9.17 Tensor Basis Independent Tensor Behavior

Tensor Basis Independent Tensor Behavior is the collection of properties and identities belonging to a tensor that hold true regardless of which basis is chosen to express its components, in direct contrast to the basis dependent behavior exhibited by the components themselves. It describes what remains fixed and objective about a tensor once all reference to any particular basis is set aside.

The Independence Being Described

The Tensor as an Object Beyond Its Components

A tensor is defined as an abstract multilinear object acting on vectors and covectors, and this definition makes no reference to any basis. Basis independent behavior refers to properties that follow directly from this abstract definition, holding regardless of how the tensor happens to be expressed numerically.

Contrast with Component Behavior

Where basis dependent component behavior describes how numbers change under a change of basis, basis independent tensor behavior describes what does not change at all, providing the stable core around which the changing components are organized.

Categories of Basis Independent Behavior

Multilinearity

A tensor's defining property of being linear in each of its arguments separately holds regardless of basis, since multilinearity is a statement about how the tensor acts on arbitrary vectors and covectors, not about any particular representation of those vectors and covectors.

Type of the Tensor

The classification of a tensor as being of a particular type, with a fixed number of contravariant and covariant slots, is a basis independent property, since this classification describes the tensor's action on arguments rather than any numerical array associated with a basis.

Full Contractions

Scalars produced by fully contracting all of a tensor's indices, whether alone or against other tensors, are basis independent, taking the same numerical value no matter which basis was used to compute the components entering the contraction.

Symmetry and Antisymmetry Properties

Whether a tensor is symmetric or antisymmetric in a given pair of indices is a basis independent property, since these properties are statements about the tensor's behavior under permutation of its arguments, which does not depend on how those arguments happen to be expressed in components.

Why These Properties Are Basis Independent

Rooted in the Abstract Definition

Each of these properties can be verified or defined by appeal only to the tensor's action as a multilinear map, without ever introducing coordinates, which is precisely why they survive unchanged through any basis change.

Surviving the Cancellation in Transformation

Where a property does depend on components, such as a full contraction, its basis independence follows from the fact that the contravariant and covariant transformation factors cancel exactly, leaving the numerical result unaffected by the choice of basis.

Practical Importance

Identifying Meaningful Quantities

Recognizing which quantities exhibit basis independent behavior is essential for distinguishing results that carry genuine, objective significance from results that are merely artifacts of a particular choice of basis and would differ if a different basis had been used.

Guiding the Formulation of Physical and Geometric Laws

In applications where tensors describe physical or geometric quantities, expressing laws in terms of basis independent behavior, such as full contractions and multilinear relationships, ensures that those laws hold true universally, independent of the observer's or analyst's arbitrary choice of coordinate axes.