Tensors2 - What Are Tensors Shina Tan Geometry in...

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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 coordinate-less 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 1-leg (vector) quantities. A 2-leg 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, 2-leg 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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Tensors2 - What Are Tensors Shina Tan Geometry in...

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