Mean curvature is extrinsic measure

Question

Mean curvature is 1/2 of the trace of the shape operator, the shape operator can be found through a multiplication of the matrices $\begin{pmatrix} E& F\\ F&G \end{pmatrix}^{-1}\begin{pmatrix} e & f\\ f& g \end{pmatrix} $ and $e,f,g$ can be found through the derivatives of the first fundamental form, so they are intrinsic. How is the Mean curvature an extrinsic measure then ?
The multiplication and addition of intrinsic values isn't necessary intrinsic ?