Taha Sochi - Solutions of Exercises of Principles of Tensor Calculus
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| nabla differential operator | |
| ; and ; | covariant and contravariant differential operators |
| f | gradient of scalar f |
| A | divergence of tensor A |
| A | curl of tensor A |
| , ii , ii | Laplacian operator |
| v , ivj | velocity gradient tensor |
| , (subscript) | partial derivative with respect to following index(es) |
| ; (subscript) | covariant derivative with respect to following index(es) |
| hat (e.g. Ai , Ei ) | physical representation or normalized vector |
| bar (e.g. ui , Ai ) | transformed quantity |
| inner or outer product operator | |
| perpendicular to | |
| 1D, 2D, 3D, n D | one-, two-, three-, n -dimensional |
| t | absolute derivative operator with respect to t |
| i and i | partial derivative operator with respect to ith variable |
| ;i | covariant derivative operator with respect to ith variable |
| [ij,k] | Christoffel symbol of 1st kind |
| A | area |
| B , Bij | Finger strain tensor |
| B1 , Bij1 | Cauchy strain tensor |
| C | curve |
| Cn | of class n |
| d , di | displacement vector |
| det | determinant of matrix |
| diag [ ] | diagonal matrix with embraced diagonal elements |
| dr | differential of position vector |
| ds | length of infinitesimal element of curve |
| d | area of infinitesimal element of surface |
| d | volume of infinitesimal element of space |
| ei | ith vector of orthonormal vector set (usually Cartesian basis set) |
| er,e,e | basis vectors of spherical coordinate system |
| err,er,,e | unit dyads of spherical coordinate system |
| e,e,ez | basis vectors of cylindrical coordinate system |
| e,e,,ezz | unit dyads of cylindrical coordinate system |
| E , Eij | first displacement gradient tensor |
| Ei , Ei | ith covariant and contravariant basis vectors |
| i | ith orthonormalized covariant basis vector |
| Eq./Eqs. | Equation/Equations |
| g | determinant of covariant metric tensor |
| g | metric tensor |
| gij , gij , g ji | covariant, contravariant and mixed metric tensor or its components |
| g11,g12,gnn | coefficients of covariant metric tensor |
| g11,g12,gnn | coefficients of contravariant metric tensor |
| hi | scale factor for ith coordinate |
| iff | if and only if |
| J | Jacobian of transformation between two coordinate systems |
| J | Jacobian matrix of transformation between two coordinate systems |
| J1 | inverse Jacobian matrix of transformation |
| L | length of curve |
| n , ni | normal vector to surface |
| P | point |
| P(n,k) | k -permutations of n objects |
| qi | ith coordinate of orthogonal coordinate system |
| qi | ith unit basis vector of orthogonal coordinate system |
| r | position vector |
| Ricci curvature scalar | |
| Rij , R ij | Ricci curvature tensor of 1st and 2nd kind |
| Rijkl , R ijkl | Riemann-Christoffel curvature tensor of 1st and 2nd kind |
| r,, | coordinates of spherical coordinate system |
| S | surface |
| S , Sij | rate of strain tensor |
| S , Sij | vorticity tensor |
| t | time |
| T (superscript) | transposition of matrix |
| T , Ti | traction vector |
| tr | trace of matrix |
| ui | ith coordinate of general coordinate system |
| v , vi | velocity vector |
| V | volume |
| w | weight of relative tensor |
| xi , xi | ith Cartesian coordinate |
| x i , xi | ith Cartesian coordinate of particle at past and present times |
| x,y,z | coordinates of 3D space (mainly Cartesian) |
| , ij | infinitesimal strain tensor |
| rate of strain tensor | |
| kij | Christoffel symbol of 2nd kind |
| Kronecker delta tensor | |
| ij , ij , ji | covariant, contravariant and mixed ordinary Kronecker delta |
| ijkl , ijklmn , i1inj1jn | generalized Kronecker delta in 2D, 3D and n D space |
| , ij | second displacement gradient tensor |
| ij , ijk , i1in | covariant relative permutation tensor in 2D, 3D and n D space |
| ij , ijk , i1in | contravariant relative permutation tensor in 2D, 3D and n D space |
| ij , ijk , i1in | covariant absolute permutation tensor in 2D, 3D and n D space |
| ij , ijk , i1in | contravariant absolute permutation tensor in 2D, 3D and n D space |
| , | coordinates of plane polar coordinate system |
| ,,z | coordinates of cylindrical coordinate system |
| , ij | stress tensor |
| vorticity tensor | |
| region of space |
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