MATH 3220-002 FALL 2025 - Week 10 - Integral Calculus 1 скачать в хорошем качестве

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

--- 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... --- 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 --- 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...   --- 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/