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outer products v5
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An old version of a section from Phil's tensor wedge document. It covers covariant and contravariant components in a possibly non-Cartesian basis, raising and lowering indices with the metric tensor g_ij, and how the metric makes V a normed, metric and Hilbert space. It defines length, distance and the inner product, and notes that g_ij equals e_i dot e_j. The text shown is only the opening of the section.
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1.3 Outer Products
In this section we are going to discuss the components of vectors and tensors. As mentioned above (1.1.2), we shall assume
ei = basis elements of V dim(V) = n i = 1,2...n
e'j = basis elements of W dim(V) = n' j = 1,2...n'
Since we allow the ei to form a non-Cartesian basis, we must deal with the metric tensor and covariant notation. In this notation, a vector has two kinds of components. Contravariant components are written up as for example vi, while covariant components are written down, vi. The two kinds of components are related by the metric tensor gij:
vi = Σjgijvj gij raises a component index
vi = Σjgijvj gij lowers a component index
Once we assume a metric tensor for V, the raw vector space V because also a normed linear space, a metric space, and a Hilbert space, if we use the following "natural" definitions. It is a normed linear space because we can define the norm || a || of a vector a ϵ V in this manner
|| a || 2 = Σij gijaiaj = Σiaiai
where ||a|| is the "length" of vector a. It is a metric space because we can then define a metric d(a,b) according to
[d(a,b)]2 = || a - b || 2 = Σij gij (a-b)i(a-b)j = Σi (a-b)i(a-b)i
where d(a,b) is the "distance between" a and b. It is a Hilbert space because we can define an inner (scalar, dot) product of two vectors in V by
a b = Σij gij aibj = Σi aibi = Σi aibi
Given a general basis ei, the metric tensor is determined by
ei ej = gij