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INTRODUCTION TO TENSORS
Tensors pervade all the branches of physics. In mechanics, the moment of inertia tensor is used to describe the rotation of rigid bodies, and the stress-strain tensor describes the deformation of elastic bodies. In electromagnetism, the conductivity tensor extends Ohm’s law to handle current flow in nonisotropicmedia, and Maxwell’s stress tensor is the most elegant way to deal with electromagneticforces. The metric tensor of relativistic mechanics describes the strange geometry of space and time. This chapter presents an introduction to tensors and their manipulations, first strictly in Cartesian coordinates and then in generalized curvilinear coordinates. We will limit the material in this chapter to orthonormal coordinate systems. Tensors in nonorthonormal systems are discussed later, in Chapter 14. At the end of the chapter, we introduce the so-called ‘‘pseudo’’-objects,which arise when we consider transformations between right- and left-handed coordinate systems.
4.1 THE CONDUCTIVITY TENSOR AND OHM’S LAW
The need for tensors can easily be demonstrated by considering Ohm’s law. In an ideal resistor, Ohm’s law relates the current to the voltage in the linear expression
I = - V
R ’ (4.1)
In this equation, Z is the current through the resistor, and V is the voltage applied across it. Using MKS units, Z is measured in amperes, V in volts, and R in ohms. 67
68 INTRODUCTION TO TENSORS
Equation 4.1 describes the current flow through a discrete element. To apply Ohm’s law to a distributed medium, such as a crystalline solid, an alternative form of this equation is used:
5 = UE. (4.2) Here 5 is the current density, E is the electric field, and u is the material’s conductivity. In MKS units, 5 is measured in amperes per meter squared, E in volts per meter, and u in inverse ohm-meters.
Equation 4.2 describes a very simple physical relationship between the current
density and the electric field, because the conductivityhas been expressed as a scalar. With a scalar conductivity, the amount of current flow is governed solely by the magnitudes of (+ and E, while the direction of the flow is always parallel to E. But in some materials, this is not always the case. Many crystalline solids allow current to flow more easily in one direction than another. These nonisotropic materials must have different conductivities in different directions. In addition, these crystals can even experience current flow perpendicular to an applied electric field. Clearly Equation 4.2, with a scalar conductivity, will not handle these situations. One solution is to construct an array of conductivity elements and express Ohm’s law using matrix notation as
u 1 1 u 1 2 (+
[ = [ ;:: Uz2 Uz3] [.
u 3 2 a 3 3 In Equation 4.3, the current density and electric field vectors are represented by column matrices and the conductivity is now a square matrix. This equation can be written in more compact matrix notation as
or in subscriptlsummationnotation as
All these expressions produce the desired result. Any linear relationship between 3 and E can be described. The 1-componentof the current density is related to the 1 -component of the electric field via u 11, while the 2-component of the current density is related to the 2-component of the electric field through u 2 2. Perpendicular flow is described by the off-diagonal elements. For example, the u 1 2 element describes flow in the 1-direction due to an applied field in the 2-direction. The matrix representation for nonisotropic conductivity does, however, have a fundamental problem. The elements of the matrix obviously must depend on our choice of coordinate system. Just as with the components of a vector, if we reorient our coordinate system, the specific values in the matrix array must change. The matrix array itself, unfortunately, carries no identificationof the coordinate system used. The way we solved this problem for vector quantities was to incorporate the basis vectors
