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\title{\bf Pictorial Representations of Vectors and One-Forms}  %@@
\author{\large Roberto B. Salgado}  %@@
\maketitle  %@@
\abstract{We present well-known pictorial representations of vectors  %@@
and co-vectors (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.}  %@@


\section{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 metrically-induced 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.  %@@


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%{\scriptsize {\bf Figure 2c.} The pictorial representation of  %@@
%$0^2+1^2+2^2+3^2+4^2+5^2\approx\frac{1}{3} 6^3$.}\\  %@@
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\begin{thebibliography}{9}  %@@

\bibitem{crc} {\it CRC Standard Mathematical Tables}, 26th-ed  %@@
(Boca Raton, Florida, CRC Press.) 1981.  %@@

\bibitem{mtw}  %@@
Misner, C. W. / Thorne, K. / Wheeler, J. A.  %@@
{\it Gravitation}  %@@

\bibitem{burke-adg}  %@@
Burke, W. L.,  %@@
{\it Applied Differential Geometry},  %@@
(??Menlo Park, California, W.A. Benjamin, Inc.) 1972.  %@@

\bibitem{burke-sgc}  %@@
Burke, W. L.,  %@@
{\it Spacetime, Geometry and Cosmology},  %@@
(??Menlo Park, California, W.A. Benjamin, Inc.) 1972.  %@@

\bibitem{schouten-ricci}  %@@
Schouten, J. A.,  %@@
{\it Ricci-Calculus},  %@@
(Berlin, Springer-Verlag) 1954.  %@@

\bibitem{schouten-tensor}  %@@
Schouten, J. A.,  %@@
{\it Tensor Analysis for Physicists},  %@@
(??Menlo Park, California, W.A. Benjamin, Inc.) 1972.  %@@

\bibitem{bamberg-sternberg}  %@@
Bamberg, P. / Sternberg, S.  %@@
{\it A course in mathematics for students of physics},  %@@
(Cambdrige, Cambridge University Press) 1990.  %@@

\bibitem{vandanzig}  %@@
van Danzig, ?,  %@@
{\it ? },  %@@
(??Menlo Park, California, W.A. Benjamin, Inc.) 1972.  %@@

\bibitem{vnr}Gellert / K\"ustner / Hellwich / K\"astner,  %@@
{\it The VNR Concise Encyclopedia of Mathematics}  %@@
(New York, Van Nostrand Reinhold) 1975.  %@@

\end{thebibliography}  %@@

\ni{\em Department of Physics\\  %@@
        Syracuse University\\  %@@
        Syracuse NY 13244-1130\\  %@@
   {\tt salgado@suhep.phy.syr.edu}  %@@
   }  %@@

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