I am proving a statement and the truth of the following proposition would help me with it. If anyone could say whether the proposition is true and give a hint how to prove it I would be very much thankful. I do not know where to start.
Proposition: Assume that $\exists$ a pair $(a,b)$ such that $f(a,b)=g(a,b)$ for the functions $f(x,y)$ and $g(x,y)$ both mapping from $R^2$ to $R^2$. Can I say that if so, then $d/dx f(x,y)=d/dx g(x,y)$ at those values?
$\mathbf A$$\mathbf D$$\mathbf D:$
$a\neq b$
Thanks in advance!
No, you cannot. Let $f(x,y)=xy$ and $g(x,y)=x^2y^2$. Then $f(1,1)=g(1,1)$, but $\frac{\partial f}{\partial x}(1,1)=1$ and $\frac{\partial g}{\partial x}(1,1)=2$
Addendum: Letting $f(x,y)=x(y-1)$, $g(x,y)=x^2(y-1)^2$, and $(a,b)=(1,2)$ gives a counterexample where $a\neq b$. In fact, the property you want is usually false; the values $f(a,b)$ and $g(a,b)$ say nothing about their partial derivatives at that point.