Lecture Notes

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====== Math 104: Introduction to Real Analysis (Spring 2021) ======

Instructor: Peng Zhou \\
Email: pzhou.math@berkeley.edu

Welcome to Math 104, introduction to real analysis. We plan to cover the following topics: 

>>The real number system. Sequences, limits, and continuous functions in R and R. The concept of a metric space. Uniform convergence, interchange of limit operations. Infinite series. Mean value theorem and applications. The Riemann integral.

I am planning to use two textbooks. Most likely, I will follow Ross and assign problems from it. 
  * Elementary Analysis: The Theory of Calculus, by Kenneth A. Ross.  You can download a copy of this book [[https://link.springer.com/book/10.1007/978-1-4614-6271-2 | here]] (UC account login required).
  * Principles of Mathematical Analysis, by Walter Rudin 

Here is a problem book with solutions. The scope of this book does not exactly match with this course, which follows closely with Rudin.
  * //A Problem Book in Real Analysis//, available to [[https://link.springer.com/book/10.1007/978-1-4419-1296-1 | download from springer]] 


I will be teaching two sections, at Tuesday and Thursday  9:30 - 11:00 (section 5) and 12:30 - 14:00 (section 6). You can attend either of the lectures, they are supposedly to be the same. ( We are going to use 'Berkeley time', so the actual lecture starts at 10 minutes later. )
  * Zoom link:  [[https://berkeley.zoom.us/j/97213430388?pwd=dHJCRG5mNnhsb2tzQTBQd1A4b09LZz09
 | here]] (Meeting ID: 972 1343 0388, Passcode: 591757). 

==== Logistic ====
A tentative grading policy: 20% homework, two midterms each 20%, and final (accumulative) 40%. If you didn't do well in one of the midterm, you have the option to drop it, and final will have a 60% weight. The lowest homework grades will be dropped. 

Midterms (Feb 18, Apr 1) and final (May 12 or 13) will be done through gradescope. Midterm will be 90 minutes, and final will be 3 hours. 10 minutes extra will be given for scanning and uploading. If you are in a different time zone and prefer an alternative time slot, please contact me in advance. 

There will be weekly homework, to be submitted on gradescope. Entry code is BPKBXR



==== Office Hours and Q&A ====
Time: By appointment (everyday after 9pm, or Wednesday 1-2pm, Friday morning 9-12noon) \\
Zoom: https://berkeley.zoom.us/j/98628027987
 

We also have a GSI (Graduate Student Instructor) [[https://math.berkeley.edu/~dalal/ | Rahul Dalal]], he is available at 10:30-12:30 M, 11am-1pm WF and 5:30p-7:30p TuTh. [[https://berkeley.zoom.us/j/96793003738 | zoom link 96793003738 ]] and the password is epsilon. 

A faster way to communicate is through online platform, we can do either
  * Piazza. [[https://piazza.com/berkeley/spring2021/math104 | Sign Up]] (access code: cauchy). You can ask questions anonymously here. 
  * ** Zoom chat channel **. You may join the channel "Math 104, by Peng Zhou", and ask question there. 
  * Zoom direct message: you can search for my name "Peng Zhou" in zoom chat, and ask me questions directly there. Email is also fine. 

==== Misc Links ====
  * Prof Hutchings:[[https://math.berkeley.edu/~hutching/teach/proofs.pdf |  how to write proof.]] 
  * Latex: try https://overleaf.com, and here is [[https://courses.wikinana.org/_media/math185-3/hw1-template.tex | a simple latex template]]
  * [[./s/]]


==== Homeworks ====
There are weekly homeworks, grades will be published the next Wednesday night. There is always a late due date, 24h after the normal due date, only for emergency use. 
  * [[HW1]], Due Jan 29th 6pm (Friday). {{ math104-s21:hw1.pdf |solution}}
  * [[HW2]], Due Feb 5th 6pm (Friday). {{ math104-s21:hw2.pdf |solution}}
  * [[HW3]], Due Feb 12th 6pm (Friday) {{ math104-s21:hw3.pdf |solution}}
  * [[HW4]], Due Feb 20th 6pm **(Saturday)** {{ math104-s21:hw4.pdf |solution}}
  * [[HW5]], Due Feb 27th 6pm **(Saturday)**{{ math104-s21:hw5.pdf |solution}},  [[hw5-mistakes | common mistakes]]
  * [[HW6]], Due Mar 5th  6pm (Friday){{ math104-s21:hw6.pdf |solution}}
  * [[HW7]], Due Mar 12th  6pm (Friday){{ math104-s21:hw7.pdf |solution}}
  * ** [[about solution manual]] **
  * [[HW8]], Due Mar 19th  6pm (Friday) {{ math104-s21:hw8.pdf |solution}} 
  * [[HW9]], Due Mar 28th  6pm (Sunday) {{ math104-s21:hw_9.pdf |solution}}
  * [[HW10|HW10]], Due Apr 16th 6pm (Friday) {{ math104-s21:hw10.pdf |solution}}
  * [[hw11]], Due Apr 23 6pm (Friday) {{ math104-s21:hw11.pdf |solution}}
  * [[hw12]], Due Apr 30 6pm (Friday) {{ math104-s21:hw_12.pdf |solution}}(last homework)
==== Lectures ====
Here I put lecture video (valid for 30 days) and notes. 1 means section 9:30 and 2 means section 12:30. They are more or less the same.  ** video will expire after 30 days **. Videos can also be found on bcourse, which will not be deleted. 

| Date | Content | Notes  | Videos |  
| Jan 19, Tue | Logistic and Number system  |{{ math104-s21:note_jan_19_2021.pdf |1}}, {{ math104-s21:note_jan_19_2021_2_.pdf |2}}  |[[https://berkeley.zoom.us/rec/share/NOH2ydlIp0ymPUlI-6d8bChkRs_1A0o2ZwQxQsYqa_ugMCGQ43BIS4B-E0mRPuC6.1mBqBPWV2a7GunG8 | 1]], [[https://berkeley.zoom.us/rec/share/z-mJDKSzOQDe0pISKp9ZZYu7z-Vw68ov5h4r3_26AqqrcN58iXVe3AaRIeIa9va_.kSOa3HBJUI2H1vIS
 | 2 ]]  |  
| Jan 21, Thu | Completeness Axiom and Sequence  |{{ math104-s21:note_jan_21_2021.pdf |1}}, {{ math104-s21:note_jan_21_2021_3_.pdf |2}} | [[https://berkeley.zoom.us/rec/share/m8bU04REYsokMpaRQBScFhdkMKUOb5_9NqEpbJLrUIx9S32FOgNFlkadpKk4uJDW.5FmvHV3FeiTNo5BJ
 |1]], [[https://berkeley.zoom.us/rec/share/pEwZk4gadMV557Gq8L44Hb0XuSxk5cHx-uFRCdCKUu5b1Hztm6AqHFC1i0S1w29_.eLOGyEtbsuEiBeF8
 |2]] |    
| Jan 26, Tue  | Ross §7,§9 | {{ math104-s21:note_jan_26_2021.pdf |1}}, {{ math104-s21:note_jan_26_2021_2_.pdf |2}}| [[https://berkeley.zoom.us/rec/share/jSPmJCqt0g7ThCfc3CzJYsqyJJi-36rpI0aaa--jTDw-Uq_4IgJ4a0Yy3R8WFjrS.D-Fv7k8hET995EIc |1]], [[https://berkeley.zoom.us/rec/share/lrclJ9eB7ECfGa_TaAy0dk1bdb_9vFTWVN0tAYvyssyvs_YKVQp6lrcvD8qQ2XrN.OY0cTxS5r9Vagtn6|2]] | 
| Jan 28, Thu | Ross §10, Monotone sequence  | {{ math104-s21:note_jan_28_2021.pdf |1}}, {{ math104-s21:note_jan_28_2021_2_.pdf |2}} | [[https://berkeley.zoom.us/rec/share/oVzbK_ssi43tzEjT__mR3BwAzSiCmy_AmqT9QvgWdQkJjevtsHPveZyL-N2KPCzw.idmISrVB42f5DHpy|1]], [[https://berkeley.zoom.us/rec/share/UjlSif93v7Cso2CW8hIrV2CSX1aDRikVW0AaKGG0KspQ11YlFdc5_XsBe17LH8Mj.so90R_HCga58NDAF|2]] | 
| Feb 2, Tue  | Cauchy Sequence, limsup and liminf | {{ math104-s21:note_feb_2_2021.pdf |1}}, {{ math104-s21:note_feb_2_2021_2_.pdf |2}}  | [[https://berkeley.zoom.us/rec/share/T9qrdsXbfoGqua9IEG79sV1g1R7Xmi6H7w_xA9WLnCD3CTk5sevwEc3qheL0P7CR.amLZR7OlS6hF5Zri |1]],  [[https://berkeley.zoom.us/rec/share/pJGVF3zGDN_cr0d7VSUiqVJWGww38wtnFkFNX5IZ_XQr4LsXYUOL8OyJRkOVHB7m.U0HkUes84nDpZJkq |2]]   | 
| Feb 4, Thu | Subsequence   | {{ math104-s21:note_feb_4_2021.pdf |1}}, {{ math104-s21:note_feb_4_2021_2_.pdf |2}} | [[https://cloud.yiye.page/s/8o3G6CZFPrWbtQb |1]], [[https://berkeley.zoom.us/rec/share/l8TrHZxymA4r-QB14_UPt2Op2kpv7keANaW-0CLn9fgLgkVdNXnVit5sHxT4lKqo.tOWd20Qbb8OtDsM8|2]] |    
| Feb 9, Tue  | Subsequence(2) | {{ math104-s21:note_feb_9_2021.pdf |1}}, {{ math104-s21:math_104_2020-02-09_2.pdf |2}} |[[https://berkeley.zoom.us/rec/share/HwnPcU-Ep3Oe2Mijfy7toy_a86QxDdAFTrKgB2bIIoab5kz1uhtcpq-V7gL4w_ZH.cluH6bTl9F64B-Z4|1]]  [[https://berkeley.zoom.us/rec/share/gUkTOYqm4z8CvEr8Eit6gVfcAPWJ3FoipfRS1yu-FmhkvIq2Q7VBI2FDq9TO4EKx.lPcG8YtYQlnaBtFA|2]]  | 
| Feb 11, Thu | §12 limsup, liminf and review  |{{ math104-s21:note_feb_11_2021.pdf |1}}, {{ math104-s21:note_feb_11_2021_2_.pdf |2}}  |[[https://berkeley.zoom.us/rec/share/HwJxYBgAgA5fqKoZJJPzGFPjqBL7EhXQgKjePZwJFGLM3xcgEaMNSkALQ_rBBT_o.vpvnGkeQKnRb_sxi|1]],  [[https://berkeley.zoom.us/rec/share/BveMD5p8lQvkXn-zdrjKcvloIV-mP08ABs2bmFul-WX_bHzpQiLhPk_jAf9EPmVC.wrs9A7BmiQMi_d-N|2]] |    
| Feb 16, Tue  | §13 Metric Space and Topology | {{ math104-s21:note_feb_16_2021.pdf |1}}, {{ math104-s21:note_feb_16_2021_2_.pdf |2}} | [[https://berkeley.zoom.us/rec/share/D07cu2-L5xs7Df4SZ2OoB4xwVYR2JrtT5pKW9dIasqvQxyiX3iGsDLw6tvrLDbjt.lEIOQ3IFII75JkaU|1]], [[https://berkeley.zoom.us/rec/share/SgiCs_eZGZflHtjI8xZL2PjSG7RyMtfZ5jzjoPWWtLZZObUY0LopEUqZIlkDKAom.Aa-cYswKbAfNh6do|2]]    | 
| Feb 18, Thu | [[Midterm-1]] and [[suggestions | suggestions]]  | {{ math104-s21:math_104-mt1a.pdf |A}}, {{ math104-s21:math_104-mt1b.pdf |B}} | {{ math104-s21:midterm_1_solution.pdf |solution}} [[mt1-stat|stat]] |    
| Feb 23, Tue  | §13 Compactness |{{ math104-s21:note_feb_23_2021.pdf |1}}, {{ math104-s21:note_feb_23_2021_2_.pdf |2}}  | [[https://berkeley.zoom.us/rec/share/zHbY73kUh-HpHutiavc_PVoH8U-aEoNiJZK4H9iEbvUYL0Eefhm4NTmwcHnGKrVO.fH2144ANsHlAdK40|1]], [[https://berkeley.zoom.us/rec/share/UJMCwgRWGT6ZGwulm9EKDryJGGkjU1OTWgftugoLR2qoIWhwHkk3UGWBpA2wX8oe.3neJhcfjCwMi6smW |2]]   | 
| Feb 25, Thu | §14, 15 Series   |{{ math104-s21:note_feb_25_2021_2_.pdf |1}}, {{ math104-s21:note_feb_25_2021_3_.pdf |2}} |[[https://berkeley.zoom.us/rec/share/5c9C6MbE17MhEME2K_MRH806qdkhr8Dbf7famYD17ChtcnIT8JABRC6YM7aZeWel.NC_qMH1H6Oteg8Mx|1]],  [[https://berkeley.zoom.us/rec/share/T8LsLDNmoI0rIOQzRwW7LunHRKjZcWW7Ll4ifEAa8c4jtO9mOEwonSMLO43k8CN9.Ktu1i8FhqzKIscl1|2]] |   
| Mar 2, Tue  | Continuity of functions[Ru, p83-89] |{{ math104-s21:note_mar_2_2021.pdf |1}}, {{ math104-s21:note_mar_2_2021_2_.pdf |2}}  | [[https://berkeley.zoom.us/rec/share/a2vCEX6TAPKtSkfVgqV_4YJQHMDtVlZfYREwrdvyA0NEkVbe2TkKPHEF63MgwGOd.AMkTzLF0XShUl-RT|1]], [[https://berkeley.zoom.us/rec/share/p3NScENade0764dRPtYlLotM825wpCiBqXtijfIHUpwQKhe5R0-5plQP77_7FFYa.5jlySlVMA8fGrCNF|2]]  | 
| Mar 4, Thu | Continuity and Compactness   |{{ math104-s21:note_mar_4_2021.pdf |1}}, {{ math104-s21:note_mar_4_2021_2_.pdf |2}} |[[https://berkeley.zoom.us/rec/share/zd7rwqtGd_S-RAYFGKOQqqRhzR4Ew__8DoBhYyArMKul5UBag2Skjxx6x064sLGD.gab308Da7PFpKQ3R |1]],  [[https://berkeley.zoom.us/rec/share/dEGfKcSeVnkEw3BMTUPNvgzMGQiZKEJ9NYtCUC3SVkCCBNR6VpTCXfJgpYRjm6U0.oxiXToup2y3H5mc7|2]] |    
| Mar 9, Tue  | Uniform Continuity, Connectedness  | {{ math104-s21:note_mar_9_2021.pdf |1}}, {{ math104-s21:note_mar_9_2021_2_.pdf |2}}  | [[https://berkeley.zoom.us/rec/share/OoMZAwLALEHGGu6yww0AIwGkLVXBZvzAWhJNHLWOzDJT5S4P6Y9L8OTqEUBBGb8B.KzklsWQdbgM2PmHf|1]], [[https://berkeley.zoom.us/rec/share/-7oWSb_6VFybUkqC_BsBfq5XB0VJ-5S0Y1oVM9E7AGOuZqzXCj71jyMUgoXjZzpG.pHvXaAX1ngPVNgp5|2]]   | 
| Mar 11, Thu | Finishing Connectedness, Discussion and Examples  | {{ math104-s21:note_mar_11_2021.pdf |1}}, {{ math104-s21:note_mar_11_2021_2_.pdf |2}}  | [[https://berkeley.zoom.us/rec/share/8uxrUZGRLSO9NVJZwMI0av9QmTzxf7venGB58IhkFJicFUfBrNPQdPbVKDzGw5Ya.BHDlHy4C7aQEeKfd|1]], [[https://berkeley.zoom.us/rec/share/q4o6v-ThBRsoGm5RCWxlpw4WVi96lFx1RzgctllE6mjvFWjy903zsHqnKjUZQ8Rs.AEZ7vwmICm12C7ue|2]] |    
| Mar 16, Tue  | Sequence and Convergence of functions |{{ math104-s21:note_mar_16_2021.pdf |1}}, {{ math104-s21:note_mar_16_2021_2_.pdf |2}}  | [[https://berkeley.zoom.us/rec/share/JPieI9ADi3xL6FlToT1DFeQbVDTykFxSjoRQf_PL3Vt3Lp9PVzsncA6OUQ-rIhBa.-F48ZyX01iHF4vel|1]],  [[https://berkeley.zoom.us/rec/share/SqZptt1dEKvq-TYCZ9dv4N7aj_RgpUUC5dUpJ1oVopS-plr0VLTNitY60i-Nx9Lq.hRo_Ic9-MqMMv6Hb |2]]  | 
| Mar 18, Thu | Uniform Convergence and Continuity  | {{ math104-s21:note_mar_18_2021.pdf |1}},  {{ math104-s21:note_mar_18_2021_2_.pdf |2}}  |[[https://berkeley.zoom.us/rec/share/Gv0NqUlqX48Hc6Ta-g5_4MvZLBdm-CHqmgTkprCerw2L1haxU3sk0jhgnt5I4Bhe.rBWup3O5r1sYLZSs|1]], [[https://berkeley.zoom.us/rec/share/LufTnDXMRyK6KUlCzTiRDL6V9csPKVfmq4YAa6YebCCOh67anwoHTM6jb6Iqbjs8.8QWog7s1eg8VELld|2]]  |    
| Mar 23, 25 | Spring Break!   |  |  |   
| Mar 30, Tue | Review  |{{ math104-s21:note_mar_30_2021.pdf |1}}, {{ math104-s21:note_mar_30_2021_2_.pdf |2}}  |[[https://berkeley.zoom.us/rec/share/OiknqGIJ4XkLVCk2koHgpbeH3SyP-snF0Ixi9yYp06MX6RySizUAO7hYn7cmt5L3.WT9PagI4vYH3Ulhf|1]],  [[https://berkeley.zoom.us/rec/share/l--oudXJQ4nEUdP4XrOsyoKT_PK1YJ2Iox1j4RHxZyKlEyRL2eY_b0_xDoiCqlmk.kyEd8AcSCKln4Xxn|2]]|   
| Apr 1, Thu | [[Midterm-2]]   | {{ math104-s21:math104-mt2-a.pdf |A}}, {{ math104-s21:math104-mt2-b.pdf |B}} | {{ math104-s21:math104-mt2-solution.pdf | solution}} [[math104-s21:mt2-stat|stat]]|   
| Apr 6, Tue  | Differentiation, Mean Value Theorem | {{ math104-s21:note_apr_6_2021.pdf |1}}, {{ math104-s21:note_apr_6_2021_2_.pdf |2}} |[[https://berkeley.zoom.us/rec/share/HKCyZlk78aNiEc66sjoSYlkEkUtHhy1MZ6aSCrhiITivd-9ZykoLBjOQe6SGtihd.dAtvRkxtfGIzLRfo|1]], [[https://berkeley.zoom.us/rec/share/35G9cyuB7UNNIfi3YA14LrYTAQOodpZl9l2GGjd3GvskP0sxMPKDOKCw7cAZGj_C.6-OWMVnr0ErqQPgZ|2]]    |
| Apr 8, Thu  |Mean Value Theorem, L'Hopital Rule  | {{ math104-s21:note_apr_8_2021.pdf |1}}, {{ math104-s21:note_apr_8_2021_2_.pdf |2}} |[[https://berkeley.zoom.us/rec/play/XXvhNz5NF1zs4-3jmPLDcCRgSpS9e9sygYhsdCe1Uap2OGqoDFgePj_zY8En9R-eSSJ4UUGhrXLftqNR._zOQLOHoMoGVL1Yh?continueMode=true&_x_zm_rtaid=YZIbZyuTQXmVmseMDF2gUg.1617916229849.b11bff0202de40e2293378f93528da77&_x_zm_rhtaid=73|1]], [[https://berkeley.zoom.us/rec/share/JiMFHLQ3P0apQ-fa72kDuiKgE7bAnVzT6Eovdse6vL5C1rb8Au5yj8JYObY7VlfI.obGtrSOScmNbQGz0|2]]   |
| Announcement: [[tutor | drop-in tutor available ]] ||||
| Apr 13, Tue  | Taylor's Theorem  | {{ math104-s21:note_apr_13_2021.pdf |1}}, {{ math104-s21:note_apr_13_2021_3_.pdf |2}} | [[https://berkeley.zoom.us/rec/share/GLUsik9CXrGTajmu1hFlEl0EstxroYmTs27K2pgvqXa2IG8lztWopuK2F2onhcc.P6o1t0CxuDj542nh|1]],  [[https://berkeley.zoom.us/rec/share/AqN0ibzEASMFAD8h2ajpvDDY2C4RkxI0RnA1-8N7_8cdsgk5docbHlUCR1t1UCX1.pmvrTVE8UlVqzw6f|2]] |
| Announcement: 10 pts extra credits available: [[extra-credits | sharing notes and questions]] ||||
| Apr 15, Thu  | Finish Taylor Series. Begin Riemann integral  | {{ math104-s21:note_apr_15_2021.pdf |1}}, {{ math104-s21:note_apr_15_2021_2_.pdf |2}}  | [[https://berkeley.zoom.us/rec/share/pFDh-jWArJ3DYO5YW5Bj0uGoNMHSjj6C9KVGKTyWqunn4hU2Jq5EeFkjp9W3Chq8.g_KZ36A4MR2Ugd3p|1]], [[https://berkeley.zoom.us/rec/share/_sXMu3WIIvXXtGXtEBwMLQzTqPsYuKTD4oNkujpxbLrkyXYaUWK3oYOZ-x6uPse5.HVQSTdphE27CIWLF|2]]   |
| Apr 20, Tue  | Discussion of Integrability | {{ math104-s21:note_apr_20_2021.pdf |1}}, {{ math104-s21:note_apr_20_2021_2_.pdf |2}}  | [[https://berkeley.zoom.us/rec/share/iGZXxYRmqA-mgbcqLSIKo0UZijXzHofUP0Fu7Dei5debW6JMXvIoLlQ9pEYCqkLz.XMHnjtJDvoHBbsnU|1]], [[https://berkeley.zoom.us/rec/share/cN3IaTxLqWVsDhS4OW5y2wd7I3nNqp9vFAO8h8A1pXPEalf6vC8Jqm7fi2HoNKDP.uXr8lUXxiDrcoQWS|2]]   |
| Apr 22, Thu  | More on integrability | {{ math104-s21:note_apr_22_2021.pdf |1}}, {{ math104-s21:note_apr_22_2021_2_.pdf |2}}  | [[https://berkeley.zoom.us/rec/share/QhuQlsYCFZW5XdMHwIjQKvkQ_aazMoWgRbQleMxtY32yYndruJH2slUJ-xHdFso.pMgH7JwlgKss-zj5 |1]], [[https://berkeley.zoom.us/rec/share/txW1vqZ0dbQEa0eDaBkhl7pGXK8YRAxCzCHlhAEVpZbiLhfac8dY16tcRNM9aS91.DQTNzLwqPP7ASxsm |2]]   |
|In the last 20 minutes of video #2 above, I discuss Rudin 6.13 - 6.17. People from morning session please also watch it.  ||||
| Apr 27, Tue  | Integration and Differentiation  | {{ math104-s21:note_apr_27_2021.pdf |1}}, {{ math104-s21:note_apr_27_2021_2_.pdf |2}}  | [[https://berkeley.zoom.us/rec/share/njhk3tdPPg6ZnEs_142yxuamK0JOk6J2Wq3Oj3chgKIcUboJYF8EQt7YdoydygNz.WeinUujfp29Q_S2B |1]], [[https://berkeley.zoom.us/rec/share/Oct_fPRhdd5QXUIKZHYre55f_4AGpAjaebXPgoZsa4BJMHuxB9TZQWNO_7-h44ar.XVczqp2y46105Oua|2]] |
| Apr 29, Thu  | Uniform Convergence and review | {{ math104-s21:note_apr_29_2021.pdf |1}}, {{ math104-s21:note_apr_29_2021_2_.pdf |2}} | [[https://berkeley.zoom.us/rec/share/dsT6RcQrRPgqVqb-7S7kiOwpYLnlWw6zndt-2loTcRJTxGpFQUL-4F4mFxht_l8.OQzg1kxK7i1kiuRI|1]], [[https://berkeley.zoom.us/rec/share/y08PTTvVbC6DP6KilHYWT77ERQg1yV5bBO7UOEFAi26lDXgBOIchC_yelodBIgVi.ASqR4yFUNUuDU0me|2]]   |
| May 12-13 | [[Final ]] |{{ math104-s21:math_104-2.pdf |exam}}  | {{ math104-s21:math104-s21-final.pdf |solution}} [[final-stat|stat]]|


===== Other instances of Math 104 =====
  * [[https://ywfan-math.github.io/104s21.html | Yu-wei Fan (2021 Spring)]]
  * [[https://chenhi.github.io/math104-u14/ | Harrison Chen (2014 Summer) ]]
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