A Gentle Introduction to Tensors
                       Boaz Porat
                 Department of Electrical Engineering
                Technion – Israel Institute of Technology
                     boaz@ee.technion.ac.il
                       May 27, 2014
          Opening Remarks
          This document was written for the benefits of Engineering students, Elec-
          trical Engineering students in particular, who are curious about physics and
          would like to know more about it, whether from sheer intellectual desire
          or because one’s awareness that physics is the key to our understanding of
          the world around us. Of course, anybody who is interested and has some
          college background may find this material useful. In the future, I hope to
          write more documents of the same kind. I chose tensors as a first topic for
          two reasons. First, tensors appear everywhere in physics, including classi-
          cal mechanics, relativistic mechanics, electrodynamics, particle physics, and
          more. Second, tensor theory, at the most elementary level, requires only
          linear algebra and some calculus as prerequisites. Proceeding a small step
          further, tensor theory requires background in multivariate calculus. For a
          deeper understanding, knowledge of manifolds and some point-set topology
          is required. Accordingly, we divide the material into three chapters. The
          first chapter discusses constant tensors and constant linear transformations.
          Tensors and transformations are inseparable. To put it succinctly, tensors are
          geometrical objects over vector spaces, whose coordinates obey certain laws
          of transformation under change of basis. Vectors are simple and well-known
          examples of tensors, but there is much more to tensor theory than vectors.
          The second chapter discusses tensor fields and curvilinear coordinates. It is
          this chapter that provides the foundations for tensor applications in physics.
          The third chapter extends tensor theory to spaces other than vector spaces,
          namely manifolds. This chapter is more advanced than the first two, but all
          necessary mathematics is included and no additional formal mathematical
          background is required beyond what is required for the second chapter.
          Ihaveusedthecoordinateapproachtotensors,asopposedtotheformal
          geometrical approach. Although this approach is a bit old fashioned, I still
          find it the easier to comprehend on first learning, especially if the learner is
          not a student of mathematics or physics.
          All vector spaces discussed in this document are over the field R of real
          numbers. We will not mention this every time but assume it implicitly.
          Iwouldappreciatefeedback,comments,corrections,andcriticisms.Please
          e-mail to boaz@ee.technion.ac.il.
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              Chapter 1
              Constant Tensors and Constant
              Linear Transformations
              1.1  Plane Vectors
              Let us begin with the simplest possible setup: that of plane vectors. We
              think of a plane vector as an arrow having direction and length, as shown in
              Figure 1.1.
              The length of a physical vector must have physical units; for example: dis-
              tance is measured in meter, velocity in meter/second, force in Newton, elec-
              tric field in Volt/meter, and so on. The length of a ”mathematical vector” is
              apurenumber.Lengthisabsolute,butdirectionmustbemeasuredrelative
              to some (possibly arbitrarily chosen) reference direction, and has units of ra-
              dians (or, less conveniently, degrees). Direction is usually assumed positive
              in counterclockwise rotation from the reference direction.
              Vectors, by definition, are free to move parallel to themselves anywhere in
              the plane and they remain invariant under such moves (such a move is called
              translation).
              Vectors are abstract objects, but they may be manipulated numerically and
              algebraically by expressing them in bases. Recall that a basis in a plane is
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                       Figure 1.1: A plane vector having length and direction
                apairofnon-zeroandnon-collinearvectors(e1,e2). When drawing a basis,
                it is customary to translate e1 and e2 until their tails touch, as is shown in
                Figure 1.2.
                               Figure 1.2: A basis in the plane
                The basis depicted in Figure 1.2 happens to be orthonormal; that is, the two
                vectors are perpendicular and both have unity length. However, a basis need
                                                      ˜ ˜
                not be orthonormal. Figure 1.3 shows another basis (e1,e2), whose vectors
                are neither perpendicular nor having equal length.
                Let x be an arbitrary plane vector and let (e1,e2)besomebasisintheplane.
                Then x can be expressed in a unique manner as a linear combination of the
                basis vectors; that is,
                                     x=ex1+e x2                   (1.1)
                                         1    2
                                  1 2
                The two real numbers (x ,x)arecalledthecoordinates of x in the basis
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