9.5.1 Tensor Coordinate Basis Vector Field Role

Tensor Coordinate Basis Vector Field Role is the part played by each coordinate basis vector when it is regarded not merely as a single vector attached to one point but as a smoothly varying assignment of a vector to every point of a coordinate patch, so that e_i becomes a vector field defined throughout the region where the coordinates are valid, rather than an isolated object belonging to a single local frame; it is this role that allows coordinate basis vectors to be added, differentiated, and combined with other vector fields across the whole patch, not just compared pointwise.

From a Single Vector to a Field of Vectors

One Definition, Applied at Every Point

A coordinate basis vector e_i = ∂/∂x^i is defined by the same formula at every point of the coordinate patch: at each point it picks out the tangent direction along which the i-th coordinate increases. Because this definition applies uniformly across the whole patch, e_i is naturally a vector field, assigning a tangent vector to every point rather than to one point alone.

A Frame Field Across the Whole Patch

Taken together, the full collection e_1, …, e_n forms what is called a frame field: at every point of the patch, these vectors supply a basis of the tangent space at that point, varying smoothly as the point varies, so that the entire patch is furnished with a coherent, continuously changing family of local frames rather than a single fixed one.

Special Properties That Follow From the Vector Field Role

Coordinate Vector Fields Commute

Because each e_i arises as a partial derivative with respect to a coordinate function, and partial derivatives with respect to distinct coordinates commute, the Lie bracket of any two coordinate basis vector fields vanishes identically.

A Test for Whether a Frame Is Coordinate-Induced

This vanishing bracket property is not automatic for an arbitrary choice of frame field: a general frame field made of arbitrarily chosen vector fields need not have commuting members. The vector field role of the coordinate basis therefore also serves as a diagnostic — a frame field arises from an actual coordinate system precisely when all of its members commute pairwise.

Applying Other Vector Fields Along Coordinate Directions

Differentiating Functions Along a Coordinate Basis Vector Field

Regarded as a vector field, e_i can be applied to any smooth function f defined on the patch, producing another function equal to the partial derivative of f with respect to x^i, evaluated at each point; this operation is exactly what it means to differentiate a function along the i-th coordinate direction throughout the patch.

Expanding Any Vector Field in the Coordinate Frame

Any vector field defined on the patch can be expressed, at every point, as a combination of the coordinate basis vector fields with coefficient functions that themselves vary smoothly over the patch, so that vector fields inherit their own component functions directly from this same frame-field role of the coordinate basis.

Diagram of the Vector Field Role

Consequences of the Vector Field Role

It Supplies the Building Blocks for Vector Field Calculus

Because coordinate basis vectors are themselves vector fields, all of the standard operations of vector field calculus — Lie brackets, flows, directional derivatives of functions — apply directly to them, making the coordinate basis a working set of tools rather than a fixed static reference.

It Underlies the Definition of Coordinate Derivatives of Tensor Fields

When a tensor field's components are differentiated with respect to position, the coordinate basis vector field role is what identifies the direction of differentiation at every point consistently across the patch, ensuring that the resulting derivative is itself a well-defined field rather than a quantity meaningful only at an isolated point.