In this notation, the summations are restricted as a computational convenience. This is useful when the expression is completely antisymmetric in each of the two sets of indices, as might occur on the tensor product of a p -vector with a q -form. More than one group can be summed in this way, for example:. When using multi-index notation, an underarrow is placed underneath the block of indices: [9].

By contracting an index with a non-singular metric tensor , the type of a tensor can be changed, converting a lower index to an upper index or vice versa:. The base symbol in many cases is retained e.

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This table summarizes how the manipulation of covariant and contravariant indices fit in with invariance under a passive transformation between bases, with the components of each basis set in terms of the other reflected in the first column. The barred indices refer to the final coordinate system after the transformation. The Kronecker delta is used, see also below. Tensors are equal if and only if every corresponding component is equal; e.

## Bibliography on tensors

Consequently, there are facets of the notation that are useful in checking that an equation makes sense an analogous procedure to dimensional analysis. Indices not involved in contractions are called free indices. Indices used in contractions are termed dummy indices , or summation indices. Since the indices take various integer values to select specific components of the tensors, a single tensor equation represents many ordinary equations.

If a tensor equality has n free indices, and if the dimensionality of the underlying vector space is m , the equality represents m n equations: each index takes on every value of a specific set of values. Three of these are:.

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This illustrates the compactness and efficiency of using index notation: many equations which all share a similar structure can be collected into one simple tensor equation. Replacing any index symbol throughout by another leaves the tensor equation unchanged provided there is no conflict with other symbols already used. This can be useful when manipulating indices, such as using index notation to verify vector calculus identities or identities of the Kronecker delta and Levi-Civita symbol see also below. An example of a correct change is:. The free indices in a tensor expression always appear in the same upper or lower position throughout every term, and in a tensor equation the free indices are the same on each side.

Dummy indices which implies a summation over that index need not be the same, for example:. In other words, non-repeated indices must be of the same type in every term of the equation. When applying a rule to a number of indices differentiation, symmetrization etc. If the brackets enclose covariant indices — the rule applies only to all covariant indices enclosed in the brackets , not to any contravariant indices which happen to be placed intermediately between the brackets. Similarly if brackets enclose contravariant indices — the rule applies only to all enclosed contravariant indices , not to intermediately placed covariant indices.

The symmetrization is distributive over addition;.

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This does not hold for other than two indices. For compactness, derivatives may be indicated by adding indices after a comma or semicolon. While most of the expressions of the Ricci calculus are valid for arbitrary bases, the expressions involving partial derivatives of tensor coordinated apply only with a coordinate basis : a basis that is defined through differentiation of the coordinates.

With the same constraints on the space and on the choice of coordinate system, the partial derivatives with respect to the coordinates yield a result that is effectively covariant. Aside from use in this special case, the partial derivatives of components of tensors are useful in building expressions that are covariant, albeit still with a coordinate basis if the partial derivatives are explicitly used, as with the covariant and Lie derivatives below.

These components do not transform covariantly, unless the expression being differentiated is a scalar. This derivative is characterized by the product rule and the derivatives of the coordinates.

## Ricci calculus

For an arbitrary tensor: [15]. The components of this derivative of a tensor field transform covariantly, and hence form another tensor field. The Lie derivative is another derivative that is covariant, but which should not be confused with the covariant derivative. It is defined even in the absence of a metric tensor. The Kronecker delta is like the identity matrix.

The Kronecker delta is one of the family of generalized Kronecker deltas.

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The generalized Kronecker delta of degree 2 p may be defined in terms of the Kronecker delta by a common definition includes an additional multiplier of p! It also gives the duration of any time-like curve. See also line element. This can be generalized to get the commutator for two covariant derivatives of an arbitrary tensor as follows:.