I am trying to understand how to get the discrete gradient of a mesh that is being used as the input of some function $f$. In other words for every vertex $v$ there is a scalar quantity $s$ associated with it.
I am trying to understand how to compute the discrete gradient of $f$ on the surface. For that purpose I was checking these slides:
But it's not clicking. The piece wise notation I am assuming is just an attempt at formalizing "we have no idea what the values at the triangles would be, so we are just going to linearly interpolate using Barycentric coordinates".
I kinda understand the bottom part, which seems to say the gradient basis at $i$ is a vector orthogonal to the opposite edge divided by 2 times the area of the triangle (I assume), but how was the top formula derived?