Say I have a manifold $M$ and another manifold $N$ and a map from one to the other $\phi$.
Say I am given a point $p \in M$, I can get $\phi(p)$. Now say I can numerically approximate tangent vectors around $p$ and $\phi(p)$. Is there a way for me to approximate the differential map?
More practically I have a 2-Manifold embedded in 3D represented implicitly as $f(x,y,z) = 0$ which I am sampling in a computer. I have a map from it to 2D. I am trying to measure the distortion at a point that my map produces but I don't have explicit lines of curvature.