Unit base vectors in a new coordinate system

Question

Let's assume we have a function $f:\Omega =R^2 \rightarrow R $

$f(x,y)=x+2xy+x^2y$. Obviously our unit base vectors on $\Omega$ are $e_x=\hat{i}$ and $e_y=\hat{j}$.

Now we want to change the variables $u=x$ and $v=xy$. As a result;

$f(u,v)=u+2v+uv$

How should the unit vectors $e_u$ and $e_v$ be defined in the new coordinate $(u,v)$, so that for example, we can compute directional derivative $\nabla f. \hat{e}_u=\Large(\frac{\partial f}{ \partial u}\hat{e}_u+\frac{\partial f}{ \partial v}\hat{e}_v). \hat{e}_u$ in the new coordinate?

Thanks

Answer 1

This can be done in a pretty standard way. $$\vec r = (x, y)$$ $$ x= u$$ $$ y = \frac vu$$ $$ \vec r_u=(1, -\frac v{u^2})$$ $$ \vec r_v=(0, \frac 1{u})$$ $$||\vec r_u ||=\sqrt{1+{(\frac v{u^2})}^2}$$ $$||\vec r_v ||=\sqrt{{(\frac 1{u})}^2}$$ $$\vec e_u = \frac {(1, -\frac v{u^2})}{\sqrt{1+{(\frac v{u^2})}^2}} $$ $$\vec e_v = \frac {(0, \frac 1{u})}{\sqrt{{(\frac 1{u})}^2}} $$