Ae-ce-am 108a – Computational Mechanics 0 Vector and Tensor Algebra 0.1 Vectors and the Summation Convention

or, using Einstein’s summation convention, as u = vi gi. (3) The summation convention implies that repeated indices appearing exactly twice in a single term (by term we mean any single vector or tensor quantity or any product of vector and/or tensor quantities) are to be summed from 1 through d (where d is the number of dimensions), i.e., for example in three dimensions, v = vi gi = v1 g1 + v2 g2 + v3 g3. (4) As a common convention, one uses Greek letters for summations in two dimensions (e.g., in plane strain in problems of solid mechanics). Therefore, in two dimensions one writes v = vα gα = v1 g1 + v2 g2. (5) A repeated index to be summed is called a dummy index, while those indices only appearing once (and hence not requiring summation) are known as free indices. To avoid ambiguity, no index is allowed to appear more than twice in in a single term. For example, the definition