Editing Tensor (section)

== Notation ==

There are several notational systems that are used to describe tensors and perform calculations involving them.

=== Ricci calculus ===

[[Ricci calculus]] is the modern formalism and notation for tensor indices: indicating [[inner product|inner]] and [[outer product]]s, [[covariance and contravariance of vectors|covariance and contravariance]], [[summation]]s of tensor components, [[symmetric tensor|symmetry]] and [[antisymmetric tensor|antisymmetry]], and [[partial derivative|partial]] and [[covariant derivative]]s.

=== Einstein summation convention ===

The [[Einstein summation convention]] dispenses with writing [[summation sign]]s, leaving the summation implicit.  Any repeated index symbol is summed over: if the index {{mvar|i}} is used twice in a given term of a tensor expression, it means that the term is to be summed for all {{mvar|i}}.  Several distinct pairs of indices may be summed this way.

=== Penrose graphical notation ===

[[Penrose graphical notation]] is a diagrammatic notation which replaces the symbols for tensors with shapes, and their indices by lines and curves.  It is independent of basis elements, and requires no symbols for the indices.

=== Abstract index notation ===

The [[abstract index notation]] is a way to write tensors such that the indices are no longer thought of as numerical, but rather are [[Indeterminate (variable)|indeterminates]].  This notation captures the expressiveness of indices and the basis-independence of index-free notation.

=== Component-free notation ===

A [[component-free treatment of tensors]] uses notation that emphasises that tensors do not rely on any basis, and is defined in terms of the [[Tensor product|tensor product of vector spaces]].