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L, M and N are the second fundamental coefficients, and E, F and G are the first fundamental coefficients.

Since $\kappa~d\vec r+d\vec N$ is perpendicular to both $\vec r_1$ and $\vec r_2$ I can say that it is along normal to the tangent plane at that point since $\vec r_1$ and $\vec r_2$ are the basis for the tangent plane.

As is given in the book that it lies in the tangent plane it has to be a zero vector since it can't be both normal and tangential to the surface.

I have a problem with the statement that it lies in the tangent plane.

$d\vec r=du~\vec r_1+dv~\vec r_2$, so $d\vec r$ lies in the tangent plane. Now, the vector $\kappa~d\vec r+d\vec N$ will also lie in the tangent plane if $d\vec N$ lies in the tangent plane. How can I prove that $d\vec N$ lies in the tangent plane?

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$\vec N\cdot \vec N = 1$, so $2d\vec N\cdot\vec N = 0$.

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    $\begingroup$ I apologize for my lazy review, I admit I was not careful on this one. Indeed, there is no issue. $\endgroup$
    – Suzet
    Commented Sep 5, 2018 at 19:05

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