Maths - Tensor Algebra Terminology Table

The tensor terminology (see this page for more definitions)

operation

super &
subscript

expansion
(using Einstein convention)
equivalent vector and matrix notation

dot product

ei Ai e0 A0 + e1 A1
e0 e1
a0
a1
=
e0*a0 + e 1*a1
dot product Aijbi A0jb0 + A1jb1
a00 a01
a10 a11
b0
b1
=
a00*b0 + a01*b1
a10*b0 + a11*b1
=
a00*b0
a10*b0
+
a01*b1
a11*b1
dot product AijBki A0jBk0 + A1kBk2
a00 a01
a10 a11
b00 b01
b10 b11
=
a00*b00 a00*b01
a10*b00 a10*b01
+
a01*b10 a01*b11
a11*b10 a11*b11
       
metric tensor ds²=gp gdupdup  
g00 g01 g02
g10 g11 g12
g20 g21 g22
u0
u1
u2
u0
u1
u2

cross product

a×b

εijkajbk  
permutation
a0
a1
a2
b0
b1
b2
       
gram matrix Gij viT vj (vi | vj)
a0
a1
b0 b1
=
a0*b0 a0*b1
a1*b0 a1*b1
       

 


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