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70 INTRODUCTION TO TENSORS
This is analogous to how a vector can be expanded in terms of its basis vectors as
v = X V i & = V1& + v,e, + v3g. (4.1 1) I
Let’s see how this new notation handles Ohm’s law. Using the conductivitytensor,
we can write it in coordinate-independent“vectorltensor” notation as
Notice the dot product between the conductivity tensor and the electric field vector
on the RHS of this expression.We can write this out in subscript/summationnotation
as
By convention, the dot product in Equation 4.13 operates between the second basis vector of and the single basis vector of E. We can manipulate Equation 4.13 as follows:
(4.14) (4.15) (4.16)
The quantities on the left- and right-hand sides of Equation 4.16 are vectors. The ith components of these vectors can be obtained by dot multiplying both sides of Equation 4.16 by 4 to give
which is identical to Equations 4.3-4.5. Keep in mind that there is a difference between.^ and E - ??, The order of the terms matters because, in general,
$j& * # 61 * ej&. (4.18)
The basis vectors in this tensor notation serve several functions:
- They establish bins to separate the tensor components.
- They couple the components to a coordinate system.
- They set up the formalism for tensor algebra operations.
- As shown later in the chapter, they also simpllfy the formalism for transforma- tions between coordinate systems.
Now that we have motivated our investigation of tensors with a specific example,
we proceed to look at some of their more formal properties.
TRANSFORMATIONS BETWEEN COORDINATE SYSTEMS 71
4.2 GENERAL TENSOR NOTATION AND TERMINOLOGY
The conductivity tensor is a specific example of a tensor that uses two basis vectors and whose elements have two subscripts. In general, a tensor can have any number of subscripts, but the number of subscripts must always be equal to the number of basis vectors. So in general,
The number of basis vectors determines the rank of the tensor. Notice how the tensor
notation is actually a generalizationof the vector notation used in previous chapters.
Vectors are simply tensors of rank one. Scalars can be considered tensors of rank
zero. Keep in mind that the rank of the tensor and the dimension of the coordinate system are different quantities. The rank of the tensor identifies the number of basis vectors on the right-hand side of Equation 4.19, while the dimension of the coordinate system determines the number of different values a particular subscript can take. For a three-dimensional system, the subscripts (i, j , k, etc.) can each take on the values (1,233). This notation introduces the possibility of a new operation between vectors called the dyadic product. This product is either written as K:E or just A B. The dyadic product between vectors creates a second-ranktensor,
(4.20)
This type of operation can be extended to combine any two tensors of arbitrary rank. The result is a tensor with rank equal to the sum of the ranks of the two tensors in the product. Sometimes this operation is referred to as an outer product, as opposed
to the dot product, which is often called an inner product.
4.3 TRANSFORMATIONS BETWEEN COORDINATE SYSTEMS
The new tensor notation of Equation 4.19 makes it easy to transform tensors between differentcoordinate systems. In fact, many texts formally define a tensor as “an object that transforms as a tensor.”While this appears to be a meaningless statement, as will be shown in this section, it is right to the point. In this chapter, only transformations between orthonormal systems are consid- ered. First, transformations between Cartesian systems are examined and then the results are generalized to curvilinear systems. The complications of transformations in nonorthonormal systems are deferred until Chapter 14.
4.3.1 Vector Transformations Between Cartesian Systems
We begin by looking at vector component transformations between two simple, two- dimensional Cartesian systems. A primed system is rotated by an angle @, with
