Taha Sochi - Solutions of Exercises of 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-dimensional, two-dimensional, three-dimensional, 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 ith variables |
| a | determinant of surface covariant metric tensor |
| a | surface covariant metric tensor |
| a11,a12,a21,a22 | coefficients of surface covariant metric tensor |
| a11,a12,a21,a22 | 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 |
| b11,b12,b21,b22 | coefficients of surface covariant curvature tensor |
| b , b , b | surface curvature tensor or its components |
| C | curve |
| CB , CN , CT | spherical indicatrices of curve C |
| Ce , Ci | evolute and involute curves |
| Cn | of class n |
| c , c , c | tensor of third fundamental form or its components |
| d | Darboux vector |
| d1 , d2 | unit vectors in Darboux frame |
| det | determinant of matrix |
| ds | length of infinitesimal element of curve |
| dsB,dsN,dsT | length of line element in binormal, normal and 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 |
| Ei , Ej | covariant and contravariant space basis vectors |
| E , E | covariant and contravariant surface basis vectors |
| Eq./Eqs. | Equation/Equations |
| f | function |
| g | topological genus of closed surface |
| gij , gij | space metric tensor or its components |
| H | mean curvature |
| IS , IIS , IIIS | first, second and third fundamental forms |
| IS , IIS | 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 |
| Kt | surface 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 respect to subscripted variables |
| R1 , R2 | principal radii of curvature |
| n | n -dimensional space (usually Euclidean) |
| Rij , R ij | Ricci curvature tensor of 1st and 2nd kind for space |
| R , R | Ricci curvature tensor of 1st and 2nd kind for surface |
| Rijkl | Riemann-Christoffel curvature tensor of 1st kind for space |
| R | Riemann-Christoffel curvature tensor of 1st kind for surface |
| R ijkl | Riemann-Christoffel curvature tensor of 2nd kind for space |
| R | Riemann-Christoffel curvature tensor of 2nd kind for surface |
| R | radius of curvature |
| R | radius of torsion |
| r,, | spherical coordinates of 3D space |
| s | natural parameter of curve representing arc length |
| S | surface |
| ST | tangent surface of space curve |
| t | general parameter of curve |
| T | function period |
| T | tangent unit vector of space curve |
| TPS | tangent space of surface S at point P |
| tr | trace of matrix |
| u | geodesic normal vector |
| u1,u2 | surface coordinates |
| u | surface coordinate |
| u,v | surface coordinates |
| xi | space coordinate |
| x i | surface basis vector in full tensor notation |
| x,y,z | coordinates in 3D space (usually Cartesian) |
| [ij,k] | Christoffel symbol of 1st kind for space |
| [,] | Christoffel symbol of 1st kind for surface |
| kij | Christoffel symbol of 2nd kind for space |
| Christoffel symbol of 2nd kind for surface | |
| ij , ij , ji | covariant, contravariant and mixed Kronecker delta |
| ijkl | generalized Kronecker delta |
| discriminant of quadratic polynomial | |
| ijk , ijk | covariant and contravariant relative permutation tensor in 3D space |
| ijk , ijk | covariant and contravariant absolute permutation tensor in 3D space |
| , | covariant and contravariant relative permutation tensor in 2D space |
| , | covariant and contravariant absolute permutation tensor in 2D space |
| angle or parameter | |
| s | sum of interior angles of polygon |
| curvature of curve | |
| , | principal curvatures of surface at a given point |
| B , T | curvature of binormal and tangent spherical indicatrices |
| g , n | geodesic and normal curvatures |
| gu , gv | geodesic curvatures of u and v coordinate curves |
| nu , nv | normal curvatures of u and v coordinate curves |
| K | curvature vector of curve |
| Kg , Kn | geodesic and normal components of curvature vector of curve |
| direction parameter of surface |
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