In Relativity, both the old Galilean theory or Einstein's Special Relativity, one of the most important things is the discussion of whether or not physical laws are invariant. Einstein's theory then states that they are invariant in all inertial frames of reference.
The books usually states that invariant means that the equations take the same form. So, for example, if in one frame $\mathbf{F} = m\mathbf{a}$ holds one expect this same equation to hold on other inertial frame.
If one studies first relativity and then differential geometry, there seems to be really important to postulate that: there's no guarantee whastoever that the equations will be equal. I, however, studied differential geometry first and this led me to this doubt.
On differential geometry everything is defined so that things don't depend on coordinates. So for example: vectors are defined as certain differential operators, or as equivalence classes of curves. Both definitions makes a vector $V$ be one geometrical object, that although has representations on each coordinate system, is independent of them.
Because of that, any tensor is also defined without coordinates and so equalities between vectors and tensors automatically are coordinate-independent. This of course, is valid for vector and tensor fields.
Scalar functions follow the same logic: a function $f : M\to \mathbb{R}$ has one coordinate representation $\tilde{f} : \mathbb{R}^n\to \mathbb{R}$ which is just $\tilde{f} = f\circ x^{-1}$ but still, $f$ is independent of the coordinates. So if $f = g$ this doesn't refer to a coordinate system, but to the functions themselves.
So, it seems that math guarantees that objects are coordinate-independent by nature. So in that case, what are examples where a Physical law is not invariant and why my reasoning fails for those examples?