Thus it should not be confused with the Ricci calculus.
2.
In an unrelated use, superscripts also indicate contravariant tensors in Ricci calculus.
3.
Ricci calculus is the modern formalism and notation for tensor indices : indicating partial and covariant derivatives.
4.
The description of tensor fields and operations on them in terms of their components is the focus of the Ricci calculus.
5.
Where the second expression is the contraction of the vector field valued 1-form with itself and the last expression is the traditional coordinate expression from Ricci calculus.
6.
See also Ricci calculus for a summary of tensor index notations, and raising and lowering indices for definition of superscript and subscript indices, and how to switch between them.
7.
In the 1950s Schouten completely rewrote and updated the German version of " Ricci-Kalk�l " and this was translated into English as " Ricci Calculus ".
8.
Which is only valid in three dimensions, or an antisymmetric tensor field of second order via antisymmetrization of indices, indicated by delimiting the antisymmetrized indices by square brackets ( see Ricci calculus ):
9.
Expressions in the notation of the Ricci calculus may generally be interpreted as a set of simultaneous equations relating the components as functions over a manifold, usually more specifically as functions of the coordinates on the manifold.
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