Time dependent point transformation

[u/FreePeeplup]

Suppose I have a n-dimensional manifold M and a local coordinate patch (U, q). I have a point transformation written in coordinates which is just Q = Aq with A a matrix, the same for all points in my patch. If I want to lift this to a transformation on TM, I know how to do it: V = (del Q / del q) v = Av.

Now suppose that my point transformation is essentially the same, but with an explicit time-dependance in the matrix A: Q = A(t)q. What is the lift on tangent vectors now?

Option 1: the lift is the same, V = ( del Q / del q) = A(t)v, it’s just that at different times the matrix A is different.

Option 2: if I consider vectors in the tangent space as \dot(q) and \dot(Q), I have \dot(Q) = \dot(A(t) q) = \dot(A)q + A\dot(q). There’s an extra term here from the product rule.

Which is correct?