Answer to Question 55: Are there pictorial examples that distinguish covariant and contravariant vectors?

TL;DR

This paper introduces visual methods to differentiate covariant vectors from contravariant vectors using pictorial representations, aiding understanding of their geometric and physical distinctions.

Contribution

It provides a novel pictorial approach to distinguish covariant and contravariant vectors, with examples from physics illustrating their natural occurrences.

Findings

Pictorial depiction of vectors as directed segments.

Linear forms represented by equidistant parallel planes.

Examples of physical quantities as vectors or linear forms.

Abstract

We present pictorial means of distinguishing contravariant vectors (or simply vectors) from covariant vectors (or linear forms). When one depicts vector as the directed segment, then the pictorial image of a linear form is a family of equidistant parallel planes with an arrow joining the neighbouring planes and showing the direction of increase of the form. First of these planes is the linear subspace of dimension two on which the linear form gives value zero. Several examples of physical quantities are given which are naturally vectors, and others which are naturally linear forms.