Calculus on Manifolds-Integration, guaranteed and verified 100% Pass

1


Integration
Suppose that 𝐴 ⊆ ℝ𝑛 is a closed rectangle, i.e.
𝐴 = {(𝑥1 , … , 𝑥𝑛 )| 𝑥𝑖 ∈ [𝑎𝑖 , 𝑏𝑖 ], 𝑖 = 1, … , 𝑛}

We want to first discuss the definition of: ∫𝐴 𝑓 , where 𝑓: 𝐴 → ℝ.


A partition of an interval, [𝑎, 𝑏], is a sequence, 𝑡0 , 𝑡1 , 𝑡2 , … , 𝑡𝑛 , where:
𝑎 = 𝑡0 ≤ 𝑡1 ≤ 𝑡2 ≤ ⋯ ≤ 𝑡𝑛 = 𝑏

A partition of a rectangle, [𝑎1 , 𝑏1 ] × [𝑎2 , 𝑏2 ] × … × [𝑎𝑛 , 𝑏𝑛 ], is a
collection: 𝑃 = {𝑃1 , 𝑃2 , … , 𝑃𝑛 } where each 𝑃𝑖 is a partition of [𝑎𝑖 , 𝑏𝑖 ].




A Partition in ℝ3

, 2


Suppose 𝑓: 𝐴 → ℝ is a bounded function and 𝑃 is a partition of 𝐴. For each
subrectangle, 𝑆, of the partition let:
𝑚𝑆 (𝑓) = inf{𝑓(𝑥)|𝑥 ∈ 𝑆}
𝑀𝑆 (𝑓) = sup{𝑓(𝑥)|𝑥 ∈ 𝑆}
and let 𝑣(𝑆) = volume of 𝑆.

Define the lower and upper sums of 𝑓 for 𝑃 by:


𝐿(𝑓, 𝑃 ) = ∑ 𝑚𝑠 (𝑓) ∙ 𝑣(𝑆)
𝑆


𝑈(𝑓, 𝑃 ) = ∑ 𝑀𝑆 (𝑓) ∙ 𝑣 (𝑆).
𝑆



Upper Sum= 𝑈(𝑓, 𝑃) Lower Sum= 𝐿(𝑓, 𝑃)




Notice 𝐿(𝑓, 𝑃 ) ≤ 𝑈(𝑓, 𝑃) since 𝑚𝑠 (𝑓) ≤ 𝑀𝑆 (𝑓).