differential geometry w notes from teacher_Part_85

# differential geometry w notes from teacher_Part_85 - 169...

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7.2. SINGULAR CHAINS 169 7.2 Singular Chains The standard Euclidean p -simplex in R p is the convex set Δ p R p gener- ated by an ordered ( p + 1)-tuple ( P 0 , P 1 , . . . , P p ) of points in R p P 0 = (0 , . . . , 0) , P i = (0 , . . . , 0 , 1 , 0 , . . . , 0) , i = 1 , . . . , p , all of whose components are 0 except for the i -th component, which is equal to 1. We use the following notation Δ p = ( P 0 , P 1 , . . . , P p ) . Let M be an n -dimensional manifold. A singular p -simplex in M is a di erentiable map σ p : Δ p M . By slightly abusing notation we denote the image σ p ( Δ p ) of Δ p in M under the map σ p just by σ p . Let α be a p -form on M and σ p be a p -simplex in M . We deﬁne the integral of α over σ p by Z σ p α = Z Δ p σ * p α . The k -th face of a standard p -simplex Δ p = ( P 0 , P 1 , . . . , P p ) (a face opposite to to the vertex P k ) is the convex set in R p generated by an ordered p -tuple of points in R p Δ ( k ) p - 1 = ( P

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differential geometry w notes from teacher_Part_85 - 169...

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