7.6.4 Tensor Component Coordinate Assigned Value

Tensor Component Coordinate Assigned Value is the number a tensor's component takes on once a coordinate system, rather than merely a basis at a single point, has been fixed across a region, so that the assigned value can in general vary from point to point according to how the coordinate system itself varies over that region.

Definition and Scope

From a Fixed Basis to a Coordinate System

A basis assigned value fixes a single set of basis vectors at one point and reads off a number. A coordinate assigned value extends this by fixing a coordinate system, an assignment of a basis at every point of a region, so that the same tensor field produces a coordinate assigned value at each point, generally different from point to point even for a single fixed tensor field:

with the explicit dependence on the point (x) distinguishing a coordinate assigned value from a value assigned at a single, isolated point.

Coordinate Bases Induced by a Chart

In a coordinate system given by functions (x^1, \dots, x^n), the coordinate assigned value at a point is computed using the basis vectors (\partial/\partial x^i) naturally induced by that coordinate system at that point, a basis that itself changes direction and scale from point to point whenever the coordinates are curvilinear rather than affine.

Structural Properties

Variation Across a Region

Because the coordinate-induced basis can change from point to point, a coordinate assigned value can vary across a region even for a tensor field that is, in an appropriate sense, constant; a tensor that is genuinely constant in Cartesian coordinates may have coordinate assigned values that vary with position once expressed in polar or spherical coordinates, reflecting the changing basis rather than any change in the underlying field.

Transition Between Coordinate Systems

Passing from one coordinate system to another reassigns every value at every point according to the Jacobian of the coordinate transformation, generalizing the fixed change-of-basis matrix used for a single point to a matrix that itself varies with position:

with the partial derivatives playing the role that a constant matrix (A) plays in the single-point case.

Comparing Values at Different Points

A coordinate assigned value at one point and a coordinate assigned value at a different point are not directly comparable through simple subtraction or equality checking, since they are expressed relative to different, locally varying bases; comparing tensor values meaningfully across distinct points generally requires additional structure, such as parallel transport, beyond what the coordinate assignment alone supplies.

Role Within Tensor Algebra

Foundation for Tensor Fields

Coordinate assigned values are what turn an abstract tensor field, a rule assigning a tensor to every point of a region, into an explicit set of numerical functions of position, the form in which tensor fields are actually written down, differentiated, and integrated in applied settings such as physics and differential geometry.

Necessity for Correct Differentiation

Because the coordinate-induced basis changes from point to point, ordinary differentiation of coordinate assigned values with respect to position does not, by itself, produce another tensor's components; this observation motivates the covariant derivative, which corrects for the changing basis so that differentiating a tensor field yields a result that is again a properly behaved tensor.