Calculus on Manifolds-Singular-n-Chains, guaranteed and verified 100% Pass

1


Singular 𝑛-Chains

Def. A singular 𝒏-cube in 𝐴 ⊆ ℝ𝑛 is a continuous function:
𝑐: [0, 1]𝑛 → 𝐴
([0, 1]𝑛 = [0, 1] × [0, 1] × … × [0, 1], 𝑛-times)



Ex. 𝑐: [0, 1]2 → 𝐷 ⊆ ℝ2 , where 𝐷 is the unit disk 𝑥 2 + 𝑦 2 ≤ 1.
𝑐(𝑟, 𝜃) = (𝑟 cos 2𝜋 𝜃, 𝑟 sin 2𝜋 𝜃).


Def. The standard 𝒏-cube is the identity map on [0, 1]𝑛 :
𝐼 𝑛 : [0, 1]𝑛 → ℝ𝑛 ; 𝐼 𝑛 (𝑥 ) = 𝑥.


We will consider the formal sums of singular 𝑛-cubes of the form:
3𝑐1 − 4𝑐2 + 5𝑐3 , where 𝑐1 , 𝑐2 , 𝑐3 are singular 𝑛-cubes in 𝐴. A formal sum like
this is called an 𝒏-chain in 𝐴.


𝑛
Def. Let 𝑥 ∈ [0, 1]𝑛−1 . Define 𝐼(𝑖,0) (𝑥) = 𝐼 𝑛 (𝑥1 , … , 𝑥𝑖−1 , 0, 𝑥𝑖 , … , 𝑥𝑛−1 )
and 𝐼(𝑛𝑖,1) (𝑥) = 𝐼 𝑛 (𝑥1 , … , 𝑥𝑖−1 , 1, 𝑥𝑖 , … , 𝑥𝑛−1 ).
𝑛 𝑛
𝐼(𝑖,0) is called the (𝒊, 𝟎) face and 𝐼(𝑖,1) is called the (𝒊, 𝟏) face.



We define the boundary of 𝐼 𝑛 by:


𝑛 1
𝑛
𝜕𝐼 𝑛 = ∑ ∑ (−1)𝑖+𝛼 𝐼(𝑖,𝛼) .
𝑖=1 𝛼=0

, 2


Ex. For 𝐼 2 = [0, 1]2 :
2
2 −𝐼(2,1)
𝐼(1,0) = 𝐼 2 (0, 𝑥1 ) = (0, 𝑥1 )
2
𝐼(1,1) = 𝐼 2 (1, 𝑥1 ) = (1, 𝑥1 )
2
2 2(
−𝐼(1,0) 2
𝐼(1,1)
𝐼(2,0) = 𝐼 𝑥1 , 0) = (𝑥1 , 0)
2
𝐼(2,1) = 𝐼 2 (𝑥1 , 1) = (𝑥1 , 1)



2
𝐼(2,0)

2 1
2
𝜕𝐼 2 = ∑ ∑ (−1)𝑖+𝛼 𝐼(𝑖,𝛼) = −𝐼(21,0) + 𝐼(21,1) + 𝐼(22,0) − 𝐼(2,1)
2

𝑖=1 𝛼=0




Def. For a general singular 𝑛-cube, 𝑐: 𝐼 𝑛 → 𝐴, we define the (𝑖, 𝛼) face as:

𝒄(𝒊,𝜶) = 𝒄 ∘ 𝑰𝒏(𝒊,𝜶) .


And we define the boundary of 𝑐 by:
𝒏 𝟏

𝝏𝒄 = ∑ ∑ (−𝟏)𝒊+𝜶 𝒄(𝒊,𝜶) .
𝒊=𝟏 𝜶=𝟎