I'm having a hard time understanding some stuff about the first fundamental form. Since I think I have decently understood what the first fundamental form means (inner product restricted to the tangent space of a surface) I think what confuses me is the notation.. Specifically I'm having a hard time understanding how to calculate the angle of two curves on a surface
Let's take this problem as an example
Let $g_{11} = 1, g_{12}=0, g_{22}=4+u^2$ be the coefficients of the first fundamental form of some surface $Σ$ calculate the angle between the curves that lie on the surface given by the equations:
$u+v=0$ and $u-v=0$
So typically a general type of solution for this would be to calculate the tangent vectors of each of their curves at their point of intersection and use their inner product to find their angle. Let's try to do this here:
Let $c_1(t)=(t,t)$ and $c_2(t)=(t,-t)$ be a parametrization of the two curves. These two curves have tangent vectors $x_1 = (1,1) $ and $x_2 = (1,-1)$.
Now let $θ$ be their angle, it can be found as:
$cosθ = \frac{\left \langle x_1,x_2 \right \rangle}{\left \| x_1 \right \| \left \| x_2 \right \|}$
Of course since we are talking about angles on a surface we have to use the first fundamental form. In my textbook this inner product is written as: $\left \langle a,b \right \rangle = g_{ij} a^i b^j$
Can someone please use the example to help me understand this notation?
Thanks in advance!