430 CHAPTER 16. STONE’S THEOREM

Proof: From the cofactor expansion theorem,

det(Dg) =n

∑i=1

gi, j cof(Dg)i j

and so∂ det(Dg)

∂gi, j= cof(Dg)i j (16.3.2)

which shows the last claim of the lemma. Also

δ k j det(Dg) = ∑i

gi,k (cof(Dg))i j (16.3.3)

because if k ̸= j this is just the cofactor expansion of the determinant of a matrix in whichthe kth and jth columns are equal. Differentiate 16.3.3 with respect to x j and sum on j.This yields

∑r,s, j

δ k j∂ (detDg)

∂gr,sgr,s j = ∑

i jgi,k j (cof(Dg))i j +∑

i jgi,k cof(Dg)i j, j .

Hence, using δ k j = 0 if j ̸= k and 16.3.2,

∑rs(cof(Dg))rs gr,sk = ∑

rsgr,ks (cof(Dg))rs +∑

i jgi,kcof(Dg)i j, j .

Subtracting the first sum on the right from both sides and using the equality of mixedpartials,

∑i

gi,k

(∑

j(cof(Dg))i j, j

)= 0.

If det(gi,k)̸= 0 so that

(gi,k)

is invertible, this shows ∑ j (cof(Dg))i j, j = 0. If det(Dg) = 0,let

gk = g+ εkI

where εk→ 0 and det(Dg+ εkI)≡ det(Dgk) ̸= 0. Then

∑j(cof(Dg))i j, j = lim

k→∞∑

j(cof(Dgk))i j, j = 0

Definition 16.3.2 Let h be a function defined on an open set, U ⊆ Rn. Then h ∈Ck(U)

ifthere exists a function g defined on an open set, W containng U such that g = h on U andg is Ck (W ) .

Lemma 16.3.3 There does not exist h ∈ C2(

B(0,R))

such that h : B(0,R)→ ∂B(0,R)which also has the property that h(x)= x for all x∈ ∂B(0,R) . That is, there is no retractionof B(0,R) to ∂B(0,R) .