# Introduction to differential 2-forms - UCB Mathematics wodzicki/H185.S11/podrecznik/ ...

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Math 53M, Fall 2003 Professor Mariusz Wodzicki

Introduction to differential 2-formsJanuary 7, 2004

These notes should be studied in conjunction with lectures.1

1 Oriented area Consider two column-vectors

v1 =(v11

v21

)and v2 =

(v12

v22

)(1)

anchored at a point x R2 . The determinant

(x; v1, v2) det(v11 v12

v21 v22

)= v11v22 v21v12 (2)

equals, up to a sign, the area of the parallelogram spanned by v1 and v2 . We will denote

x(v1, v2)

x

v2

7

v1

two ways of ordering a pair of vectors v1and v2 correspond to two ways of orient-ing parallelogram x(v1, v2): if v1 comesfirst then one traverses the boundary ofx(v1, v2) by following the direction of v1 ;if v2 comes first then one follows the di-rection of v2 .

this parallelogram by x(v1, v2) and call quantity (2) its oriented area.

Note the following properties of :

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

(x;au + bv, w) = a(x; u, w) + b(x; v, w) (3)

(x; u,av + bw) = a(x; u, v) + b(x; u, w) (4)

1Abbreviations DCVF and LI stand for Differential Calculus of Vector Functions and Line Integrals, re-spectively.

1

Math 53M, Fall 2003 Professor Mariusz Wodzicki

(b) Antisymmetry: (x; v, u) = (x; u, v) ,

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

2 Differential 2-forms Any function : D Rm Rm R satisfying the above twoconditions will be called a differential 2-form on a set D Rm . By contrast, differentialforms of LI will be called from now on differential 1-forms.

3 Exterior product Given two differential 1-forms 1 and 2 on D , the formula

(x; v1, v2) det(

1(x; v1) 1(x; v2)2(x; v1) 2(x; v2)

)(5)

gives us a differential 2-form. We denote it 1 2 and call it the exterior product of1-forms 1 and 2 .

Note that

2 1 = 1 2 . (6)

Indeed,

(2 1)(x; v1, v2) = det(

2(x; v1) 2(x; v2)1(x; v1) 1(x; v2)

)

= det(

1(x; v1) 1(x; v2)2(x; v1) 2(x; v2)

)= (1 2)(x; v1, v2) .

In particular, for any 1-form one has

= 0 . (7)

W Exercise 1 Verify that for any differential 1-forms , , and 2 scalars a and b, one has:(a1 ) (a + b) = a + b ;

(a2 ) (a+ b) = a + b .

2Greek letter is called khee while letter is called ypsilon.

2

Math 53M, Fall 2003 Professor Mariusz Wodzicki

4 Example Let us calculate df1 df2 where f1 and f2 are two functions D R on asubset of R2 . We have

df1 =f1

x1dx1 +

f1

x2dx2

df2 =f2

x1dx1 +

f2

x2dx2

(it is more instructive to use notation x1 and x2 instead of x and y), and

df1 df2 =

(f1

x1dx1 +

f1

x2dx2

)

(f2

x1dx1 +

f2

x2dx2

)

=f1

x1

f2

x2dx1 dx2 +

f2

x1

f1

x2dx2 dx1 (since dxi dxi = 0)

=

(f1

x1

f2

x2f1

x2

f2

x1

)dx1 dx2 (since dx2 dx1 = dx1 dx2)

= (det Jf (x))dx1 dx2 . (8)

where f(f1

f2

)denotes the vector function D R2 having f1 and f2 as its components.

5 dx dy Note that

dx dy (x; v1, v2) = det(v11 v12

v21 v22

)(9)

which is the right-hand-side of (2) and, up to a sign, the area of parallelogram formed bycolumn-vestors v1 and v2 at point x R2 . We call the differential 2-form on R2 , dx dy,the oriented-area element.

6 Basic differential forms dxidxj Differential forms dxidxj , i 6= j, on Rm are calledbasic differential 2-forms. What is their meaning?

If u =

u1...um

and v = v1...vm

, thendxi dxj (x; u, v) = det

(ui vi

uj vj

)(10)

3

Math 53M, Fall 2003 Professor Mariusz Wodzicki

which is the (oriented) area of the parallelogram

x(u, v) (11)

where the column-vectors

u(ui

uj

)and v

(vi

vj

), (12)

and the point

x(xi

xj

)(13)

are projections of column-vectors u and v , and point x, respectively, onto the plane R2xixjspanned by xi - and xj -axes.3

7 2-forms on R2 Let be any differential 2-form on a set D R2 . For a pair ofcolumn-vectors

v1 = v11 i + v12 j (14)

v2 = v21 i + v22 j (15)

to calculate value (x; v1, v2) we plug first (14) and use Property (a1 ) from Exercise 1:

(x; v1, v2) = (x; v11 i + v12 j, v2) = v11(x; i, v2) + v12(x; j, v2) , (16)

and then plug (15) into the right-hand-side of (16) and use Property (a2 ) from the sameexercise:

= v11(v21(x; i, i) + v22(x; i, j)) + v12(v21(x; j, i) + v22(x; j, j))

= (v11v22 v21v12)(x; i, j)

= (x; i, j) (dx dy)(x; v1, v2)

= ((x; i, j)dx dy)(x; v1, v2) . (17)

3In other words, parallelogram x(u, v) is obtained by projecting parallelogram x(u, v) onto xi xj -plane.

4

Math 53M, Fall 2003 Professor Mariusz Wodzicki

In other words, any differential 2-form on a subset of R2 can be represented as a multipleof the oriented-area element:

= f dx dy where f(x)(x; i, j) . (18)

The function-coefficient f in (18) is, for obvious reasons, denoted

dx dy. (19)

8 2-forms on R3 A similar, completely straightforward, calculation shows that any 2-form on a subset D R3 can be represented as

= f1 dy dz+ f2 dz dx+ f3 dx dy (20)

or,

= f1 dx2 dx3 + f2 dx3 dx1 + f3 dx1 dx2 (21)

if one uses notation x1 , x2 , x3 instead of x, y, z.

W Exercise 2 Verify thatf1(x) = (x; j, k) , f2(x) = (x; k, i) and f3(x) = (x; i, j) . (22)

A very important observation follows from formulae (2021):

on subsets of R3 , and of R3 only, both differential 1-forms and differential2-forms are given in terms of three function-coefficients f1 , f2 and f3

. (23)

5

Math 53M, Fall 2003 Professor Mariusz Wodzicki

9 Area element The function that associates with a pair of column-vectors v1 and v2anchored at apoint x Rm , the area of parallelogram x(v1, v2) will be called the areaelement and denoted .

We already know that in R2 the area element coincides with the absolute value of basicdifferential 2-form

= |dx1 dx2| = |dx dy| . (24)

In general, for vectors in Euclidean space Rm , we have the formula

(x; v1, v2) = v1 v2 sin (]v2v1) = (v1 v2)2 (1 cos2 (]v2v1))

=

(v1 v2)2 (v1 v2)2 (25)

Let us see what does this formula look like in R3 . We have:

(v1 v2)2 = (v211 + v212 + v213)(v221 + v222 + v223) (26)

and(v1 v2)2 = (v11v21 + v12v22 + v13v23)2 . (27)

After expanding the right-hand side of (27) and subtracting it from (26), we get the follow-ing formula for the area element in R3 :

(x; v1, v2) =

(det ( v21 v32v31 v22

))2+

(det

(v31 v12

v11 v32

))2+

(det

(v11 v22

v21 v12

))2(28)

Recognizing that the 2 2 determinants are just the values of basic forms dx2 dx3 ,dx2 dx3 and dx2 dx3 , we can rewrite (28) in more legible (as well as more easilymemorizable!) form:

=

(dx2 dx3)2 + (dx3 dx1)2 + (dx1 dx2)2 . (29)

This is Pythagoras Theorem4 for the area function, since identity (29) can be expressed

4PUAGORAS (6th Century BC), one of the most mysterious and influential figures in Greek, and there-fore also our, intellectual history. He was born in Samos in the mid-6th century BC and migrated to Croton

6

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Pythagoras.html

Math 53M, Fall 2003 Professor Mariusz Wodzicki

also as saying:

The square of the area of a parallelogram is the sumof the squares of areas of orthogonal projections ofthat parallelogram onto all coordinate planes.

(30)

As stated, Theorem (30) holds for any n. For n = 2 , formula (29) reduces to formula (24).

For n = 3 , the coordinate planes are R3x2x3 , R3x3x1

and R3x1x2 , respectively.

10 Example: cross-product of vectors in R3 Let us calculate the exterior product of two1-forms on R3

(a1dx1 + a2dx2 + a3dx3) (b1dx1 + b2dx2 + b3dx3) (31)

with constant coefficients a1 , a2 , a3 , b1 , b2 , b3 . The result is the sum of 3 3 = 9 formsaibjdxi dxj . However, three of them are zero, since dxi dxi = 0 . For the remaining six,one has aibjdxi dxj = bjaidxj dxi , so the final result is the following combinationof three basic 2-forms on R3 :

(a2b3 a3b2)dx2 dx3 + (a3b1 a1b3)dx3 dx1 + (a1b2 a2b1)dx1 dx2 (32)

in around 530 BC. There he founded the sect or society that bore his name, and that seems to have playedan important role in the political life of Magna Graecia for several generations. Pythagoras himself is said tohave died as a refugee in Metapontum. Pythagorean political influence is attested well into the 4th century,with Archytas of Tarentum.

The name of Pythagoras is connected with two parallel traditions, one religious and one scientific. Pythago-ras is said to have introduced the doctrine of transmigration of souls into Greece, and his religious influenceis reflected in the cult organization of the Pythagorean society, with periods of initiation, secret doctrines andpasswords (akousmata and symbola), special dietary restrictions, and burial rites. Pythagoras seems to havebecome a legendary figure in his own lifetime and was identified by some with the Hyperborean Apollo. Hissupernatural status was confirmed by a golden thigh, the gift of bilocation, and the capacity to recall his pre-vious incarnations. Classical authors imagine him studying in Egypt; in the later tradition he gains universalwisdom by travels in the east. Pythagoras becomes the pattern of the divine man: at once a sage, a seer, ateacher, and a benefactor of the human race.

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