early outline

********************* Section 1: state basis elements ei of V and ei' of W and then basis of V x W state "rules" for the direct product in this V x W context state rules which are wrong state most general element of W x V as T = ....Tij.... specialize to V x V and describe T as a "tensor" show the four covariant notations if ei are tangent base vectors Section 2 state basis elements λi of V* and λi' of W* develop some facts about these λi functionals, introducing the qi vectors. state α and β as most general vectors in V* and W* introduce the direct product space V* x W* and show its basis elements assume the relation (λiλ'j)(v,w) = λi(v)λj'(w) state the "rules" for write most general element of V* x W* as Φ = .... evaluate Φ at a point, so for first time have a function Φ(v,w) expand α β onto basis and also show as a function finally restrict to W* = V* semantically associate both Φ and Φ(u,v) with a rank-2 tensor in V*2. Let's try to make a dual-space vectors be Greek letters. Section 3 Wedge in the V2 world (a) def of the wedge a definition of wedge product a ^ b in usual way develop "rules" for the wedge operator using a,b,c. state that those rules also apply for ei etc. write most general element of W 2 as F = , then show it includes a ^ b show how you could break basis vectors into two groups (b) how big is W 2 ? (c) how do determinants appear? geometric interpretations Hodge relation to the cross product Section 4 Wedge in the V*2 world (a) def of the wedge a definition of wedge product a ^ b in usual way develop "rules" for the wedge operator using a,b,c. state that those rules also apply for ei etc. write most general element of W 2 as F = , then show it includes a ^ b show how you could break basis vectors into two groups (b) how big is W 2 ? (c) how do determinants appear? geometric interpretations Hodge relation to the cross product