Pictorial Representations of Vectors and One-Forms

Roberto B. Salgado

June 15, 1995

Abstract

We present well-known pictorial representations of vectors and covectors (one-forms). Using a pictorial representation of a metric tensor, we represent the operations of ”raising and lowering of indices” and ”computing the square-norm of a vector”. In particular, we provide an analytic formula for these pictorial constructions. We use this formula in a simple computer program to help visualize these operations for the case of a Minkowski metric.

1 Introduction

Vectors are pictorially represented as directed line-segments. Covectors are pictorially represented as an ordered-pair of parallel-planes.

A metric tensor is pictorially represented by a conic-section: a circle for a cartesian metric, an ellipse for a euclidean metric, a hyperbola for a lorentzian metric.

The operation of computing the interior-product (transvection) between a vector and a covector is pictorially represented by counting the bongs / piercings of the planes by the vector.

The operation of computing the square-norm of a vector requires the lowering the index of the vector with the metric to obtain the metricallyinduced covector, then computing the transvection.

The operation of lowering the index of a vector with a metric tensor corresponds to finding the polar hyperplane of the tip of the vector with respect to the conic-section of the metric.

References

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Department of Physics Syracuse University Syracuse NY 13244-1130

salgado@suhep.phy.syr.edu