Tensors Beyond the Array: A Teaching Note on Karras's Account, Set Against Abstract Geometry and Fractal Algebra

This note isolates the claims in Dimitrios A. Karras's short professional exposition of the tensor, and reads them against two bodies of theory that a research student is liable to conflate with it: the abstract-geometric theory of tensors, from the absolute differential calculus to the modern language of multilinear maps and tensor bundles; and the several programmes gathered, loosely, under fractal algebra and non-integer geometry. Karras is right on the point that matters for a first encounter. A tensor is not a multidimensional array. It is an object whose components transform by a definite rule, so that the quantity represented does not depend on the coordinates used to write it down. That criterion is the classical one of Ricci and Levi-Civita, and it correctly separates the geometric object from the storage format used in computation and in machine learning. The account is nevertheless incomplete as a definition, and it is silent on the setting in which the definition can fail. It counts indices rather than distinguishing covariant from contravariant slots; it treats a change of basis and a change of coordinates as one operation; it does not say that a tensor field is a section, nor that the coefficients of a connection fail the tensor test. Abstract geometry takes the multilinear map, or the element of a tensor product, as primary, and derives the transformation law. Fractal and fractional geometries suspend the hypothesis on which that law depends: a tangent space of integer dimension, with a Jacobian at a typical point. The teaching consequence is sharp. Karras is the right door into the component calculus. It is the wrong place to stop if the student is to understand curvature, or to understand why a snowflake curve does not carry an ordinary stress tensor. KSCCN teaching note, 10 October 2026. Dr. Syed Muntasir Mamun, ORCID 0000-0001-6845-2853. The views expressed are those of the author.