9.5.3 Tensor Coordinate Basis Local Frame

The Tensor Coordinate Basis Local Frame defines a local coordinate system for tensors, enabling precise representation in a specific geometric context.

Tensor Coordinate Basis Local Frame is the complete set of coordinate basis vectors and their dual covectors attached to a single point of a space, treated as the frame of reference in which tensors located at that point are given coordinates, distinct from the frame attached to any other point; it is the "local" refinement of a tensor coordinate basis system, recognizing that in a curvilinear or curved setting the basis itself is a function of position rather than a single fixed set of vectors valid everywhere.

Why the Frame Must Be Local

Coordinate Basis Vectors Generally Depend on Position

In a curvilinear coordinate system, the coordinate basis vectors e_i = ∂/∂x^i are computed as tangent directions to the coordinate lines passing through a given point, and these tangent directions typically differ from point to point. A tensor coordinate basis local frame captures the basis exactly as it exists at one chosen point, rather than presuming a single global set of directions.

e_{i} (p) = \frac{\partial}{\partial x^{i}} |_{p}

Tensors Are Assigned Coordinates Frame by Frame

A tensor located at a point p receives its coordinates strictly from the local frame at p: the same tensor located at a different point q, even if it is in some sense "the same" tensor field value transported there, must be assigned coordinates using the local frame at q, since the frame at p has no vectors situated at q to pair against it.

Components of a Local Frame

The Local Primal Basis

The local frame includes the primal coordinate basis vectors e_1(p), …, e_n(p), spanning the tangent space at p, each pointing along the coordinate direction through p associated with one coordinate function.

The Local Dual Basis

The local frame also includes the dual covectors e^1(p), …, e^n(p), satisfying the pairing condition at that same point, used to assign upper-index coordinates to any tensor located at p.

e^{i} (p) (e_{j} (p)) = δ_{j}^{i}

Relating Local Frames at Different Points

No Direct Comparison Without Additional Structure

Because a local frame belongs to a single point, comparing a vector's coordinates at one point directly against a vector's coordinates at a different point, term by term, carries no intrinsic meaning unless some additional structure — such as a connection providing a rule for identifying nearby frames — is introduced to relate the two frames.

Smooth Variation Across Neighboring Points

Although each local frame is defined pointwise, the coordinate basis vectors making up the frame vary smoothly as the point moves continuously through the space, so that nearby points have local frames that are close to one another in a precise sense, even though they remain formally distinct frames.

Diagram of Local Frames at Different Points

Consequences of Working With Local Frames

Component Equations Hold Only Within a Single Frame

Any equation relating the components of tensors, written using a tensor coordinate basis local frame, is understood to hold at the single point supplying that frame; extending such an equation to a neighborhood or to the whole space requires the equation to be verified at each point using that point's own local frame.

Differentiating Tensor Fields Requires Tracking the Frame Itself

Because the local frame changes from point to point, differentiating the components of a tensor field with respect to position must account for the change of the frame itself as well as the change of the coordinates, which is the origin of the extra terms appearing whenever ordinary differentiation of tensor components is replaced by a frame-aware derivative.