Brouwer Degree in Two Dimensions

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Brouwer Degree in Two Dimensions

Bob Lutz

Pomona College ’13

August 29, 2011

1 Introduction

Having developed out of early 20th

century work in topology by J. Hadamardand L. E. J. Brouwer, degree theory is fundamentally concerned with the exis-tence, number, and nature of solutions to the equation

f (x) = y

in a suitable space. For our purposes, this will be R2.Because its value can guarantee solutions within a domain, the Brouwer de-

gree in particular is a useful tool in the study of nonlinear differential equations.The objective of these notes is to motivate a basic comprehension of degree the-ory to this end by providing an analytical discussion of the Brouwer degree intwo dimensions. We begin by introducing the Kronecker index, an analogousconcept with a tractable integral formulation.

Let F := P (x, y)ı+Q(x, y) ∈

C 2(U,R2) where U ⊆R2 is open, and consider

θ, the angle of F at (x, y) relative to the positive x-axis, given by

θ = tan−1

Q

P

for P = P (x, y) and Q = Q(x, y). Consider also

σ := dθ =P dQ − Q dP

P 2 + Q2, (1)

which describes the change in θ as (x, y) varies. When integrated over a closedcurve, σ measures the total change in the angle of F . We formalize this conceptwith the planar notion of the Kronecker index.

Definition 1.1 (Kronecker index). Let F be defined as above, and let D ⊂D ⊆ U be open such that ∂D is a positively oriented C 1 Jordan curve. Supposethat F (x, y) = (0, 0) for all (x, y) ∈ ∂D. The Kronecker index of F around ∂Dis given by

indexK(F,∂D) ≡ 1

2π

∂D

σ. (2)

1



This expression calculates the number of times the curve F (∂D) “winds”around the origin; fittingly, this is called the winding number of F (∂D) in the

complex plane. More pertinently, however, the integral expression in (2) alsodescribes the planar Brouwer degree for the case in which F is C 1. Theseconnections (particularly the latter) will be developed throughout the notes aswe expand the above definition by using the Kronecker index to address theBrouwer degree generally for the case of continuous F and ∂D.

2 Examples

We present several examples and provide computations of the Kronecker indexfor each.

2.1 Identity map

Let id be the identity map defined by

id(x, y) = xı + y for all (x, y) ∈ R2, (3)

and let D be the unit disk. From (2) we have that


2π

S1

x dy − y dx

x2 + y2.

Parameterizing S 1 by (x, y) = (cos t, sin t) with t ∈ [0, 2π] gives

1

2π

S1

x dy − y dx

x2 + y2=

1

2π

2π0

cos2 t + sin2 t

cos2 t + sin2 tdt

=1

2π

2π0

dt

= 1.

Hence, indexK(F,∂D) ≡ 1.

2.2

Let F (x, y) := −yı + x for all (x, y) ∈ R2, and let D be the unit disk. From

(2) we have that

indexK(F,∂D)

≡1

2π S1x dy − y dx

x2

+ y2

.

As in 2.1, the parameterization (x, y) = (cos t, sin t) of S 1 gives


2π

2π0

dt = 1.

2



2.3

Let F (x, y) := (x2

− y2

)ı + 2xy for all (x, y) ∈ R2

, and let D be the unit disk.From (2) we have that


2π

S1

(x2 − y2)d(2xy) − 2xy d(x2 − y2)

(x2 − y2)2 + (2xy)2. (4)

Note thatd(2xy) = 2x dy + 2y dx (5)

andd(x2 − y2) = 2x dx − 2y dy. (6)

After a bit of algebra, combining (4), (5), and (6) yields

indexK(F,∂D)

≡

1

2π S1(2x3 + 2xy2) dy − (2x2y + 2y3) dx

(x2

+ y2

)2

≡ 1

π

S1

x dy − y dx

x2 + y2.

As in 2.1 and 2.2, the parameterization (x, y) = (cos t, sin t) of S 1 gives


π

2π0

dt = 2.

Alternatively, note that F (sin t, cos t) = (sin 2t, cos2t) on S 1.

3 Properties of the Kronecker index

It is necessary to establish several properties of the Kronecker index in R2. We

denote by · the Euclidean norm on R2, and by · C 1(U ) the C 1 norm on U given by

GC 1(U ) = sup(x,y)∈U

G + sup(x,y)∈U

DG

for G ∈ C 1(U ), where DG denotes the derivative of G. We refer to Stokes’theorem as given in [5, p. 124], i.e.

∂D

ω =

D

dω,

where ω is a differential 1-form on D.

3.1 Winding number

Given the introductory explanation and examples, it may seem intuitive thatthe Kronecker index is integer-valued. To prove this fact, however, we mustexamine the index in the complex plane. Expressed as a special case of theCauchy integral formula, the complex winding number is readily comparable toour integral formulation of the Kronecker index.

3



Definition 3.1. Suppose that F ∈ C 1(U,C) where U ⊂ C is open, and letD

⊂U be open such that ∂D is a positively oriented C 1 Jordan curve. The

winding number of F around ∂D is given by

1

2πi

∂D

dF

F .

Following this definition, suppose that F , U , and D are defined as above,and write F (x + iy) ≡ P (x, y) + iQ(x, y), so that dF = dP + i dQ. We havethat ln F = ln |F | + i arg F ; a brief calculation yields

1

2πi

∂D

dF

F =

1

2πi

∂D

d(ln F )

=1

2πi

∂D

d(ln |F |) +1

2π

∂D

d(arg F )

= 12π

∂D

dtan−1 QP

= indexK(F,∂D),

noting that ∂D

d(ln |F |) = 0, since exact differential forms are closed.Using this formulation, it can be shown that the winding number (or, equiv-

alently, the Kronecker index) is an integer. We adapt the argument from [4,p. 115] and [3, p. 10].

Lemma 3.2. Let F ∈ C 1(U,R2), and suppose that F = (0, 0) for all (x, y) ∈∂D. Then, indexK(F,∂D) is an integer.

Proof. Let φ : [0, 2π] → ∂D be a parameterization defined by φ(t) = (x(t), y(t)),

and let Φ(t) := x(t) + iy(t). Consider the function

h(t) :=

ta

ddτ

F (Φ(τ ))

F (Φ(τ ))dτ for all t ∈ [a, b].

By the fundamental theorem of calculus along curves, we have that

h(t) =ddtF (Φ(t))

F (Φ(t)).

Consider also g(t) := e−h(t)F (Φ(t)). We have that

g(t) = ddt [F (Φ(t))] e−h(t) − h(t)e−h(t)F (Φ(t))

= e−h(t) ddt

F (Φ(t))

−h(t)F (Φ(t))= 0 for all t ∈ [a, b].

Thus, g(t) is constant, so g(a) = F (Φ(a)) = g(t) for all t ∈ [a, b]. It follows that

eh(t) =F (Φ(t))

F (Φ(a)).

4



Since Φ(a) = Φ(b), we have that eh(b) = 1. Hence, h(b) = 2πmi for some m ∈ Z,so that

indexK(F,∂D) = h(b)2πi

= m.

3.2 Basic results

The behavior of indexK(F,∂D) depends on the existence and location of equi-librium points of F : i.e., points (x, y) ∈ U where F (x, y) = (0, 0). The followingseveral theorems describe this relationship explicitly. Initially, we address onlythe case of C 2 vector fields and C 1 curves, but the concepts are subsequentlygeneralized as we address analogous properties of the Brouwer degree.

Unless stated otherwise, U and D are defined as in Definition 1.1. Webegin with the solution property, a result which can guarantee the existence of equilibrium points of F in D.

Theorem 3.3 (Solution property of the Kronecker index). Let F = P ı + Q ∈C 2(U,R2) be such that F = (0, 0) for all (x, y) ∈ ∂D. If indexK(F,∂D) = 0,

then there exists ( p,q) ∈ D such that F ( p,q) = (0, 0).

Proof. We prove the contrapositive. Suppose that F (x, y) = (0, 0) for all (x, y) ∈D, and let σ be defined as in (1). Then σ is a differential 1-form on D, sinceP 2 + Q2 = 0 on D. We have by Stokes’ theorem that

indexK(F,∂D) =

D

dσ, (7)

where

dσ = d

P

P 2 + Q2

∧ dQ − d

Q

P 2 + Q2

∧ dP, (8)

since d2Q = d2P = 0 for C 2 fields P and Q. Note that

d

P

P 2 + Q2

=

dP

P 2 + Q2− 2

(P 2 + Q2)2(P 2 dP + P Q dQ) (9)

and

d

Q

P 2 + Q2

=

dQ

P 2 + Q2− 2

(P 2 + Q2)2(P Q dP + Q2 dQ). (10)

Combining (8), (9), and (10) and applying the anti-commutativity of the wedgeproduct yields

dσ

2=

1

P 2 + Q2dP ∧ dQ +

P Q dP + Q2 dQ

(P 2 + Q2)2 ∧ dP − P 2 dP + P Q dQ

(P 2 + Q2)2 ∧ dQ.

We use the fact that dQ ∧ dQ = dP ∧ dP = 0 to obtain

dσ = 2

1

P 2 + Q2− 1

P 2 + Q2

dP ∧ dQ = 0.

Hence, the statement follows from (7).

5



Much of differential equation theory investigates conditions which guaranteesolutions within a domain; given this statement of the solution property, then,

its application to the study of differential equations is more clear. We make useprimarily of the contrapositive in subsequent proofs, most notably in that of theexcision property.

Parameterizing the boundary of a given D as in Section 2 is often difficult,if not impossible, when computing the Kronecker index. The excision propertyallows us to excise open subsets of D not containing equilibrium points of F while maintaining the value of the index, in order to attain a region with a moremanageable boundary.

Theorem 3.4 (Excision property of the Kronecker index). Let D1 ⊂ D be open

such that ∂D1 is a positively oriented C 1 Jordan curve. Suppose F (x, y) = (0, 0)on D \ D1. Then,

indexK(F,∂D) = indexK(F,∂D1).

Proof. We have by the Jordan curve theorem that D1 lies in the interior of ∂D. Since F (x, y) = (0, 0) on D \ D1, it follows from the contrapositive of thesolution property that

indexK(F,∂D \ D1) ≡ 1

2π

∂D−∂D1

σ = 0.

Note that ∂ (D \ D1) = ∂D − ∂D1, since ∂D1 is traversed as illustrated below.

Figure 1: Positively orienting ∂ (D \ D1).

By Stokes’ theorem, we have that

∂D−∂D1

σ =

Ddσ

− D1

dσ = 0.

Hence, ∂D

σ =

∂D1

σ,

from which the statement follows.

6



In some cases, D may be open but disconnected. However, it is not imme-diately clear from Definition 1.1 how to calculate the index of F around the

corresponding ∂D. This discrepancy is addressed by the additivity property,which provides a convenient sum formulation for the index in such cases.

Theorem 3.5 (Additivity property). Suppose that D is the union of disjoint

open sets D1, . . . , Dm where ∂Di is a C 1 Jordan curve for i = 1, . . . , m. Addi-

tionally, suppose that F (x, y) = (0, 0) for all (x, y) ∈ ∂Di. Then,

indexK(F,∂D) =mi=1

indexK(F,∂Di).

Proof. Suppose that φi : [a, b] → ∂Di is a parametrization for i = 1, . . . , m.Then we have for φi = φi(t) that

indexK(F,∂D) =1

2π ∂DP (x, y) dQ − Q(x, y) dP

P (x, y)2 + Q(x, y)2

=1

2π

mi=1

ba

P (φi)ddt

Q(φi) − Q(φi)ddt

P (φi)

P (φi)2 + Q(φi)2dt

=mi=1

indexK(F,∂Di).

While it is difficult to generalize about the behavior of the Kronecker index,the normalization property provides a simple evaluation for a specific case of F .

Theorem 3.6 (Normalization property). Let id : R2 → R2 be defined as in

Example 2.1. Then,

indexK(id,D) = 1 if (0, 0)

∈D

0 if (0, 0) /∈ D.

Proof. Suppose (0, 0) ∈ D. Since D is open, there exists ε > 0 such thatDε = {(x, y) : x2 + y2 < ε} ⊂ D with F (x, y) = (0, 0) on S 1ε = ∂Dε. Note thatS 1ε is C 1, and that (0, 0) /∈ id(D \ Dε). We have that

indexK(F,∂Dε) ≡ 1

2π

S1ε

x dy − y dx

x2 + y2.

Parametrizing S 1ε by (x, y) = (ε cos t, ε sin t) gives


2π

2π0

ε2

ε2dt = 1.

Hence, by the excision property,

indexK(F,∂D) = indexK(F,∂Dε) = 1.

Next, suppose (0, 0) /∈ D. Then we have that id(x, y) = (0, 0) for all (x, y) ∈ D.It follows from the contrapositive of the solution property that

indexK(F,∂D) = 0.

7



4 Defining the Brouwer degree

We begin our approach of the Brouwer degree with a proof of the continuityof the Kronecker index. This will allow us to properly address the use of closeapproximations of F in defining the degree.

Lemma 4.1. Let F, G ∈ C 1(U,R2), and suppose that F, G = 0 for all (x, y) ∈∂D. Then, for every ε > 0, there exists δ > 0 such that

F − GC 1(∂D) < δ ⇒ |indexK(F,∂D) − indexK(G,∂D)| < ε.

Proof. Let F := F 1(x, y)ı + F 2(x, y) and G be defined similarly. We have that

|indexK(F,∂D) − indexK(G,∂D)|=

1

2π ∂DF 1 dF 2 − F 2 dF 1

F 21 + F

22 − ∂D

G1 dG2 − G2 dG1

G21 + G

22

Let φ : [a, b] → ∂D be a parameterization defined by φ(t) = (x(t), y(t)), and letψ : R4 → R be defined by

ψ( p,q,r,s) =ps − qr

p2 + q2. (11)

Writing ψF,x = ψ(F 1, F 2, (F 1)x, (F 2)x) and similarly for ψF,y, ψG,x, and ψG,y,we have after grouping terms that

1

2π

∂D

F 1 dF 2 − F 2 dF 1

F 21 + F 22− G1 dG2 − G2 dG1

G21 + G2

2

= 12π

ba [(ψF,x − ψG,x)x(t) + (ψF,y − ψG,y)y(t)] dt≤ 1

2π

ba

|(ψF,x − ψG,x, ψF,y − ψG,y) · φ(t)| dt.

Let Ψ : C 1(∂D,R2) → R2 be defined by Ψ(F ) = (ψF,x, ψF,y). We have by the

Cauchy-Schwarz inequality that

1

2π

ba

|(Ψ(F ) − Ψ(G)) · φ(t)| dt ≤ 1

2π

ba

Ψ(F ) − Ψ(G)φ(t) dt

≤ (∂D)

2πmax

(x,y)

∈∂D

Ψ(F ) − Ψ(G),

where (∂D) is the arclength of ∂D. We claim that for every ε > 0, there existsδ > 0 such that

F − GC 1(∂D) < δ ⇒ max(x,y)∈∂D

Ψ(F ) − Ψ(G) <2πε

(∂D). (12)

8



Note that ψ = ψ( p,q,r,s) defined in (11) is continuous for p2 + q2 = 0. Con-sequently, ψ is uniformly continuous on any compact K

⊆R4

\ { p2 + q2 = 0

}.

Thus, for every ε1 > 0, there exists δ1 > 0 such that

( p1, q1, r1, s1) − ( p2, q2, r2, s2) < δ1

⇒ |ψ( p1, q1, r1, s1) − ψ( p2, q2, r2, s2)| < ε1, (13)

for ( p1, q1, r1, s1), ( p2, q2, r2, s2) ∈ K . In particular, let ε1 =√2πε

(∂D) , and let

K =

( p,q,r,s) : δo ≤ ( p,q) ≤ δ1 + F C 1(∂D), (r, s) ≤ δ1 + F C 1(∂D)

.

Note that K is compact, and that if F − GC 1(∂D) < δ1, then

(F 1, F 2, (F 1)α, (F 2)α), (G1, G2, (G1)α, (G2)α) ∈ K

for α = x, y and all (x, y)

∈∂D, since

G

≤δ1 +sup(x,y)

∈∂D

F

, and

DG

≤δ1 + sup(x,y)∈∂D DF . Set

ν α = (F 1, F 2, (F 1)α, (F 2)α) − (G1, G2, (G1)α, (G2)α).

We have thatν x2 + ν y2 ≤ F − G2C 1(∂D) < δ21,

so ν α < δ1. We have by (13) that |ψF,α− ψG,α| < ε1 for all (x, y) ∈ ∂D; thus,

Ψ(F ) − Ψ(G)2 = |ψF,x − ψG,x|2 + |ψF,y − ψG,y|2 < 2ε21.

Hence,

F − GC 1(∂D) < δ1 ⇒ Ψ(F ) − Ψ(G) <√

2ε1 =2πε

(∂D),

so the claim in (12) is proven.

An immediate consequence of Lemma 4.1 is the homotopy invariance of theKronecker index. We are particularly concerned with C 1 homotopies.

Definition 4.2. Let G ∈ C 1(U,R2), and suppose G(x, y) = (0, 0) for all (x, y) ∈∂D. A C 1 homotopy between F and G is a function H : D × [0, 1] → R

2 whereH t denotes the function (x, y) → H ((x, y), t) such that

(i) H 0 = F , H 1 = G, and H t ∈ C 1(D) for t ∈ [0, 1]

(ii) lims→t H s − H tC 1(D) = 0.

Homotopy invariance allows us to deform F while maintaining the value of the Kronecker index.

Theorem 4.3 (Homotopy invariance of the Kronecker index). Let F and Gbe defined as above, and let H ((x, y), t) be a C 1 homotopy between them. If

H ((x, y), t) = (0, 0) for all ((x, y), t) ∈ ∂D × [0, 1], then

indexK(F,∂D) = indexK(G,∂D).

9



Proof. Let h : [0, 1] → Z be defined by h(t) = indexK(H t, ∂D). We have byLemmata 3.2 and 4.1 that h maps [0, 1] to a connected component in Z; thus,

we have that h(t) = n for some integer n and all t ∈ [0, 1]. In particular,indexK(H 0, ∂D) = indexK(H 1, ∂D), from which the statement follows.

For F whose Kronecker index is difficult to calculate, it is often simpler toprove that F is homotopic to a vector field with a more easily calculable index.Homotopy invariance allows us this convenience.

In order to extend homotopy invariance and other important properties of the Kronecker index to the Brouwer degree for continuous F , we require a C 2

approximation. The following result, adapted from [2, p. 4], allows us to choosesuch a function in any neighborhood of F .

Proposition 4.4. Suppose that F ∈ C (U,R2), and let D ⊂ D ⊆ U be open

such that ∂D is compact. For any ε > 0, there exists G ∈ C ∞(U,R2) such that

F − GC (∂D) < ε.With this, we have finally established the results necessary to define the

Brouwer degree explicitly for continuous F and D whose boundary is C 1.

Definition 4.5 (Brouwer degree). Let F ∈ C (U,R2) be such that F (x, y) =(0, 0) for all (x, y) ∈ ∂D, and let G ∈ C 2(U,R2) be such that

G − F C (∂D) < min(x,y)∈∂D

F (x, y). (14)

The Brouwer degree of F in D over (0, 0), denoted by degB(F,D, (0, 0)), is givenby

degB(F,D, (0, 0)) = indexK(G,∂D).

In order to justify the above definition, it suffices to show that any two C 2

approximations of F yield the same Kronecker index. We adapt the argumentfrom [6, p. 18].

Let F ∈ C (U,R2), and set

µ = min(x,y)∈∂D

F .

Let Gi ∈ C 1(U,R2) for i = 1, 2 be such that F − GiC (∂D) < µ. Note thatGi = 0 for all (x, y) ∈ ∂D, and consider the C 1 homotopy H defined by

H t(x, y) = tG1(x, y) + (1 − t)G2(x, y) for all (x, y) ∈ D, ∈ [0, 1],

where H t(x, y) = H ((x, y), t). We have that

H t − F C (∂D) = t(G1 − F ) + (1 − t)(G2 − F )C (∂D)

< tµ + (1 − t)µ = µ,

so if (x, y) ∈ ∂D, then

H t(x, y) ≥ F (x, y) − F (x, y) − H t(x, y) > 0 for all t ∈ [0, 1].

10



Thus, H ((x, y), t) = 0 for all ((x, y), t) ∈ ∂D × [0, 1]. We have by homotopyinvariance that

indexK(G1, ∂D) = indexK(G2, ∂D).

Consequently, indexK(G,∂D) is constant for all G ∈ C 1(U,R2) with G = 0on ∂D such that F − GC (∂D) < µ. Our definition of the Brouwer degree istherefore constant for any C 1 approximation G of F which satisfies the estimatein (14).

5 Properties of the Brouwer degree

With a definition of the Brouwer degree now established, we wish to extend sev-eral key properties of the Kronecker index to the Brouwer degree for continuousF , beginning with the solution property. In particular, we write degB(F, D) =deg

B(F,D, (0, 0)).

Theorem 5.1 (Solution property of the Brouwer degree). If degB(F, D) = 0,

then there exists ( p,q) ∈ D such that F ( p,q) = (0, 0).

Proof. We prove the contrapositive. Suppose that F (x, y) = (0, 0) for all (x, y) ∈D, and approximate F by G ∈ C 2(U,R2) such that

G − F C (D) < min(x,y)∈D

F (x, y).

Observe that G(x, y) = (0, 0) for all (x, y) ∈ D. We have by Theorem 3.3 that

indexK(G,∂D) = 0.

Hence, it follows from Definition 4.5 that degB(F, D) = 0.

The excision property follows naturally.

Theorem 5.2 (Excision property of the Brouwer degree). Let D1 ⊂ D be

open such that ∂D is a C 1 Jordan curve. Suppose that F (x, y) = 0 for all

(x, y) ∈ D \ D1. Then,

degB(F, D) = degB(F, D1).

Proof. Approximate F by G ∈ C 2(U,R2) such that

G − F C (D\D1)< min

(x,y)∈D\D1

F (x, y).

We have thatG(x, y) = (0, 0) for all (x, y) ∈ D \ D1.

Hence, by Theorem 3.4 and Definition 4.5,

degB(F, D) = degB(F, D1).

11



As was the case for the Kronecker index, we must first establish the conti-nuity of the Brouwer index before discussing homotopy invariance.

Lemma 5.3. Let G ∈ C (U,R2), and suppose that F, G = 0 for all (x, y) ∈ ∂D.

Then, for every ε > 0, there exists δ > 0 such that

F − GC (∂D) < δ ⇒ |degB(F,∂D) − degB(G,∂D)| < ε.

Proof. Set ε > 0, and let

µ = min(x,y)∈∂D

F (x, y).

Suppose that G − F C (∂D) < µ/4. Let G be a C 1 approximation of G such

that

G − GC (∂D) < µ/4. It follows that

G = (0, 0) for all (x, y) ∈ ∂D.

Furthermore, since

G − F C (∂D) ≤ G − GC (∂D) + G − F C (∂D) < µ/2,

we have thatdegB(F, D) = indexK(G∂D) = degB(G, D).

Hence,

F − GC (∂D) < µ/4 ⇒ |degB(F,∂D) − degB(G,∂D)| = 0 < ε.

Note that, as a consequence of Lemma 3.2 and Definition 4.5, the Brouwerdegree is integer-valued. This fact allows us to prove the homotopy invarianceof the Brouwer degree.

Theorem 5.4 (Homotopy invariance of the Brouwer degree). Let G ∈ C (U,R2

),and let H : D × [0, 1] → R

2 be a continuous homotopy between F and G. If

H ((x, y), t) = (0, 0) for all ((x, y), t) ∈ ∂D × [0, 1], then

degB(F,∂D) = degB(G,∂D).

Proof. Let H : U ×[0, 1] → R2 be a continuous map. Suppose that H ((x, y), t) =

(0, 0) for all ((x, y), t) ∈ ∂D × [0, 1], and define the map f : [0, 1] → Z by

f (t) = degB(H t, D, (0, 0) for all t ∈ [0, 1],

where H t(x, y) = H ((x, y), t) for all ((x, y), t) ∈ D × [0, 1]. We have by Lemma5.3 that f maps [0, 1] to a connected component in Z. In particular, f (0) = f (1),from which the statement follows.

6 Generalizing for continuous ∂D

We wish to generalize our definition of the Brouwer degree to address D witha continuous boundary. In order to do so, we must construct several curves

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approximating ∂D; however, we first require a neighborhood of ∂D on whichF (x, y)

= (0, 0).

Let D ⊂ D ⊆ U be open such that ∂D is a Jordan curve. Additionally, sup-pose that F = (0, 0) for all (x, y) ∈ ∂D. Then there exists an open neighborhoodV of the boundary of D such that F = (0, 0) for all (x, y) ∈ V .

To see why this is the case, note that if ( x, y) ∈ ∂D, then there exists anopen disk around (x, y) of radius δ(x, y) > 0 such that F > 0 on the closureof the disk. The collection of disks around all (x, y) ∈ ∂D forms an open coverof ∂D. Since ∂D is compact, there exists a finite subcover of disks whose unionwe take as V .

With our neighborhood defined, we require a piecewise smooth approxima-tion of ∂D which lies in V . The following result, adapted from [1, pp. 588–589],allows guarantees the existence of a polygon in any neighborhood of ∂D.

Lemma 6.1. Suppose that J

⊂R2 is a Jordan curve with parameterization φ :

[a, b] → J . For any ε > 0, there exists a polygon P ⊂ R2 with parameterization ψ : [a, b] → P such that φ(t) − ψ(t) < ε for all t ∈ [a, b].

Let φ(t) : [a, b] → ∂D be a parameterization. By Lemma 6.1, there exists apolygon P with parameterization ψ : [a, b] → P such that

φ(t) − ψ(t) < dist(∂D,∂V ) for all t ∈ [a, b].

Note that F = (0, 0) for all (x, y) in some neighborhood of P , since P ⊂ V . Byreplacing all pi with arcs, it is possible to construct a differentiable curve in thisneighborhood, thereby “smoothing” the polygon. Denote the n vertices of P by pi for i = 1, . . . , n, and denote by ωi the smaller of the interior and exteriorangles of each pi.

The smoothing procedure is as follows. For each pi, we draw a circle in ωi

with two points of tangency to P , as illustrated in Figure 2. A curve is thenconstructed by removing the vertex as well as the line segments connecting itto either point of tangency and replacing these with the minor arc of the circlebetween the same two points. After this has been done for all n vertices, i.e. P has been smoothed at all pi, the resulting curve P is C 1.

Observe that the quantity ri tan ωi corresponds to the length of the linesegments removed when smoothing P at pi, also as illustrated in Figure 2. Wemust choose the radius ri of each circle such that

ri tan ωi < min

pi − pi−12

, pi+1 − pi

2, dist(P,∂V )

,

adjusting the indices appropriately for the cases i = 1 and i = n. The restric-

tions ri tan ωi <1

2 pi − pi−1 and ri tan ωi <1

2 pi − pi+1 guarantee that thesum of the lengths of line segments removed from each edge of P does not exceedthe length of the edge itself. The restriction ri tan ωi < dist(P, ∂V ) guarantees

that P ⊂ V , and furthermore that F = (0, 0) for all (x, y) ∈ P .

The curve P is a C 1 approximation of ∂D with respect to F . By excision,the Brouwer degree of F over (0, 0) is equal in the interior of any two smooth

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Figure 2: P , in blue, replaces red segments with an arc, smoothing P at pi.

approximations of a Jordan curve ∂D. With this result, we may state thefollowing definition.

Definition 6.2. Let F ∈ C (U,R2), and let D ⊂ D ⊂ U be open such that∂D is a positively oriented Jordan curve. Suppose that F (x, y) = (0, 0) for

all (x, y) ∈ ∂D, and let P be a positively oriented C 1 Jordan curve such thatP ⊂ V for some neighborhood V of ∂D on which F = (0, 0). The Brouwerdegree of F in D over (0, 0) is given by

degB(F, D) = degB(F, int( P )).

7 Exercises

1. Suppose that F ∈ C 2(U,R2), and that F (∂D) ⊆ S 1. Prove that


π

D

det(DF ) dx dy.

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References

[1] C. Jordan, Cours d’analyse de l’ Ecole polytechnique, vol. 3, Gauthier-Villars,Paris, 1887.

[2] D. O’Regan et al., Topological Degree Theory and Applications, Chapman &Hall, Boca Raton, 2006.

[3] G. Dinca and J. Mawhin, Brouwer Degree and Applications, Birkhauser,Basel, 2012 (preprint).

[4] L. Ahlfors, Complex analysis: an introduction to the theory of analytic func-

tions of one complex variable, McGraw- Hill, New York, 1953.

[5] M. Spivak, Calculus on Manifolds: a modern approach to classical theorems

of advanced calculus, Addison- Wesley, Reading, 1965.

[6] N. G. Lloyd, Degree theory , Cambridge University Press, Cambridge, 1978.

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Appendix

A Solutions to exercises

1. Note thatdet(DF ) = P xQy − P yQx. (15)

We can write


2π

∂D

P dQ − Q dP

≡ 1

2π

∂D

[(P Qx − QP x) dx + (P Qy − QP y) dy],

since

F 2 = P 2 + Q2 = 1 on ∂D. An application of Green’s theorem

yields


2π

D

(P xQy − QxP y − P yQx + QyP x) dx dy

≡ 1

2π

D

2(P xQy − P yQx) dx dy.

The statement then follows from (15).

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Brouwer Degree in Two Dimensions

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