788 CHAPTER 23. DEGREE THEORY

Thus for small ε, the Vi are disjoint open sets in Ω and∫Ω

φ ε

(θ−1g(x)−θ

−1y)

detDg(x)∣∣detθ

−1∣∣dx

=m

∑i=1

∫Vi

φ ε

(θ−1 (g(x)−y)

)detDg(x)

∣∣detθ−1∣∣dx

Now just let z = g(x)−y and change the variables.

=m

∑i=1

∣∣detθ−1∣∣∫

g(Vi)−yφ ε

(θ−1z)

detDg(g−1 (y+ z)

)∣∣detDg−1 (y+ z)∣∣dz

By the chain rule, I = Dg(g−1 (y+ z)

)Dg−1 (y+ z) and so

detDg(g−1 (y+ z)

)∣∣detDg−1 (y+ z)∣∣

= sgn(detDg

(g−1 (y+ z)

))·

∣∣detDg(g−1 (y+ z)

)∣∣ ∣∣detDg−1 (y+ z)∣∣

= sgn(detDg

(g−1 (y+ z)

))= sgn(detDg(x)) = sgn(detDg(xi)) .

and so it all reduces to

m

∑i=1

sgn(detDg(xi))∫

g(Vi)−yφ ε

(θ−1z)

dz

=m

∑i=1

sgn(detDg(xi))∫

θB(0,ε)

∣∣detθ−1∣∣φ ε

(θ−1z)

dz

=m

∑i=1

sgn(detDg(xi))∫

B(0,ε)φ ε (w)

∣∣detθ−1∣∣ |detθ |dw

=m

∑i=1

sgn(detDg(xi)) = d (g,Ω,y) .

What about functions which have values in finite dimensional vector spaces?

Theorem 23.8.3 Let Ω be an open bounded set in V a real normed n dimensional vectorspace. Then there exists a topological degree d ( f ,Ω,y) for f ∈C

(Ω̄,V

),y /∈ f (∂Ω) which

satisfies all the properties of the degree for functions having values in Rn described above,

1. d (id,Ω,y) = 1 if y ∈Ω.

2. If Ωi ⊆Ω,Ωi open, and Ω1∩Ω2 = /0 and if y /∈ f(Ω\ (Ω1∪Ω2)

), then d ( f ,Ω1,y)+

d ( f ,Ω2,y) = d ( f ,Ω,y).

788 CHAPTER 23. DEGREE THEORYThus for small €, the V; are disjoint open sets in Q and[9 (0 'g(x) — 6 |y) detDg (x) |detO~!| dxQm Q= Y [| % (0~' (g(x) —y)) det Dg (x) |deto~'| dxi=17ViNow just let z = g(x) — y and change the variables.m=) |deto'| La , (@ 'z) detDg (g~' (y+z)) |\detDg”! (y+z)| dzi=1 ZEVi)-YBy the chain rule, J = Dg (g~! (y+z)) Dg”! (y +z) and sodet Dg (g"' (y +z)) |detDg! (y +-z)|= sgn (detDg (g (y+z)))-|det Dg (g-' (y+z))| |detDg™! (y +2)|= sgn (detDg (g"' (y +z)))= sgn (det Dg (x)) = sgn (det Dg (x;)).and so it all reduces tom¥ sen (detDg(xi)) [ , (0 'z) dzi=l g(Vi)—-ym= det Dg (x; detO~'|@, (0-'z)dYisen(detDa(xi)) [ [det "| 6. (0° '2) aei=1= Y" sgn (det Dg (xi) [ 0, (Ww) |det ~'| |\det 0|dwi=1 B(0,e)m= y sgn (det Dg (x;)) =d(g,Q,y).i=lWhat about functions which have values in finite dimensional vector spaces?Theorem 23.8.3 Let Q be an open bounded set in V a real normed n dimensional vectorspace. Then there exists a topological degree d(f,Q,y) for f EC (Q, Vv) sy € f (0Q) whichsatisfies all the properties of the degree for functions having values in R" described above,1. d(id,Q,y) =1ifyeQ.2. If Q) C QQ; open, and Q| NQ2 =O and ify € f (Q\ (Qi UQ»)), then d (f,Q1,y) +d(f,Q2,y) =d(f,Q,y).