# Derivative matrices

1. Mar 12, 2009

### Appa

1. The problem statement, all variables and given/known data
Suppose that the mapping F:Rn$$\rightarrow$$Rm is continuously differentiable and that there is a fixed mxn matrix A so that
DF(x)=A for every x in Rn.

Prove that then F is a mapping such that F(x)=Ax+c for some c$$\in$$Rm

2. Relevant equations

DF(x)ij= $$\partial$$Fi(x)/$$\partial$$xj (the ijth entry of the derivative matrix)

3. The attempt at a solution
I tried to solve this with the first-order approximation F(x)=F(xo)+F´(xo)(x-xo) +$$\epsilon$$(x-xo)||x-xo||

Where $$\epsilon$$(x-xo)||x-xo|| approaches naught.
But it didn't amount to anything sensible and I'm not sure what to do... What should I try next?

2. Mar 12, 2009

### CompuChip

Actually, when you write F´(xo)(x-xo) like in a Taylor series, you mean
DF(xo) . (x-xo)
which is a multiplication of a matrix by a vector. You know that DF(xo) = A... if your expansion is otherwise correct that should enable you to actually identify c explicitly.