and that differential dϕ of a 1-form is a 2-form (see Section 11 of
2F). Now, it is time to extend this operation to 2-forms. For any differential 2-form ψ on
a set D ⊆ Rn , which is represented as in (14), we set
X
dψ ˜ dgjk ∧ dxj ∧ dxk (27)
j,k
X ∂gjk
= dxi ∧ dxj ∧ dxk . (28)
i,j,k
∂xi

10 A calculation: For any function f : D → R and a 2-form ψ on D, one has

d(fψ) = df ∧ ψ + fdψ . (29)

Indeed, it suffices to verify (29) for ψ = g dxj ∧ dxk :

d(fψ) = d(fg dxj ∧ dxk ) = d(fg) ∧ dxj ∧ dxk = (gdf + fdg) ∧ dxj ∧ dxk
= df ∧ (g dxj ∧ dxk ) + f(dg ∧ dxj ∧ dxk )
= df ∧ ψ + fdψ . (30)

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 Math 53M, Fall 2003 Professor Mariusz Wodzicki

11 Another calculation: For any 1-forms ϕ and χ on D, one has

d(ϕ ∧ χ) = dϕ ∧ χ − ϕ ∧ dχ . (31)

Similarly, it suffices to verify (31) for ϕ = f dxj and χ = g dxk :

d(ϕ ∧ χ) = d(f dxj ∧ g dxk ) = d(fg) ∧ dxj ∧ dxk = (gdf + fdg) ∧ dxj ∧ dxk
= (df ∧ dxj ) ∧ (g dxk ) + f(dg ∧ dxj ∧ dxk )
= (df ∧ dxj ) ∧ (g dxk ) − (f dxj ) ∧ (dg ∧ dxk )
= dϕ ∧ χ − ϕ ∧ dχ . (32)

In the last equality in (32), we have used identity (6) from 2F.

12 Yet another calculation: If coefficients of a 1-form ϕ = f1 dx1 + · · · + fn dxn satisfy

the condition
∂2 fi ∂2 fi
= (for all i, j and k), (33)
∂xk ∂xj ∂xj ∂xk
then

d(dϕ) = 0 . (34)

Indeed,

(d ◦ d)(ϕ) = d(df1 ∧ dx1 + · · · + dfn ∧ dxn )

= (d(df1 ) ∧ dx1 + · · · + d(dfn ) ∧ dxn ) − (df1 ∧ d(dx1 ) + · · · + dfn ∧ d(dxn ))
= 0 (35)

in view of formula (31) above and identity (41) in 2F.

The remaining properties of the operation of differential are left to you as an exercise.

W Exercise 4 Verify that for any differential 2-forms ψ, ξ and scalars a and b, one has:

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Math 53M, Fall 2003 Professor Mariusz Wodzicki

(c1 ) (aψ + bξ) = a ψ + b ξ ;

(c2 ) d(f ∗ ψ) = f ∗ dψ.
Here f : E → Rn is a vector function sending its domain into D and the pullback of 3-forms
is defined in exactly the same manner as for 1-forms and 2-forms:

(f ∗ ω)(x; u, v, w) ˜ ω f (x); fx0 (u), fx0 (v), fx0 (w)

. (36)

13 Example: the divergence of a vector field in R3 Let us calculate the differential of an

arbitrary 2-form in R3 :
d(f1 dx2 ∧ dx3 + f2 dx3 ∧ dx1 + f3 dx1 ∧ dx2 )

= df1 ∧ dx2 ∧ dx3 + df2 ∧ dx3 ∧ dx1 + df3 ∧ dx1 ∧ dx2 )

 
∂f1 ∂f1 ∂f1
= dx1 ∧ dx2 ∧ dx3 + dx2 ∧ dx2 ∧ dx3 + dx3 ∧ dx2 ∧ dx3
∂x1 ∂x2 ∂x3
 
∂f2 ∂f2 ∂f2
+ dx1 ∧ dx3 ∧ dx1 + dx2 ∧ dx3 ∧ dx1 + dx3 ∧ dx3 ∧ dx1
∂x1 ∂x2 ∂x3
 
∂f3 ∂f3 ∂f3
+ dx1 ∧ dx1 ∧ dx2 + dx2 ∧ dx1 ∧ dx2 + dx3 ∧ dx1 ∧ dx2
∂x1 ∂x2 ∂x3

∂f1 ∂f2 ∂f3

= dx1 ∧ dx2 ∧ dx3 + dx2 ∧ dx3 ∧ dx1 + dx3 ∧ dx1 ∧ dx2
∂x1 ∂x2 ∂x3
 
∂f1 ∂f2 ∂f3
= + + dx1 ∧ dx2 ∧ dx3 . (37)
∂x1 ∂x2 ∂x3

We have used here properties (12) and (13) of the exterior product, and the fact that ϕ∧ϕ =
0 for any 1-form, see (7) of 2F.
The function-coefficient in (37) is known under the name of divergence4
∂f1 ∂f2 ∂f3
div F ˜ + + . (38)
∂x1 ∂x2 ∂x3
4
The divergence of F is often denoted ∇ ¨ F in Physics textbooks (note the “dot”).

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Math 53M, Fall 2003 Professor Mariusz Wodzicki

of the vector field  

f1
F =  f2  .
f3

In the language that avoids mentioning differential forms, identity (34) becomes the follow-
ing statement:

div(curl F) = 0 . (39)

14 Grand Picture Let ΩqD denote the the set of differential q-forms on a set D ⊆ Rn .
We are already familiar with cases q = 0, 1, 2 and 3. It is not difficult to see how to define
differential q-forms also for higher values of q (make an attempt at such a definition! it’s
worth it).
Sets of differential forms for different values of q are related to each other by means of the
operation of differential:
d d d d
Ω0D −→ Ω1D −→ Ω2D −→ Ω3D −→ · · · (40)

so that the composition of two consecutive operations of differential is zero d ◦ d = 0.

What you see in (40) is called the Rham5 complex of set D. Differential forms η whose
differential is zero: dη = 0, are called closed forms. Forms η which are equal to dξ for
some form ξ are called exact. It follows from what has been just said that

every exact form is closed. (41)

De Rham’s lifetime discovery was that

the extent to which closed forms on a given set

D are not exact provides a very precise mea- . (42)
sure of the geometrical complexity of D

This is what is called de Rham’s theory.

5
Georges de Rham (1903–1990).

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Math 53M, Fall 2003 Professor Mariusz Wodzicki

One can easily extend our definitions of exterior product to arbitrary forms, so that the
product of a p-form η and a q-form ϑ
η∧ϑ
is a (p + q)-form. Then

d(η ∧ ϑ) = dη ∧ ϑ + (−1)p η ∧ dϑ . (43)

The p-th power of −1 in (43) signals that the sign is + for all even values of p and −
for all odd valuse of p.
We already know this formula for p = q = 0 (this is the derivative-of-the-product formula
of Freshman Calculus), p = 0 and q = 1 (this is formula (b) in Section (14) of 2F), p = 0
and q = 2 (this is formula (29) above) and p = q = 1 (formula (31) above). These formulae
are collectively known under the name of Leibniz Rule.

15 Maxwell’s Equations Functions in R3 which evolve “with time” are profitably thought
of as functions on subsets of R4 . We shall denote coordinates in R4 by x0 , x1 , x2 and x3 .6
Any 2-form in R4 can be represented as
F = E1 dx0 ∧ dx1 + E2 dx0 ∧ dx2 + E3 dx0 ∧ dx3
−B1 dx2 ∧ dx3 − B2 dx3 ∧ dx1 − B3 dx1 ∧ dx2 (44)
for unique function-coefficients E1 , E2 , E3 , B1 , B2 and B3 .
Similarly, any 3-form in R4 can be represented as
J = ρ dx1 ∧ dx2 ∧ dx3 − j1 dx0 ∧ dx2 ∧ dx3 − j2 dx0 ∧ dx3 ∧ dx1 − j3 dx0 ∧ dx1 ∧ dx2 (45)
for unique function-coeeficients ρ, j1 , j2 and j3 .
In Electrodynamics, the vector functions
   
E1 B1
E ˜  E2  and B ˜  B2  (46)
E3 B3
6
The physical meaning is x0 = ct, where t stands for time and c denotes the speed of light; x1 = x,
x2 = y and x3 = z are spatial variables.

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Math 53M, Fall 2003 Professor Mariusz Wodzicki

are called the electric and, respectively, magnetic field, the vector function
 
j1
j =  j2  (47)
j3

is called the electric current, and finally, ρ is a scalar-valued function playing the role of the
density of electric charge.
It is remarkable that the whole theory of Electrodynamics7 in the language of differential
forms is contained in the following elegant pair of equations:

dF = 0 and d(∗F) = 4πJ (48)

where ∗F denotes the 2-form:

∗F ˜ B1 dx0 ∧ dx1 + B2 dx0 ∧ dx2 + B3 dx0 ∧ dx3

+E1 dx2 ∧ dx3 + E2 dx3 ∧ dx1 + E3 dx1 ∧ dx2 (49)

The closedness of 2-form F is expressed by the following four equations

∂E2 ∂E3 ∂B1



 − − =0


 ∂x3 ∂x2 ∂x0




 ∂E3 ∂E1 ∂B2

 − − =0
∂x1 ∂x3 ∂x0


(50)

 ∂E1 ∂E2 ∂B3

 − − =0



 ∂x2 ∂x1 ∂x0


 ∂B1 + ∂B2 + ∂B3 = 0



∂x1 ∂x2 ∂x3
7
in vacuum

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Math 53M, Fall 2003 Professor Mariusz Wodzicki

while equation d(∗F) = J is equivalent to the following four

∂B2 ∂B3 ∂E1



 − + = −4πj1


 ∂x3 ∂x2 ∂x0




 ∂B3 ∂B1 ∂E2

 − + = −4πj2
∂x1 ∂x3 ∂x0


(51)

 ∂B1 ∂B2 ∂E3

 − + = −4πj3



 ∂x2 ∂x1 ∂x0


 ∂E1 + ∂E2 + ∂E3 = 4πρ .



∂x1 ∂x2 ∂x3

Collectively, these eight partial differential equations are called Maxwell’s8 Equations.
Some authors of traditional textbooks of Electrodynamics express these eight equations in
the following equivalent form that is more compact:
1 ∂B 1 ∂E 4π
 
 curl E +
 =0  curl B −
 = j
c ∂t and c ∂t c (52)
 
div B = 0 div E = 4πρ ,
 

while others prefer to express the same equations by employing an alternative notation for
curl and div:
1 ∂B 1 ∂E 4π
 
 ∇× E +
 =0  ∇× B −
 = j
c ∂t and c ∂t c (53)
∇¨ B = 0 ∇ ¨ E = 4πρ .

 


16 Integration of 3-forms This is done very similarly to how we did that for 2-forms in
Sections 16–22 of 2F:

(a) rectangles in R2 are replaced by rectangular boxes;

(b) the area of plane regions is replaced by the volume of space regions;
8
In these eight equations, James Clerk Maxwell (1831–1879) gave a mathematical formulation to discov-
eries of Michael Faraday (1791–1867).
Inspired by these equations great physicist Ludwig Boltzmann (1844–1906) exclaimed, in imitation of
Romantic poet Goethe, Was it a God who traced these signs?.

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Math 53M, Fall 2003 Professor Mariusz Wodzicki

ZZ ZZZ
(c) double integrals f(x, y) dxdy are replaced by triple integrals f(x, y, z) dxdydz;
ZZZ D D

in particular, dxdydz = Vol(D);

D

(d) the equality ZZZ Z

f(x, y, z) dxdydz = f dx ∧ dy ∧ dz (54)
D
D

replaces equality (61) from 2F;

(e) the inequality ZZZ

f(x, y, z) dxdydz 6 M Vol(D) (55)
D

replaces inequality (70) from 2F; in particular,

ZZZ
f(x, y, z) dxdydz = 0 (56)
D

for any bounded function f on a set D of zero volume;

(f) “Fubini’s Theorem” for triple integrals

ZZZ Z b3 Z b2 Z b1  
f(x, y, z) dxdydz = f(x, y, z) dx dy dz (57)
a3 a2 a1
I

replaces “Fubini’s Theorem” for double integrals, see (74) in 2F.

(g) The Change of Variables Formula for Triple Integral:

ZZZ ZZZ
f(u, v, w) dudvdw = (f ◦ h)(x, y, z) | det Jh (x, y, z) | dxdydz . (58)
D0 D

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Math 53M, Fall 2003 Professor Mariusz Wodzicki

replaces the corresponding formula for double integrals, see formula (109) in 2F. Here
h : D → D 0 is a diffeomorphism9 of three-dimensional region D onto another region D 0 .

(h) Gauß’–Ostrogradski’s Theorem:10

Let D be a region in R3 whose boundary, ∂D, is a surface that

can be decomposed into regular patches, see Sections 33 and
35 of 2F. Let ψ be a differential 2-form on a region D ⊆ R3 . (59)
Then Z Z
dψ = ψ.
D ∂D

replaces Green’s Theorem (75) of 2F.

Note that the boundary, ∂D, of the region D is automatically oriented. Indeed, as was
explained in Section 34 of 2F, orienting a patch in R3 is the same as telling which ‘side’ is
‘positive’ and which one is ‘negative’. Thus, we orient the patches which are portions of
boundary ∂D, by declaring ‘positive’ the side that faces outside D.

17 Linking number An oriented curve in R3 consisting of two disjoint simple closed

curves C1 and C2 is called a link.11 A link is said to be trivial if loop C1 is contractible in
the complement to C2
E = R3 \ C2 , (60)
see Section 27 in 2F. This definition does not depend on which of the two closed curves is
labelled C1 and which is labelled C2 .
The number of times curve C2 is intertwined with curve C1 is called the linking number
and denoted Ln(C1 , C2 ). In order to determine Ln(C1 , C2 ), project the link onto a plane
P ⊆ R3 such that the ‘shadows’ of constituent curves C1 and C2 intersect transversally,
i.e. they intersect at regular points and they are not tangent when they intersect (cf. Section
29 of 2F). Think of the projected curves as being one-way roads. When they cross, one
9
See Section 32 of 2F.
10
Johann Carl Friedrich Gauß (1777–1855); Mihail Vasil~eviq Ostrogradski (1801–
1862).
11
More precisely, a 2-link. Oriented curves in R3 consisting of n disjoint simple closed curves are called
n-links. 1-links are better known as knots.

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Math 53M, Fall 2003 Professor Mariusz Wodzicki

(a) A trivial link (b) A nontrivial link

Figure 2: Simplest links (orientation not indicated).

of them, the “overpass,” goes over the other one, the “underpass.” Each time they cross

add 1 subtract 1

Figure 3: At each crossing add 1 when the underpass crosses

leftwards and subtract 1 when it crossses rightwards.

add 1 if the underpass crosses leftwards and subtract 1, if it crosses rightwards. Since both
“roads” are closed, they must cross each other an even number of times. Thus, the total is
always an even integer. This integer does not depend on the choice of plane P onto which
we projected the link. By definition,
1
Ln(C2 , C1 ) = Ln(C1 , C2 ) = 2 total .

Linking number of a trivial link is zero.

An alternative definition:
Count only those crossings where C1 is the overpass and, thus, C2 is the underpass. The
total obtained equals Ln(C1 , C2 ).

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Math 53M, Fall 2003 Professor Mariusz Wodzicki

Figure 4: At each of eight crossings the

underpass crosses leftwards, hence the
linking number equals

1
2
(1 + 1 + 1 + 1 + 1 + 1 + 1 + 1) = 4 .

Linking number is an analog of winding number dicussed in Sections 29–30 of 2F. In

particular, there is an analog of Index Formula (103) in 2F. Let γ1 : [a, b] → R3 and
γ2 : [c, d] → R3 be the corresponding parametrizations of C1 and C2 , respectively. The
function
σ(t, u) ˜ γ2 (u) − γ1 (t) (61)
is defined on rectangle [a, b] × [c, d] and its image does not contain the origin, 0, because
γ1 (t) 6= γ2 (u) for all t ∈ [a, b] and u ∈ [c, d] (curves C1 and C2 are disjoint!). One should
think of σ as being a parametric surface in R3 \ {0}. This surface is closed, i.e., does not
have a boundary,12 since curves C1 and C2 are closed.

18 Linking Number Formula The following is a close relative of Index Formula (108) in
2F:

x1 dx2 ∧ dx3 + x2 dx3 ∧ dx1 + x3 dx1 ∧ dx2

Z
1
Ln(C1 , C2 ) = . (62)
4π σ (x21 + x22 + x23 )3/2

This formula can be established similarly to how Index Formula (103) was proved in 2F.
One notes first that the differential 2-form on R3 \ {0}:

1 x1 dx2 ∧ dx3 + x2 dx3 ∧ dx1 + x3 dx1 ∧ dx2

ω2 ˜ , (63)
4π (x21 + x22 + x23 )3/2
12
More properly, one should say that the boundary is empty.

17
Math 53M, Fall 2003 Professor Mariusz Wodzicki

which is sometimes called the Gauß form, is closed, cf. sample problem ?? in Problembook.
Using Gauß’-Ostrogradski’s Theorem, one can show that the integral of a closed 2-form
over a closed surface does not change when one continuously deforms the surface—this
is exactly analogous to Theorem (101) of 2F (which was established using the parametric
form of Stokes’ Theorem, see Section 25 in 2F).
Without loss of generality, one can assume that curves C1 and C2 are parametrized by
interval [0, 1]. Then it can be shown that, if Ln(C1 , C2 ) = m, then parametric surface σ
can be deformed in R3 \ {0} to the function
 
sin(πt) sin(2πmu)
σ1 (t, u) ˜  sin(πt) cos(2πmu)  . (64)
cos t

which parametrizes unit sphere in R3 so that every point of sphere, except for the Northern
and Southern Poles, is ‘visited’ exactly m times. The integral of ω2 over σ1 is m times
the integral of ω2 over the sphere, i.e., equals m (cf., exercise ?? and sample problem ?? in
Problembook).

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