11.3. DIVERGENCE THEOREM 269

Lemma 11.3.2 Let α1, · · · ,α p be real numbers and let A(α1, · · · ,α p) be the matrixwhich has 1+α2

i in the iith slot and α iα j in the i jth slot when i ̸= j. Then detA = 1+∑

pi=1 α2

i .

Proof of the claim: The matrix, A(α1, · · · ,α p) is of the form

A(α1, · · · ,α p) =

1+α2

1 α1α2 · · · α1α pα1α2 1+α2

2 α2α p...

. . ....

α1α p α2α p · · · 1+α2p

Now consider the product of a matrix and its transpose, BT B below.

1 0 · · · 0 α10 1 0 α2...

. . ....

0 1 α p−α1 −α2 · · · −α p 1



1 0 · · · 0 −α10 1 0 −α2...

. . ....

0 1 −α pα1 α2 · · · α p 1

 (11.10)

This product equals a matrix of the form(A(α1, · · · ,α p) 0

0 1+∑pi=1 α2

i

)Therefore,

(1+∑

pi=1 α2

i)

det(A(α1, · · · ,α p)) = det(B)2 = det(BT)2. However, using row

operations,

detBT = det

1 0 · · · 0 α10 1 0 α2...

. . ....

0 1 α p0 0 · · · 0 1+∑

pi=1 α2

i

= 1+p

∑i=1

α2i

and therefore, (1+

p

∑i=1

α2i

)det(A(α1, · · · ,α p)) =

(1+

p

∑i=1

α2i

)2

which shows det(A(α1, · · · ,α p)) =(1+∑

pi=1 α2

i). ■

Now consider the case of σ on ∂U . The maps will be of the form

x̂ ∈ Qk→(

x1 · · · xi−1 g(x̂i) xi+1 · · · xp)T

= h(x̂i)

I need to describe det(Dh(x̂i)

∗Dh(x̂i))1/2 ≡ J (x̂) .

Consider an example sufficient to see what happens in general in which p= 3 and i= 2.Then in this case, J (x̂) will be the square root of the determinant of

(1 gx1 00 gx3 1

) 1 0gx1 gx30 1

=

(g2

x1+1 gx1gx3

gx1gx3 g2x3+1

).