224 CHAPTER 8. IMPLICIT FUNCTION THEOREM

Df (x)(y) = Df1 (g (x))Dg (x)(y) = D1f1 (g (x))

=π1Dg(x)(y)︷ ︸︸ ︷π1HDf (x)(y)

+D2f1 (g (x))π2Gy

We need to verify the last term equals 0. Solving for this term,

D2f1 (g (x))π2Gy = Df (x)(y)−D1f1 (g (x))π1HDf (x)(y)

As just explained, Lx ◦H is the identity on Px, the image of Df (x). Then

D2f1 (g (x))π2Gy = Lx ◦HDf (x)(y)−D1f1 (g (x))π1HDf (x)(y)

=(

Lx ◦HDf (x)−D1f1 (g (x))π1HDf (x))(y)

Factoring out that underlined term,

D2f1 (g (x))π2Gy = [Lx−D1f1 (g (x))π1]HDf (x)(y)

Now Df (x) : M→Px =Df (x)(Rn) is onto. (This is based on the assumption that Df (x)has rank m.) Thus it suffices to consider only y ∈M in the right side of the above. However,for such y,π2Gy= 0 because to be in M,ψk (y) = 0 if k ≥ m+ 1, and so the left side ofthe above equals 0. Thus it appears this term on the left is 0 for any y chosen. How canthis be so? It can only take place if D2f1 (g (x)) = 0 for every x ∈ V . Thus, since g isonto, it can only take place if D2f1 (x) = 0 for all x ∈U . Therefore on U it must be thecase that f1 depends only on x1, · · · ,xm as desired. ■

8.9 The Local Structure of C1 MappingsIn linear algebra it is shown that every invertible matrix can be written as a product ofelementary matrices, those matrices which are obtained from doing a row operation to theidentity matrix. Two of the row operations produce a matrix which will change exactly oneentry of a vector when it is multiplied by the elementary matrix. The other row operationinvolves switching two rows and this has the effect of switching two entries in a vectorwhen multiplied on the left by the elementary matrix. Thus, in terms of the effect on avector, the mapping determined by the given matrix can be considered as a composition ofmappings which either flip two entries of the vector or change exactly one. A similar localresult is available for nonlinear mappings. I found this interesting result in the advancedcalculus book by Rudin.

Definition 8.9.1 Let U be an open set in Rn and let G : U → Rn. Then G is calledprimitive if it is of the form

G(x) =(

x1 · · · α (x) · · · xn)T

.

Thus, G is primitive if it only changes one of the variables. A function F : Rn → Rn iscalled a flip if

F (x1, · · · ,xk, · · · ,xl , · · · ,xn) = (x1, · · · ,xl , · · · ,xk, · · · ,xn)T .

Thus a function is a flip if it interchanges two coordinates. Also, for m = 1,2, · · · ,n, define

Pm (x)≡(

x1 x2 · · · xm 0 · · · 0)T