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Extra info for Algebraic topology. Errata (web draft, Nov. 2004)

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44) we may write 9 [dx-^dxir9^^ (452) 37 Metric Dual and taking into account Eq. 53) that is, the g*kl are the elements of the inverse of the matrix (gki)Thus, we can write Eq. 54) with gij =g*(dx\dxj). 55) With Eq. 52) in mind, in Euclidean space described by Cartesian coordinates, the orthonormal basis g | x , . . , -^ naturally induces orthonormality for the corresponding basis of the dual space gij = g*(dx\dxj) = Sij . 56) In Minkowski space described by coordinates (x° = c ^ x ^ x ^ x 3 ) , related orthonormal basis (gfs, gfr, gjy, gfr) and metric rj = diag(l, —1, —1, —1),

21) does not vanish. • Considering an open subset U of R n , two coordinate systems with coordinates {x1} and {y-7}, determining the associated ordered frames { A } and {^7} on U, are said to define the same orientation if the Jacobian determinant J = det (^j) is positive at all points of the subset. An orientation on U is the equivalence class of these ordered frames. In 3-dimensional Euclidean space we distinguish between righthanded and left-handed orthogonal sets of vectors [CP 86] and a transformation is orientation preserving if the Jacobian is positive.

9) This being true for the coordinate basis or any other basis, when u and r are 1-forms we have A{U®T) = -(u> ® T - T ® u>). 10) 48 Geometrical Properties of Vectors and Coventors Consequently, we obtain for the antisymmetrized wedge product of 1-forms LO /\T ~U®T - T ®UJ . 12) and the wedge product of a 1-form with itself vanishes, wAw = 0. E x a m p l e . Now we show that, given UJ 6 fi*(Af) and r G ft£(M) uAr = {-)krTAuj. 13) In fact, we have for the coordinate basis or any other basis d w d d d (cu T) d = ( T

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Algebraic topology. Errata (web draft, Nov. 2004) by Hatcher A.

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