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Einstein notation, also known as the Einstein summation convention, is a notational convention used in tensor calculus where repeated indices in a term imply summation over those indices. For example, in the expression \( a_i b_i \), the repeated index \( i \) indicates that you should sum over all possible values of \( i \).
In Einstein notation, you sum over any index that appears twice in a single term, once as an upper (superscript) index and once as a lower (subscript) index. For example, in the term \( T^i_i \), the index \( i \) is repeated and thus implies summation over \( i \).
Addition of tensors in Einstein notation follows the same rules as regular addition, but you must ensure that the indices match. For example, \( A_{ij} + B_{ij} \) is valid if \( A \) and \( B \) are tensors of the same rank and dimensions. Multiplication involves summing over repeated indices, such as in the dot product \( A_i B_i \), which sums over all values of \( i \).
Common mistakes include summing over indices that should not be summed (e.g., using the same index more than twice in a term), not matching indices properly in addition or multiplication, and confusing free and dummy indices. Free indices appear only once in a term and represent the dimensions of the resulting tensor, while dummy indices are summed over.
Sure! Consider the dot product of two vectors \( A \) and \( B \). If \( A \) and \( B \) are vectors with components \( A_i \) and \( B_i \), the dot product can be written as \( A_i B_i \). According to Einstein notation, this implies summation over the index \( i \), resulting in the scalar value \( \sum_i A_i B_i \).
