9.17.2 Tensor Basis Independent Equation Form

Tensor Basis Independent Equation Form is the way of writing a relationship between tensors directly as an equality of abstract tensors, without reference to any particular basis, so that the equation remains valid under every possible choice of basis simultaneously. It is the style of formulating tensor equations that expresses a relationship as holding between the objects themselves rather than between their basis-dependent component arrays.

Structure of the Form

Equating Tensors Rather Than Numbers

A basis independent equation states that one tensor equals another tensor, or equals a combination of other tensors formed through addition, scalar multiplication, tensor product, or contraction, with no basis, index labels, or components appearing anywhere in the statement.

Contrast with Component Form

This differs from the component form of the same relationship, which would express the equation index by index, with each side carrying the same free indices and holding only relative to whichever single basis the components are expressed in.

Validity of the Equation Form

Holding Automatically in Every Basis

Because a basis independent equation asserts equality between the abstract tensors themselves, and tensor operations such as addition and tensor product are defined without reference to any basis, the truth of the equation does not depend on, and is not restricted to, any single choice of basis.

Equivalent to Its Component Form in Any Basis

A basis independent equation is true precisely when its corresponding component equation holds in some, and therefore every, basis, since the transformation law guarantees that if a component equation holds in one basis it holds, in appropriately transformed form, in every other basis as well.

Advantages of the Equation Form

Conciseness and Generality

Writing an equation in basis independent form avoids the need to introduce index notation, basis choices, or summation symbols, expressing the essential relationship between tensors as compactly and generally as possible.

Immediate Applicability Across Contexts

Because a basis independent equation makes no reference to a specific basis, it can be applied directly in any setting, coordinate system, or computational scheme without first needing to adapt or reinterpret it for that particular choice of basis.

Translating Between the Two Forms

Deriving Component Form When Needed

Whenever explicit computation is required, a basis independent equation can be translated into component form by choosing a basis, expanding every tensor in the equation using that basis, and equating components index by index according to the tensors' types.

Deriving Equation Form From Components

Conversely, if a relationship among components is found to hold in one basis and is confirmed to be consistent with the standard transformation law under a change of basis, it can be lifted to a basis independent equation among the underlying tensors themselves.

Practical Role of the Equation Form

Standard Style for Fundamental Relationships

Fundamental relationships in tensor algebra, including definitions, identities, and general laws, are most naturally and most often stated in basis independent equation form, reserving component form for the stage where explicit numerical computation is actually required.

A Check on the Generality of a Result

Successfully expressing a derived relationship in basis independent equation form serves as confirmation that the relationship is a genuine property of the tensors involved, rather than an artifact that happened to hold only in the particular basis used during its derivation.