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In Einstein notation, summation is represented by placing a subscript and a superscript on a repeated index. For example, the sum of a vector A can be represented as A_i = A^1 + A^2 + A^3.
Basic vector operations, such as addition and multiplication, can be performed in Einstein notation by using the repeated index convention. For example, the dot product of two vectors A and B can be written as A_iB^i.
In Einstein notation, covariant indices are represented with subscripts, while contravariant indices are represented with superscripts. Covariant indices are associated with basis vectors, while contravariant indices are associated with components of a vector.
Indices can be raised and lowered in Einstein notation using the metric tensor. Raising an index is done by multiplying by the metric tensor, while lowering an index is done by multiplying by the inverse metric tensor.
To perform tensor contraction in Einstein notation, you simply set two indices equal to each other and sum over them. For example, the contraction of a tensor T with indices T^ij and T_jk would be represented as T^ijT_jk.
