9.5.2 Tensor Coordinate Basis Coordinate Direction

Tensor Coordinate Basis Coordinate Direction is the geometric direction associated with each individual member of a coordinate basis, namely the direction along which that basis vector points, or equivalently the direction along which the corresponding coordinate increases while every other coordinate is held fixed; it is what gives each abstract basis vector e_i of a tensor coordinate basis system a concrete geometric meaning as "the direction of increasing coordinate i," rather than leaving it as a purely algebraic symbol.

What a Coordinate Direction Represents

The Direction of Increasing a Single Coordinate

For a coordinate system with coordinates (x^1, x^2, …, x^n), the coordinate direction associated with the i-th basis vector is the direction obtained by varying x^i alone while keeping every other coordinate constant. This is expressed as the partial derivative direction along that coordinate.

A Local Notion at Every Point

In a general curvilinear or curved setting, a coordinate direction is defined at each individual point of the space separately: the direction associated with e_i at one point need not point the same way, in any absolute sense, as the direction associated with e_i at a different point, since the coordinate lines themselves may bend or spread apart across the space.

Coordinate Directions and the Coordinate Grid

Coordinate Lines as the Visible Trace of a Direction

Following a single coordinate direction while holding all others fixed traces out a coordinate line or coordinate curve through the space. The coordinate direction at any point along this line is precisely the tangent direction to that line at that point, so the abstract basis vector and the geometric curve are two descriptions of the same underlying direction.

Directions May Be Neither Orthogonal Nor of Unit Length

Unlike an idealized set of perpendicular unit arrows, coordinate directions arising from a tensor coordinate basis system are, in general, neither mutually perpendicular nor of equal length; two coordinate directions may lean toward each other at an angle other than a right angle, and their lengths are measured only once a metric tensor is introduced separately.

Coordinate Directions for Covector Slots

Dual Directions as Level-Surface Normals

The dual basis vectors e^i, used to assign upper indices, are associated not with a direction of travel but with a direction of most rapid increase of the coordinate function x^i, which is normal to the surfaces on which x^i is constant; this dual notion of direction complements, rather than duplicates, the coordinate direction carried by the primal basis vectors.

Diagram of Coordinate Directions

Consequences of Coordinate Direction

It Enables Geometric Intuition Alongside Algebraic Coordinates

Attaching a coordinate direction to each basis vector allows a purely numerical component array to be reinterpreted geometrically: a vector's components can be read as "how far to move along each coordinate direction," restoring a spatial picture to what would otherwise be an abstract list of numbers.

It Explains Why Coordinate Directions Can Vary From Point to Point

Because a coordinate direction is defined locally, at a single point, it accounts for why the same coordinate basis vector e_i may point differently at different locations in a curvilinear coordinate system, and why quantities such as the metric components, which depend on the coordinate directions, are generally functions of position rather than fixed constants.