tensor alg section removed from 3_1

Tensor algebra section removed from Section 3.1 The Tensor Algebra Suppose we define a very large vector space as the following direct sum of vector spaces, T(V) = V0 V V2 V3 ....... on forever // = Σi=0∞ Vi (3.1.16) (The space V0 represents the space of scalars.) The elements of this space T(V) are then tensors of any rank. A vector would fit into the V part, a rank-2 tensor would fit into the V2 part, and so on. One could in fact have a linear combination of a vector and a rank-2 tensor in this space T(V). Since T(V) contains arbitrary linear combinations of tensors of different ranks, T(V) is called a "graded algebra" where the ranks of the pieces of t ϵ T(V) are the grades. The "algebra" part is that one has a set of elements with rules + and for adding and multiplying elements. The set of elements of T(V) is closed under each of these operations. Each Vk subspace is clearly closed under addition +. Although Vk is obviously not closed under , the direct sum space T(V) is closed under . For example, one can think of the product of any number of tensors as [**...][**...][**...] ..... = ***** .... ϵ Vk (if k factors) ϵ T(V) (3.1.17) where we use the associativity property (2.8.22). Then the most general elements of the space T(V) are linear combinations (with coefficients in field K) of the tensors shown in (3.1.14). This huge vector space with all its many elements is known as the tensor algebra over field K, and again, we usually think K = R = reals. Of course "vectors" in the space T(V) defined above can have an infinite number of graded components. Usually only a finite number of these components are non-zero. In the context of this large space T(V), we have two meanings for "mixed" tensors which we don't want to confuse together. A pure rank-2 tensor has the form ab. A mixed rank-2 tensor is ΣijFijeiej . But a "mixed tensor" in the space T(V) could be any linear combination of tensors of different rank, such as 5.3 + 2v + ΣijFijeiej - π abc (3.1.18) Often this kind of mixed tensor is called a multivector. Suppose t is such a multivector composed of a linear combination of tensors whose grades (ranks) range from s to r. And suppose t' has grades ranging from s' to r'. The reader can easily show that tensor tt' can only contain tensors whose grades range from s+s' to r+r' . Example: t = v + ab s = 1 r = 2 t' = 2 + cde s = 0 r = 3 tt' = ( v + ab) (2 + cde) = v2 + vcde + ab2+ abcde = 2v + vcde + 2ab + abcde grades lie in range 1 to 5 (3.1.19) Notice the convention that vα = αv = αv when α is a scalar lying in V0. The above discussion of the tensor algebra T(V) will be repeated with more detail in Section 5.4. Then in section 6.9 it will be repeated again, but for the wedge operator ^ , resulting in the exterior algebra L(V).