11.17 Tensor Upper Lower Index Meaning

Tensor Upper Lower Index Meaning is the convention by which the vertical placement of an index on a tensor symbol, either raised as a superscript or lowered as a subscript, encodes the type of transformation behavior that index follows under a change of basis or coordinate system, with upper indices marking contravariant behavior and lower indices marking covariant behavior.

Foundational Setting

Indices as Labels for Components

A tensor of a given rank is represented, once a basis is fixed, by an array of numbers called components, each identified by one or more index labels. The notation $$v^i$$ denotes the components of a vector, while $${\omega }_i$$ denotes the components of a covector, and the difference in vertical placement is not stylistic but carries precise mathematical content.

Why Position Matters

Placing an index as a superscript versus a subscript signals which of two mutually inverse transformation rules governs that component when the basis changes, allowing an expression to be understood correctly without any separate accompanying explanation.

Meaning of an Upper Index

Contravariant Behavior

An upper index marks a component that transforms with the inverse of the matrix relating old and new basis vectors. If the basis vectors are denoted $${\tilde{e}}_i$$ in a new basis related to the old by matrix $$A$$, then:

Typical Examples

Position vectors, velocity vectors, and displacement vectors are all naturally represented with upper indices, since their components must shrink when the basis vectors are stretched in order to describe the same fixed displacement.

Meaning of a Lower Index

Covariant Behavior

A lower index marks a component that transforms using the same matrix $$A$$ that relates the basis vectors directly:

Typical Examples

Gradients of scalar fields and the components of linear functionals are naturally represented with lower indices, since they measure how a quantity changes per unit of basis vector, and thus scale directly with the basis.

The Summation Convention and Index Balance

Einstein Summation

When an index appears once as an upper index and once as a lower index in the same term, summation over that index is implied without an explicit summation symbol, and the index is called a dummy or contracted index:

Free Indices

An index appearing only once in a term, not paired with a matching opposite-position index, is called a free index, and it must appear in the same vertical position on every term of a valid tensor equation, since mismatched free index positions signal an ill-formed expression.

Visual Summary

Diagram of Index Roles

Mixed Tensors and Combined Index Roles

Simultaneous Upper and Lower Indices

A single tensor may carry several indices of both types at once, written for instance as $$T_{jk}^i$$, where the upper index $$i$$ transforms contravariantly and the lower indices $$j$$ and $$k$$ each transform covariantly, independently of one another.

Rank and Valence

The total count of upper indices together with the total count of lower indices determines the rank of the tensor, while the specific split between upper and lower counts is called the valence or type of the tensor, commonly written as an ordered pair such as type $$(1,2)$$ for one upper and two lower indices.

Summary of Key Traits

Defining Characteristics

• Upper index placement denotes contravariant transformation behavior.
• Lower index placement denotes covariant transformation behavior.
• A repeated index appearing once upper and once lower within a term signals implicit summation.
• The counts of upper and lower indices together define a tensor's rank and valence.