7.1.4 Tensor Component Extraction Scope

Tensor Component Extraction Scope is the delimitation of what the operation of "extracting" a component from a tensor can legitimately produce, ranging from a single scalar (when every index slot is fixed) to a lower-order tensor-valued slice (when only some slots are fixed), together with the formal description of extraction as a linear operation built from evaluating the tensor against basis vectors and dual basis covectors. Understanding this scope clarifies that "extracting a component" is really one instance of a more general family of extraction operations parametrized by how many, and which, of the tensor's slots are held fixed.

Full Extraction: Producing a Single Scalar

Fixing Every Slot

When every one of a tensor's p + q slots is assigned a specific basis index, extraction produces a single scalar, the component T^{i₁...i_p}_{j₁...j_q} in the fullest and most familiar sense. Formally, this is achieved by evaluating T, viewed as a multilinear map, on the corresponding basis vectors and dual basis covectors:

for a simple (1,1) example, generalizing to more slots by supplying one basis vector or covector per slot.

Partial Extraction: Producing a Lower-Order Tensor

Fixing Only Some Slots

If only a subset of the slots is assigned specific indices while the rest are left free, the result of extraction is not a scalar but a tensor of reduced order, sometimes called a slice: fixing the first index of a (1,1) tensor T at i = 3 while leaving j free produces the covector T³_{~•}, a genuine order-one object, not a single number. Partial extraction therefore extends the scope of "extraction" beyond scalars to include any lower-order tensor obtainable by fixing a proper subset of the slots.

The Boundary Between Full and Partial Extraction

The distinction between full and partial extraction is precisely whether every slot has been fixed; this is the exact boundary separating "a component" (a scalar, fully extracted) from "a slice" (a tensor of positive order, partially extracted), and conflating the two — treating a partially fixed slice as though it were a single scalar component — is a common source of error when working with higher-order tensors.

Diagram of the Extraction Spectrum

Properties of Extraction as an Operation

Linearity of Extraction

Extraction, at any level between full and none, is a linear operation in T: extracting the same fixed indices from S + T gives the sum of the extractions from S and from T individually, and extracting from cT gives c times the extraction from T. This linearity is what allows component-wise reasoning about sums and scalar multiples of tensors to be valid.

Extraction Does Not Commute Freely With Basis Change

Because a component's value depends on the basis used, extracting "the same" fixed indices before and after a change of basis generally produces different scalars — extraction is basis-relative at every point along the spectrum from full to partial, and only quantities obtained by summing extractions in a specific, transformation-compatible pattern (such as a full contraction) recover basis-independent results.

Relevance to Practical Slicing Operations

The partial extraction case is the formal underpinning of the "slicing" operations found throughout numerical tensor libraries, where fixing some axes of an array while leaving others free produces a lower-dimensional array; recognizing this as tensor extraction, rather than as an unrelated programming convenience, clarifies that a slice inherits the same basis-dependence and transformation considerations that apply to any other tensor component or representation.