6.13. C1 FUNCTIONS 123

Definition 6.13.6 Let g : U ⊆∏ni=1Fri → Fq, where U is an open set. Then the map xi→

g(x) is a function from the open set in Fri ,

{x : x =(x1, · · · ,xi−1,x,xi+1, · · · ,xn) ∈U}

to Fq. When this map is differentiable, its derivative is denoted by Dig(x). To aid in thenotation, for v ∈ Fri , let θ iv ∈∏

ni=1Fri be the vector (0, · · · ,v, · · · ,0) where the v is in the

ith slot and for v ∈∏ni=1Fri , let vi denote the entry in the ith slot of v. Thus, by saying

xi→ g(x) is differentiable is meant that for v ∈ Fri sufficiently small,

g(x+θ iv)−g(x) = Dig(x)v+o(v) .

Note Dig(x) ∈L (Fri ,∏ni=1Fri) .

Here is a generalization of Theorem 6.13.5.

Theorem 6.13.7 Let g,U,∏ni=1Fri , be given as in Definition 6.13.6. Then g is C1 (U) if

and only if Dig exists and is continuous on U for each i. In this case,

Dg(x)(v) = ∑k

Dkg(x)vk (6.13.21)

where v = (v1, · · · ,vn) .

Proof: Suppose then that Dig exists and is continuous for each i. Note that ∑kj=1 θ jv j =

(v1, · · · ,vk,0, · · · ,0). Thus ∑nj=1 θ jv j = v and define ∑

0j=1 θ jv j ≡ 0. Therefore,

g(x+v)−g(x) =n

∑k=1

[g

(x+

k

∑j=1

θ jv j

)−g

(x+

k−1

∑j=1

θ jv j

)](6.13.22)

Consider the terms in this sum.

g

(x+

k

∑j=1

θ jv j

)−g

(x+

k−1

∑j=1

θ jv j

)= g(x+θ kvk)−g(x)+ (6.13.23)

(g

(x+

k

∑j=1

θ jv j

)−g(x+θ kvk)

)−

(g

(x+

k−1

∑j=1

θ jv j

)−g(x)

)(6.13.24)

and the expression in 6.13.24 is of the form h(vk)−h(0) where for small w ∈ Frk ,

h(w)≡ g

(x+

k−1

∑j=1

θ jv j +θ kw

)−g(x+θ kw) .

Therefore,

Dh(w) = Dkg

(x+

k−1

∑j=1

θ jv j +θ kw

)−Dkg(x+θ kw)