I have have an exercise where it says:
Let $M=\{x \in \mathbb{R}^2 \vert x_2>0\}$ with metric $g=\dfrac{1}{x_2^2}((dx_1)^2+(dx_2)^2)$.
Now I have trouble understanding how this metric works, let for example $a,b$ be tangent vectors in $M$. Then what is $g(a,b)$?
The form $dx_1$ selects the first component of a vector $a=(a_1,a_2).$ So $dx_1(a)=a_1.$ The notation $(dx_1)^2$ is shorthand for $dx_1\otimes dx_1$. So $(dx_1)^2(a,b) = dx_1\otimes dx_1(a,b) = a_1\cdot b_1.$
Therefore $g(a,b)=(a_1\cdot b_1 + a_2\cdot b_2)/x_2^2,$ where $x_2$ is the second coordinate of the basepoint of the tangent vectors $a$ and $b$.
By the way it may interest you to know the metric given is the Poincaré half plane.