Sochi - Introduction to Differential Geometry of Space Curves and Surfaces
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| nabla differential operator | |
| Laplacian operator | |
| ~ | isometric to |
| , subscript | partial derivative with respect to the following index(es) |
| ; subscript | covariant derivative with respect to the following index(es) |
| 1D, 2D, 3D, n D | one-, two-, three-, n -dimensional |
| overdot (e.g. r ) | derivative with respect to general parameter t |
| prime (e.g. r ) | derivative with respect to natural parameter s |
| t | absolute derivative with respect to t |
| , i | partial derivative with respect to th and i th variables |
| a | determinant of surface covariant metric tensor |
| a | surface covariant metric tensor |
| a 11 ,a 12 ,a 22 | coefficients of surface covariant metric tensor |
| a 11 ,a 12 ,a 22 | coefficients of surface contravariant metric tensor |
| a , a , a | surface metric tensor or its components |
| b | determinant of surface covariant curvature tensor |
| b | surface covariant curvature tensor |
| B | binormal unit vector of space curve |
| b 11 ,b 12 ,b 22 | coefficients of surface covariant curvature tensor |
| b , b , b | surface curvature tensor or its components |
| C | curve |
| C B , C N , C T | spherical indicatrices of curve C |
| C e , C i | evolute and involute curves |
| C n | of class n |
| c , c , c | tensor of third fundamental form or its components |
| d | Darboux vector |
| d 1 , d 2 | unit vectors in Darboux frame |
| det | determinant of matrix |
| ds | length of infinitesimal element of curve |
| ds B ,ds N ,ds T | length of line element in binormal, normal, tangent directions |
| d | area of infinitesimal element of surface |
| e ,f ,g | coefficients of second fundamental form |
| E ,F ,G | coefficients of first fundamental form |
| , , V | number of edges, faces and vertices of polyhedron |
| E i , E j | covariant and contravariant space basis vectors |
| E , E | covariant and contravariant surface basis vectors |
| Eq./Eqs. | Equation/Equations |
| f | function |
| Fig./Figs. | Figure/Figures |
| g | topological genus of closed surface |
| g ij , g ij | space metric tensor or its components |
| H | mean curvature |
| I S , II S , III S | first, second and third fundamental forms |
| I S , II S | tensors of first and second fundamental forms |
| iff | if and only if |
| J | Jacobian of transformation between two coordinate systems |
| J | Jacobian matrix |
| K | Gaussian curvature |
| K t | total curvature |
| L | length of curve |
| n | normal unit vector to surface |
| N | principal normal unit vector to curve |
| P | point |
| r , R | radius |
| Ricci curvature scalar | |
| r | position vector |
| r , r | 1st and 2nd partial derivative of r with subscripted variables |
| R 1 , R 2 | principal radii of curvature |
| n | n -dimensional space (usually Euclidean) |
| R ij , R ij |
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