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Here is the complete summary table for all 13 surfaces we have discussed, organized with their first and second fundamental coefficients and normal vectors.

Surface Type & Vector Equation \(\vec{r}\)	First Fundamental \((E, F, G)\)	Normal Vector \(\vec{N}\)	Second Fundamental \((L, M, N)\)
1. Monge's Form \((x, y, f(x,y))\)	\(E = 1+f_1^2\) \(F = f_1 f_2\) \(G = 1+f_2^2\)	\(\frac{(-f_1, -f_2, 1)}{H}\)	\(L = \frac{f_{11}}{H}, M = \frac{f_{12}}{H}\) \(N = \frac{f_{22}}{H}\)
2. Revolution \((u\cos v, u\sin v, f(u))\)	\(E = 1+f_1^2\) \(F = 0\) \(G = u^2\)	\(\frac{(-f_1\cos v, -f_1\sin v, 1)}{\sqrt{1+f_1^2}}\)	\(L = \frac{f_{11}}{\sqrt{1+f_1^2}}, M = 0\) \(N = \frac{uf_1}{\sqrt{1+f_1^2}}\)
3. Conoidal \((u\cos v, u\sin v, f(v))\)	\(E = 1\) \(F = 0\) \(G = u^2+f_2^2\)	\(\frac{(f_2\sin v, -f_2\cos v, u)}{H}\)	\(L = 0, M = \frac{-f_2}{H}\) \(N = \frac{uf_{22}}{H}\)
4. Right Helicoid \((u\cos v, u\sin v, cv)\)	\(E = 1, F = 0\) \(G = u^2+c^2\)	\(\frac{(c\sin v, -c\cos v, u)}{H}\)	\(L = 0, M = \frac{-c}{H}\) \(N = 0\)
5. Plane \((u, v, u+v)\)	\(E = 2, F = 1\) \(G = 2\)	\(\frac{(-1, -1, 1)}{\sqrt{3}}\)	\(L = 0, M = 0\) \(N = 0\)
6. Saddle (A) \((u, v, uv)\)	\(E = 1+v^2, F = uv\) \(G = 1+u^2\)	\(\frac{(-v, -u, 1)}{H}\)	\(L = 0, M = \frac{1}{H}\) \(N = 0\)
7. Saddle (B) \((u+v, u-v, uv)\)	\(E = 2+v^2, F = uv\) \(G = 2+u^2\)	\(\frac{(u+v, v-u, -2)}{H}\)	\(L = 0, M = \frac{-2}{H}\) \(N = 0\)
8. Paraboloid \((u, v, u^2+v^2)\)	\(E = 1+4u^2, F = 4uv\) \(G = 1+4v^2\)	\(\frac{(-2u, -2v, 1)}{H}\)	\(L = \frac{2}{H}, M = 0\) \(N = \frac{2}{H}\)
9. Cylinder \((a\cos u, a\sin u, v)\)	\(E = a^2, F = 0\) \(G = 1\)	\((\cos u, \sin u, 0)\)	\(L = -a, M = 0\) \(N = 0\)
10. Cone \((v\cos u, v\sin u, v)\)	\(E = v^2, F = 0\) \(G = 2\)	\(\frac{(\cos u, \sin u, -1)}{\sqrt{2}}\)	\(L = \frac{-v}{\sqrt{2}}, M = 0\) \(N = 0\)
11. Sphere \((\sin u\cos v, \dots)\)	\(E = 1, F = 0\) \(G = \sin^2 u\)	\(\vec{r}\) (Position Vector)	\(L = -1, M = 0\) \(N = -\sin^2 u\)
12. Hyperboloid \(2z=7x^2+6xy-y^2\)	\(E = 1, F = 0\) \(G = 1\)	\((0, 0, 1)\)	\(L = 7, M = 3\) \(N = -1\)
13. Minimal \(e^z\cos x = \cos y\)	\(E = \sec^2 x\) \(F = -\tan x\tan y\) \(G = \sec^2 y\)	\(\frac{(-\tan x, \tan y, 1)}{H}\)	\(L = \frac{\sec^2 x}{H}, M = 0\) \(N = \frac{-\sec^2 y}{H}\)

Note: \(H = \sqrt{EG-F^2}\) represents the magnitude of the normal vector before normalization.