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Unformatted text preview: What Are Tensors? Shina Tan Geometry in Euclid’s time had nothing to do with coordinates . A point is just a point, not a set of numbers (coordinates). A surface is just a surface, not an equation. A circle is a perfect beautiful you know what curve, not the equation x 2 + y 2 = 1. In later times, perhaps started by Descartes, people began to deal with geometry by coordinates and equations. This stimulated dramatic progress. But if we throw coordinates at a student too early, without telling him or her the real coordinateless geometrical objects, the student may be doing tons of math without knowing the simple geometrical pictures behind the equations. 1 Scalar Let there be a quantity with a magnitude but no direction. It should be the same no matter what Cartesian coordinate system you choose, or you choose nothing, like Euclid did. The length of a line segment, for instance, is a scalar. 2 Vector A vector has both a magnitude and a direction. It is there . It exists in space. No matter how you describe it, how you measure it with whatever coordinate system, it is there, unaffected, just like a scalar. The displacement from one given point to another given point in the space, for example, is a vector. Both the scalar and the vector are independent from the choice of coordinates. Any quantity that is independent from the choice of coordinates is called a tensor. In the Euclidean geometry (as we assume from now on), any two vectors v and w has a dot product: v · w . In place of two vectors there emerges a scalar carrying some information about the previ ous two. Think of the two vectors as two “legs”, which have now been “amputated”  or contracted , leaving behind a legless quantity, v · w . 1 3 Two Legs In our world, a person with two legs is not strange. In the world of geometry, however, students are used to legless (scalar) and 1leg (vector) quantities. A 2leg quantity sounds so weird since it is usually never mentioned in introductory physics. In more advanced courses, standard textbooks typically introduce tensors in terms of coordinates and coordinate trans formations. The transformations are certainly very useful. But there is a geometrical way of looking at things. Transformations have then just a derivative status , and may be omitted in the introduction. Let there be a quantity with a magnitude and two directions. This quantity is the building block of second rank tensors or, colloquially, 2leg tensors . Take any two vectors v and w , which may or may not be parallel. Instead of contracting them into a scalar, we may simpy put them together , ie juxtaposing them , without doing much extra: vw . This is named the juxtaposition product of v and w . We obtain a quantity with two directions (legs)....
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 Spring '11
 Tan
 Linear Algebra, mechanics, Vector Space, Tensor, vw

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