3.10.3 Tensor Covector Component Extraction

Tensor Covector Component Extraction is the practical technique of obtaining the individual numbers f_1, ..., f_n from a covector f that has been specified in some other form, such as an explicit formula, a geometric description, or a combination of other known covectors, by systematically applying f to each vector of a chosen basis in turn. While the definition f_i = f(e_i) is simple to state, extraction in practice involves recognizing how to apply this definition efficiently depending on how the covector was originally presented.

Extraction from an Explicit Formula

Direct Substitution

When a covector is given as an explicit formula, such as f(x, y, z) = 2x - y + 5z on R^3, extraction proceeds by substituting each standard basis vector into the formula:

confirming that, when the standard basis is used, the components of a linear formula are simply its coefficients.

General Basis Substitution

For a non-standard basis, each e_i must first be expressed in the coordinate system in which the formula for f is written, and then substituted; extraction still reduces to plugging vectors into the formula, but the intermediate step of expressing the basis vectors correctly must not be skipped.

Extraction from a Combination of Known Covectors

Linearity Simplifies Extraction

If f is given as a linear combination of covectors whose components are already known, such as f = 3g - 2h, extraction uses linearity directly:

so the components of f are obtained componentwise from the components of g and h, without recomputing any evaluation from scratch.

Extraction Using the Dual Basis Directly

Reversing the Roles

Since the dual basis satisfies e^i(e_j) = δ^i_j, the components of any covector f can equivalently be extracted by expressing f in terms of the dual basis and reading off the coefficients directly, without evaluating f on each e_i explicitly, whenever such an expansion is already available or easy to obtain from the way f was defined.

Extraction via Projection

More generally, extraction can be viewed as applying a projection: the assignment f ↦ f_i is itself a linear map V* -> F, and it is, in fact, given by evaluation against the corresponding double-dual basis vector, tying component extraction directly back to the natural pairing between V* and V**.

Extraction from Geometric or Physical Descriptions

Covectors Defined by Perpendicularity or Level Sets

When a covector is specified geometrically, for instance as the functional whose kernel is a given hyperplane and which takes a specified value at one additional point, extraction requires first converting this geometric data into an explicit formula, typically by solving a small linear system, before the component values can be read off directly.

Covectors Defined as Derivatives

In contexts where covectors arise as differentials of scalar functions, extraction of components corresponds to computing partial derivatives with respect to each coordinate direction, evaluated at a specific point, since the differential of a function assigns to each direction the corresponding rate of change, which is exactly the role a component plays for the corresponding basis vector.

Verifying an Extraction

Consistency Check

After extracting a proposed set of components f_1, ..., f_n, a reliable check is to reconstruct the covector as f_i e^i and confirm it reproduces the original description of f on a few independently chosen test vectors, not merely on the basis vectors themselves, to guard against arithmetic slips made during the extraction process.

Dimension Check

The number of extracted components must always equal the dimension n of V; obtaining too few or too many values during extraction signals an error either in identifying the basis or in the formula used for f.

Diagrammatic Summary

The diagram shows the extraction process as a sequence of individual evaluations, one per basis vector, together producing the full component list of the covector.