Tensor analysis is the type of subject that can make even the best of students shudder. My ownpost-graduate instructor in the subject took away much of the fear by speaking of an implicitrhythm in the peculiar notation traditionally used, and helped me to see how this rhythm plays itsway throughout the various formalisms.Prior to taking that class, I had spent many years "playing" on my own with tensors. I found thegoing to be tremendously difficult, but was able, over time, to back out some physical andgeometrical considerations that helped to make the subject a little more transparent. Today, it issometimes hard not to think in terms of tensors and their associated concepts.This article, prompted and greatly enhanced by Marlos Jacob, whom I've met only by e-mail, isan attempt to record those early notions concerning tensors. It is intended to serve as a bridgefrom the point where most undergraduate students "leave off" in their studies of mathematics tothe place where most texts on tensor analysis begin. A basic knowledge of vectors, matrices, andphysics is assumed. A semi-intuitive approach to those notions underlying tensor analysis isgiven via scalars, vectors, dyads, triads, and similar higher-order vector products. The readermust be prepared to do some mathematics and to think.For those students who wish to go beyond this humble start, I can only recommend myprofessor's wisdom: find the rhythm in the mathematics and you will fare pretty well. At the heart of all mathematics are numbers.If I were to ask how many marbles you had in a bag, you might answer, "Three." I would findyour answer perfectly satisfactory. The 'bare' number 3, a magnitude, is sufficient to provide theinformation I seek.If I were to ask, "How far is it to your house?" and you answered, "Three," however, I wouldlook at you quizzically and ask, "Three what?" Evidently, for this question, more information isrequired. The bare number 3 is no longer sufficient; I require a 'denominate' number - a numberwith a name.Suppose you rejoindered, "Three km." The number 3 is now named as representing a certainnumber of km. Such numbers are sometimes called scalars. Temperature is represented by ascalar. The total energy of a thermodynamic system is also represented by a scalar.If I were next to ask "Then how do I get to your house from here?" and you said, "Just walkthree km," again I would look at you quizzically. This time, not even a denominate number issufficient; it is necessary to specify a distance or magnitude, yes, but in which direction? NASA/TM2002-211716 2"Just walk three km due north." The denominate number 3 km now has the required additionaldirectional information attached to it. Such numbers are called vectors. Velocity is a vector sinceit has a magnitude and a direction; so is momentum. Quite often, a vector is represented bycomponents. If you were to tell me that to go from here to your house I must walk three blockseast, two blocks north, and go up three floors, the vector extending from "here" to "your house"would have three spatial components: - Three blocks east, - Two blocks north, - Three floors up.