u/Error400_Bad_Request

In class, I learned the proof for the conservation of mechanical energy of a simple pendulum using scalar values, as shown in the picture I attached.

However, I was wondering if it's possible to prove this rigorously by calculating the work done using the net force vector and the instantaneous displacement vector.

Here is what's confusing me:

1. The pendulum moves along a curved path, so the direction of the displacement vector changes at every single instant.
2. The net force vector (which is the vector sum of gravity and tension) is almost never parallel to the instantaneous displacement vector.

Despite these complexities, is it mathematically possible to prove it this way? Does the dot product naturally take care of the curved path and the non-parallel forces (like tension doing zero work)?

I would really appreciate it if someone could explain how the math works out or show the derivation! Thanks!