Fields have different jobs

A field assigns a value or quantity to each place in a region.

temperature 3

wind velocity

directional relations

FIELD STUDY / The landscape is interpretive art. Numbers, arrows and an ellipse distinguish quantity types; no measured spacetime geometry is shown. [37-39]

A field assigns a quantity to each location. Scalars assign numbers; vectors assign magnitude and direction. [37]

Tensor components transform by defined rules when coordinates change. Tensor fields assign such quantities throughout a region. [37]

In general relativity, the metric tensor describes spacetime geometry and enters calculations of intervals and curvature. [38-39]

A tensor can describe deformation. Neither infinite reach nor the universe's global shape follows from the word tensor alone.

The field has a job description. Infinity is not required.