This is the Theorema Egregium of Gauss.
Gaussian curvature (something defined from derivative of the normal unit vector) depends only from coefficients of the first fundamental form, so it is intrinsic. My question is "Why"?
The first fundamental form is given by these three coefficients $E=(\partial_1,\partial_1), F,G$ defined similarly.
Now, we have that $I(v_1\partial_1+v_2\partial_2)=Ev_1^2+2Fv_1v_2 + Gv_2^2$ is independent from the local chart, but its coefficients change. So, ok that $I$ is intrinsic, but why we call also its coefficients intrinsic if they are actually defined from a local chart and depend from the local chart? Why can I call everything which uses only $E,F,G$ to be intrinsic? Why is $E,F,G$ themselves intrinsic?