14.4. Normal Curvature and the Second Fun- damental Form

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It should be noted that some authors (such as do Carmo) use the notation e = N · X uu , f = N · X uv , g = N · X vv . Recalling that N = X u × X v ‖X u × X v ‖ , using the Lagrange identity we see that ( −→ u · −→ v ) 2 + ‖ −→ u × −→ v ‖ 2 = ‖ −→ u ‖ 2 ‖ −→ v ‖ 2 , ‖X u × X v ‖ = √ EG − F 2 , Normal Curvature . . . Geodesic Curvature . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 692 of 711 and L = N · X uu can be written as L = (X u × X v ) · X √ uu EG − F 2 = (X u, X v , X uu ) √ EG − F 2 , Go Back Full Screen Close Quit where (X u , X v , X uu ) is the determinant of the three vectors.
Some authors (including Gauss himself and Darboux) use the notation D = (X u , X v , X uu ), D ′ = (X u , X v , X uv ), D ′′ = (X u , X v , X vv ), Normal Curvature . . . Geodesic Curvature . . . Home Page and we also have L = D √ EG − F 2 , M = D ′ √ EG − F 2 , N = D ′′ √ EG − F 2 . Title Page ◭◭ ◮◮ ◭ ◮ Page 693 of 711 These expressions were used by Gauss to prove his famous Theorema Egregium. Since the quadratic form (x, y) ↦→ Lx 2 + 2Mxy + Ny 2 plays a very important role in the theory of surfaces, we introduce the following definition. Go Back Full Screen Close Quit
Page 1 and 2: 14.4. Normal Curvature and the Seco
Page 3 and 4: The vector −→ t is the unit tan
Page 5 and 6: Thus, it is better to pick a frame
Page 7 and 8: Using the abbreviations X uu = ∂2
Page 9: Thus, it is clear that the normal c
Page 13 and 14: Properties of the surface expressib
Page 15 and 16: Furthermore, the principal normal o
Page 17 and 18: We found that the tangential part o
Page 19 and 20: Definition 14.4.4 Given a surface X
Page 21 and 22: However, Normal Curvature . . . κ
Page 23 and 24: Doing so, and remembering that κ
Page 25 and 26: Finally, we get and A = u ′′ 1
Page 27 and 28: Looking at the formulae [α β; γ]
Page 29: From all this, we get 2[α β; γ]

14.4. Normal Curvature and the Second Fun- damental Form

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