MATH 3210-004 SPRING 2026 - Week 6 - Integral Calculus 1 скачать в хорошем качестве

MATH 3210-004 SPRING 2026 - Week 6 - Integral Calculus 1

--- This is MATH 3210-004, the advanced single variable calculus class at the University of Utah. View the complete course: https://github.com/AlpUzman/MATH_3210... --- Table of Contents: 00:00:00 1 of 4 00:03:22 recap: theorem: Let I be a compact interval, f be a real valued function on I. (ii) If im(f) is a closed and bounded interval and f is strictly increasing, then f is continuous 00:05:37 proof of (ii) 00:12:06 exercise: farm explicitly 00:12:43 proof continued 00:21:04 exercise: explicit time changes for subsequences 00:21:36 proof continued 00:32:09 discussion of theorem and proof 00:38:59 warning: continuity (and integral calculus) works well with closed and bounded intervals; differential calculus works well with open intervals 00:41:33 warning: "uniform continuity" is different from "uniform convergence" 00:42:16 theorem: Let X and Y be metric spaces, f be a function from X to Y. (i) If f is uniformly continuous, then it is continuous. (ii) if X is compact and f is continuous, then it is uniformly continuous. In particular, any real valued function defined on a closed and bounded interval is continuous iff unif continuous 00:45:54 uniform continuity 00:48:21 discussion of uniform continuity 00:49:11 example: f(x) = 1/x defined on I = ]0,1] is continuous but not uniformly continuous 00:50:19 2 of 4 00:51:16 recap: continuity versus uniform continuity 00:53:47 example continued: f(x) = 1/x defined on I = ]0,1] 00:57:39 exercise: C^0 infty-metric 01:01:54 C^0 (aka L^infty aka supremum aka uniform) metric 01:03:21 recap: reason behind the C^0 notation 01:04:10 uniform convergence of a sequence of functions 01:06:46 statements equivalent to uniform convergence 01:09:18 pointwise convergence of a sequence of functions 01:11:28 statements equivalent to pointwise convergence 01:13:55 exercise: uniform convergence implies pointwise convergence 01:15:33 example 1 [calculations will be corrected in 3 of 4] 01:31:46 example 2 01:40:23 3 of 4 01:41:06 correction: example 1 01:45:19 example 2 continued 01:58:04 comparison of uniform versus pointwise limits 01:59:57 theorem: uniform limit of a sequence of continuous functions is continuous 02:01:33 example 3 02:05:51 exercise: in theorem, "continuous" can be replaced by "uniformly continuous" or "bounded" 02:06:41 exercise: proof of theorem in the abstract metric space case 02:07:07 proof of theorem 02:18:35 comparison of uniform versus pointwise limits continued 02:22:49 plan going forward 02:25:22 preview: integration theory 02:28:51 preview: Bourbaki integration (aka regulated theory) 02:29:33 preview: two paradigms for integration: Riemann versus Lebesgue 02:30:02 4 of 4 02:30:54 integration theory 02:31:45 preferable properties for an integration theory 02:35:55 meta-representation theorem: any class is a class of functions, any scalar function on any class is an integration theory 02:37:02 natural habitat for integration is a closed and bounded interval (for our purposes) 02:38:33 two paradigms for integration: Riemann versus Lebesgue 02:41:27 Riemann paradigm for integration 02:47:57 preview: limiting procedures in Riemann paradigm; net convergence 02:50:16 Lebesgue paradigm for integration 02:55:13 comparison of two paradigms for integration 02:59:07 integration theories in Riemann versus Lebesgue paradigm 03:08:50 step function 03:11:41 example 03:14:58 integral of a step function --- 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/