MATH 3220-002 FALL 2025 - Week 10 - Integral Calculus 1

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This is MATH 3220-002, the advanced multivariable calculus class at the University of Utah.

View the complete course: https://github.com/AlpUzman/MATH_3220...

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Table of Contents:

00:00:00 1 of 2
00:03:38 three foundational theorems for metric spaces: Banach Contraction Principle, Arzela-Ascoli, Baire Category
00:05:31 Banach Contraction Principle 1: Any contraction on a complete metric space has a unique fixed point
00:06:03 recap: Lipschitz constant
00:07:01 contraction
00:08:04 BCP1 continued
00:08:18 heuristics for BCP1
00:08:42 BCP versus Brouwer Fixed Point Theorem
00:09:25 idea of the proof of BCP1
00:10:23 heuristics for BCP1 continued
00:12:15 BCP2: Let X be a complete metric space, x_0 be a point in X, R be a positive number, {K_s}_s be a continuously parameterized family of uniformly contracting contractions that moves x_0 not too much. Then the fixed point phi(s) of K_s depends continuously on s, and K_s^n(x) converges exponentially to phi(s)
00:16:54 discussion of BCP2
00:22:33 recap: implicit function theorem
00:23:26 recap: Banach space
00:23:50 recap: implicit function theorem continued
00:26:17 idea of the proof of implicit function theorem
00:47:30 2 of 2
00:48:47 recap: implicit function problem
00:51:23 claim: a C^0 solution to an implicit function problem is in fact C^k
00:51:45 proof of claim
01:01:27 exercise: if A and B are C^k, then B o A is also C^k
01:02:32 heuristics for BCP in the proof of implicit function theorem
01:12:59 exercise: formula for the solution of implicit function problem
01:16:30 neural networks
01:18:07 Newton method
01:20:00 more on BCP
01:23:50 recap: Riemann integration in 1D: proper, improper, indefinite, definite signed, definite unsigned
01:31:49 preview: Riemann integration in multivariable calculus

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Links:

More on the Banach Contraction Principle:
   • MATH 4800-001 FALL 2024 - Week 5 - Iterate...  
   • MATH 4800-001 FALL 2024 - Week 6 - Topolog...  
More on Riemann integration in 1D:
   • MATH 3210-001 SPRING 2025 - Week 8 - Integ...  

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License:

CC BY-NC-SA 4.0
Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International Public License
https://creativecommons.org/licenses/...

Alp Uzman
https://alpuzman.github.io/