Tensors California Polytechnic Stat University张量加州州立理工大学.pptVIP

Vectors and Tensors Robert Dinovo John Hostetter 10 - 20 - 99 Vector Review Adding vectors Dot Product Cross Product Gradient Divergence Curl Nomenclature for Indices Dummy index is the repeated index Non repeated index is the “Free” index Example: aijxi “i” is repeated therefore it is the “Dummy” Index “j” is not repeated, and is the “Free” Index Repetition Repetition of index = summation of term over values of index The “Dummy” index is the summation index Example: aijxi is equivalent to ? aijxi= a1jx1 +a2jx2+ a3jx3 i=1..3 Singular Index A singular index (not repeated) represents different equations This is the “Free” Index Example: aijxi is equivalent to ai1xi= ai1xi +ai1xi+ ai1xi ai2xi= ai2xi +ai2xi+ ai2xi ai3xi= ai3xi +ai3xi+ ai3xi Combine Both the free and Dummy indices used together. Example: aijxi is equivalent to: a11x1 +a21x2+ a31x3 a12x1 +a22x2+ a32x3 a13x1 +a23x2+ a33x3 Order The order of a tensor is equal to the number of indices Example: P Zero order tensor Pi First order tensor Pij Second order tensor Pijk Third order tensor Scalars Scalars are zero order tensors Example: 5 127 P Q Vectors Vectors are first order tensors Example: Pi [5,1] or [x,y] Arrays / Matrices These are second order tensors Example: Pij [x11,x21,x31] [y12,y22,y32] [z13,z23,z33] Kronecker Delta Consider a line element with components dx, dy, dz in a three dimensional Euclidean space with rectangular Cartesian coordinates x,y,z. The square of the length of the line element is: ds2=dx2+dy2+dz2 if we define dx1=dx, dx2=dy, dx3=dz Kronecker Delta (Cont.) And ?11= ?22= ?33=1 ?12= ?21= ?13 =?31= ?23= ?32=0 Then ds2=?ijdxidxj Where the symbol ?ij is the Kronecker Delta. ?ij=?xi/ ?xj={0 i?j, 1 i=j} Alternating / Permutation Tensor ?ijk is the Alternating Tensor which equals: {0 i=j=k, i=j, i=k, j=k 1 i, j, k are in cyclic order (123, 231, 312) -1 i, j, k are Not in cyclic order (132 …) } Alternating Tensor