Find $\mathrm d f_p$ for $f(x,y)=x^2+y^2$

Question

I am doing an exercise about affine map.

Let $f:\mathbb A^2_\mathbb R \to \mathbb R$ be the map such that $f(x,y) = x^2+y^2$ , and $p$ = $(1,0)$. Find d$f_p$.

But I don't know what's the meaning of d$f_p$. Any ideas?

Answer 1

As I see it the given $f$ is just a map $f: \>{\mathbb R}^2\to{\mathbb R}$ without implied scalar product in ${\mathbb R}^2$. One has $$df(p).(X,Y)={\partial f\over\partial x}\biggr|_p\,X+{\partial f\over\partial y}\biggr|_p\,Y\ ,$$ whereby $(X,Y)$ are the induced coordinates in the tangent space $T_p$. Since ${\partial f\over\partial x}=:f_x=2x$, $f_y=2y$, and $p=(1,0)$ we have $f_x(p)=2$, $f_y(p)=0$. It follows that $$df(p).(X,Y)=2X\ .$$