{"id":968,"date":"2023-06-25T19:12:57","date_gmt":"2023-06-26T02:12:57","guid":{"rendered":"https:\/\/dornsife.usc.edu\/sergey-lototsky\/?page_id=968"},"modified":"2026-04-30T22:07:21","modified_gmt":"2026-05-01T05:07:21","slug":"math-445-fall-2022","status":"publish","type":"page","link":"https:\/\/dornsife.usc.edu\/sergey-lototsky\/math-445-fall-2022\/","title":{"rendered":"Math 445: Mathematics of Physics and Engineering II, Spring 2026"},"content":{"rendered":"\n\n  \n    \n\n\n\n\n\n\n<div\n  class=\"cc--component-container cc--rich-text \"\n\n  \n  \n  \n  \n  \n  \n  >\n  <div class=\"c--component c--rich-text\"\n    \n      >\n\n    \n      \n<div class=\"f--field f--wysiwyg\">\n\n    \n  <p>Everything can change with little or no notice at any moment&#8230;<\/p>\n<p><strong>In particular, a class (lecture and\/or discussion) can be moved to on-line mode on a very short notice, so please check your e-mail before every class.<\/strong><\/p>\n<h1 align=\"center\"><span style=\"color: blue;\"> Class number 39665<\/span><\/h1>\n<p><span style=\"color: red;\"> Math 445 in Fall 2025 semester: Key dates <\/span><\/p>\n<ul>\n<li>January 12: first day of classes<\/li>\n<li>January 19: MLK Day, no class<\/li>\n<li>January 30: Last day to drop without a `W\u2019 AND with refund<\/li>\n<li>February 16: Presidents\u2019 Day, no class<\/li>\n<li>February 27: Last day to drop without a `W\u2019, BUT WITH NO refund<\/li>\n<li>March 9: Midterm Exam 1 (during regular lecture time, in the regular lecture room)<\/li>\n<li>March 13: First computer project is due<\/li>\n<li>March 16-20: Spring break<\/li>\n<li>April 10: Last day to drop with a `W\u2019<\/li>\n<li>April 20: Midterm Exam 2 (during regular lecture time, in the regular lecture room)<\/li>\n<li>April 24: Second computer project is due<\/li>\n<li>May 1:\u00a0 Last day of classes<\/li>\n<li>May 6: Final exam (2:00 &#8211; 4:00 PM, in DMC 258)<\/li>\n<\/ul>\n<h1 align=\"center\"><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2026\/01\/Sched-445-S2026.pdf\" target=\"_blank\" rel=\"noopener\">Class Schedule<\/a><br \/>\n<a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2026\/01\/Math445-HW-S2026.pdf\" target=\"_blank\" rel=\"noopener\">Homework problems<\/a><br \/>\n<a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2026\/01\/Math445-HWSol-S2026.pdf\" target=\"_blank\" rel=\"noopener\">Some homework answers<\/a><br \/>\n<a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2026\/01\/Math445-Proj1and2-S2026.pdf\" target=\"_blank\" rel=\"noopener\">Computer Projects<\/a><br \/>\n<a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2026\/01\/Crank-Nicolson-Heat.pdf\" target=\"_blank\" rel=\"noopener\">Crank-Nicolson scheme for the heat equation<\/a><br \/>\n<a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/ImplSchemeWaveEq.pdf\" target=\"_blank\" rel=\"noopener\"> Implicit method for the wave equation <\/a><\/h1>\n<p><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2026\/01\/Math445-Summary.pdf\" target=\"_blank\" rel=\"noopener\"> Extended summary<\/a><\/p>\n<ul>\n<li><strong> Instructor: <\/strong> Dr. Sergey Lototsky<br \/>\n<strong> Office:<\/strong> KAP 248D.<br \/>\n<strong> Phone:<\/strong> 213&#8211;740-2389.<br \/>\n<strong> E-mail:<\/strong> lototsky (at) USC (dot) edu<br \/>\n<strong> URL:<\/strong> https:\/\/dornsife.usc.edu\/sergey-lototsky\/<br \/>\n<strong>Lectures:\u00a0 <\/strong>MWF 1-1:50pm, DMC 258<br \/>\n<strong>Office hours:<\/strong> MWF\u00a0 \u00a0 11am-12noon<\/p>\n<h3><span style=\"color: green;\"> Please do not hesitate to talk to me about your problems, questions, or concerns in this class. <span style=\"color: green;\">We can always arrange a special zoom meeting. <\/span><\/span><\/h3>\n<\/li>\n<li><strong> Teaching Assistant:\u00a0 <\/strong>Baran \u00c7etin<br \/>\n<strong><strong> E-mail:\u00a0<\/strong><\/strong> bcetin [at] usc (dot) edu<br \/>\n<strong>Discussions:<\/strong> Tuesdays\u00a0 4-4:50pm, DMC 260<br \/>\n<strong> Office hours: <\/strong>\u00a0in Math Center (KAP 263)<\/li>\n<\/ul>\n<p><strong>Beside the discussion sections, the TA is responsible for administering and grading quizzes and collecting and grading homeworks. <\/strong><\/p>\n<ul>\n<li><strong>Textbook: <\/strong>&#8220;Advanced Engineering Mathematics&#8221; by E. Kreyszig, Wiley. Any edition will work. The official version is the custom USC edition.<\/li>\n<li><strong>Supplement: <\/strong>&#8220;Mathematics of Physics and Engineering&#8221; by Edward K. Blum and Sergey V. Lototsky, World Scientific, 2006 (ISBN-13: 978-9-812-56621-8)<\/li>\n<li><strong>The Ultimate Alternative:<\/strong> &#8220;<span id=\"productTitle\" class=\"a-size-large celwidget\" data-csa-c-id=\"f9fda6-hl4la5-2py6sa-6pq3ln\" data-cel-widget=\"productTitle\">Mathematical Methods for Physics and Engineering: A Comprehensive Guide&#8221; by K. F. Riley, M. P. Hobson, and S. J. Bence:\u00a0 <\/span><span id=\"productSubtitle\" class=\"a-size-medium a-color-secondary celwidget\" data-csa-c-id=\"d952h7-jej57z-bajvs8-y27w99\" data-cel-widget=\"productSubtitle\">3rd Edition 0f a 1300+ page textbook, published by Cambridge University Press, 2006 (paperback ISBN-13: 978-0-521-67971-8)<\/span><\/li>\n<li><strong>Course goal: <\/strong>To realize that there is a lot of beautiful and useful mathematics out there beyond calculus and ordinary differential equations. In particular, we will cover all the material promised in the catalogue description of the course (Vector field theory; theorems of Gauss, Green, and Stokes; Fourier series and integrals; complex variables; linear partial differential equations; series solutions of ordinary differential equations), although not necessarily in this order. <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Math445-BigPicture.pdf\" target=\"_blank\" rel=\"noopener\">Here<\/a> is an alternative look at it.<\/li>\n<li>Two very interesting books closely connected with the second half of the course:\n<ul>\n<li>Nicholas J. Giordano, Physics of the piano. Oxford University Press, Oxford, 2010. 184 pp.<\/li>\n<li>Barry Mazur and William Stein, Prime numbers and the Riemann hypothesis. Cambridge University Press, Cambridge, 2016. xi+142 pp.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h3><strong> Save the dates! <\/strong> <span style=\"color: red;\"> There will be two in-class one-hour (50 min) exams (March 9 and April 20, both Mondays, during regular lecture time, in the regular lecture room). The two-hour (120 min) final exam is Wednesday, May 6, 2:00 &#8211; 4:00 PM,\u00a0 in the regular lecture room DMC 258.<\/span><\/h3>\n<p><strong>Note:<\/strong> You might need a PDF Scanner, to submit your (mostly likely handwritten) work to Gradescope<br \/>\nin PDF format (one PDF per assignment or exam). There are many free phone apps that you can use<br \/>\nto scan your work to make a PDF, for instance Adobe Scan.<\/p>\n<p><strong>Homework, Quizzes, etc.:<\/strong> There will be 10 weekly quizzes (administered by the Teaching Assistant), 11 homeworks (due\u00a0 on Thursdays), and two computer projects (due Friday, March 13 and Friday, April 24). You should understand every solution to every homework problem and be ready to reproduce most of those solutions on your own and in reasonable time. You are welcome to use any help whatsoever with the homework problems and the projects, but not with the quizzes. <em><strong>For best results, do not copy the solutions you did not produce yourself; those include my notes from the top of the page, anything by your fellow students, and whatever an AI system might generate.\u00a0<\/strong><\/em><\/p>\n<p><strong>Extra note on quizzes.<\/strong> Because we have only one discussion per week, we will develop a creative\u00a0 approach to quizzes. Tentatively, the quizzes will be during weeks 2,3,4,5,6,7,8,11, 12,13. The details will be worked out by the TA and adjusted as necessary.<\/p>\n<p><strong> Note on Use of AI<\/strong><br \/>\nYou are welcome to use\u00a0 artificial intelligence (AI) powered programs as a help with homework problems and computer projects, but <em>not with quizzes and exams<\/em>. While AI tools can help you brainstorm ideas or revise work you have already written, AI text generation tools may present incorrect information, biased responses, and incomplete analyses. To adhere to our university values, you must cite any AI-generated material (such as text or images) included or referenced in your work and provide the prompts used to generate the content.\u00a0 I will not be using any AI tools in grading your work.<\/p>\n<p><strong>Grading:<\/strong><\/p>\n<ul>\n<li>Quizzes 15% total<\/li>\n<li>Homeworks, 15% total<\/li>\n<li>The projects, 10% total [5% each]<\/li>\n<li>Two Mid-Term Exams, 30% total [15% each]<\/li>\n<li>Final Two-Hour Exam, 30%<\/li>\n<\/ul>\n<p><strong> Approximate Grading Scheme. <\/strong>A: 90 and up; B: 80-89; C: 70-79. Pluses\/minuses (As in A-, B+, etc.) will mostly be decided on a case-by-case basis.<\/p>\n<p><strong> Missed work.<\/strong> The general rule: no make-up exams or quizzes, and no late submissions of homeworks or the project (but early submissions are welcome). <strong> Emergencies will be handled on a case-by-case basis. <\/strong>If you miss the final exam, with a valid excuse, you get an incomplete in the class; an incomplete is a major inconvenience for a number of people, including yourself, so, please, do not miss the final.<\/p>\n<p><strong> To encourage and reward consistent performance throughout the semester, I will not automatically drop any scores (such as the two lowest quizzes, etc.) <\/strong><\/p>\n<p><strong> Students Requiring Special Accommodation <\/strong><br \/>\nAny student requesting academic accommodations based on special needs is required to register with Office Of Student Accessibility Services (OSAS) each semester. A letter of verification for approved accommodations can be obtained from OSAS. Please be sure the letter is delivered to me (or to TA) as early in the semester as possible. OSAS is located in GFS 120. To contact OSAS: (213) 740-0776 [tel.], OSASFrontDesk@usc.edu [e-mail], <a href=\"https:\/\/osas.usc.edu\/\" target=\"_blank\" rel=\"noopener\">on the web.<\/a><\/p>\n<p><strong> Academic Integrity <\/strong><br \/>\nUSC seeks to maintain an optimal learning environment. General principles of academic honesty include the concept of respect for the intellectual property of others, the expectation that individual work will be submitted unless otherwise allowed by an instructor, and the obligations both to protect one&#8217;s own academic work from misuse by others as well as to avoid using another&#8217;s work as one&#8217;s own. All students are expected to understand and abide by these principles. Scampus (the Student Guidebook) contains the Student Conduct Code in Section 11.00, while the recommended sanctions are in Appendix A.<\/p>\n<p><strong data-olk-copy-source=\"MessageBody\">FINAL EXAMINATION POLICIES<\/strong><\/p>\n<p><strong>Student Scheduling Conflicts <\/strong><br \/>\nNo student is permitted to omit or take the final examination before the rest of the class and no instructor is authorized to permit a student to do so.<\/p>\n<p>Students should plan in advance to avoid scheduling conflicts in their final examinations. If a student is scheduled for two final examinations at the same time, the student should request to take one of the examinations on a different day or time. If a student is scheduled for more than two final examinations in one calendar day, the student may request to take one of the exams on a different day or time. In either situation, the student must contact the professors involved no later than two weeks prior to the scheduled examination date and request an accommodation. If an accommodation cannot be arranged, the student should contact USC Testing Services at <a title=\"http:\/\/contact2.esd.usc.edu\/ct.asp?id=9F382635A9E957E03BF9777033EB27072E104C870E9633A236244B67002B8CD14220FC4157E1FD0C44D25804CC7E36B5310A5C5AE71B6475917079ECAA59F48B&amp;ct=4aeUsxYAAABCWmgzMUFZJlNZGM4mCAAABh2AAAEAEEAALqeOACAAMUwAE0KPUZNMGpyWinW7gUdALLEn4u5IpwoSAxnEwQA%3d\" href=\"http:\/\/contact2.esd.usc.edu\/ct.asp?id=9F382635A9E957E03BF9777033EB27072E104C870E9633A236244B67002B8CD14220FC4157E1FD0C44D25804CC7E36B5310A5C5AE71B6475917079ECAA59F48B&amp;ct=4aeUsxYAAABCWmgzMUFZJlNZGM4mCAAABh2AAAEAEEAALqeOACAAMUwAE0KPUZNMGpyWinW7gUdALLEn4u5IpwoSAxnEwQA%3d\" target=\"_blank\" rel=\"noopener noreferrer\" data-auth=\"NotApplicable\" data-linkindex=\"10\">testing@usc.edu<\/a> or <a title=\"http:\/\/contact2.esd.usc.edu\/ct.asp?id=B455F5D5C26A9A477082088AA8408B20F30688DED2D128ACB99F81EB3785782A748A8E0C5603BDF413D5E25F80D21E5DC3FC35037D9C5F93D5AB2381B46C703B&amp;ct=4aeUsxAAAABCWmgzMUFZJlNZkcKGPgAAB5mAAAh9kAIEBAAgACIaA0BA0DQNNJpADZ3unxdyRThQkJHChj4%3d\" href=\"http:\/\/contact2.esd.usc.edu\/ct.asp?id=B455F5D5C26A9A477082088AA8408B20F30688DED2D128ACB99F81EB3785782A748A8E0C5603BDF413D5E25F80D21E5DC3FC35037D9C5F93D5AB2381B46C703B&amp;ct=4aeUsxAAAABCWmgzMUFZJlNZkcKGPgAAB5mAAAh9kAIEBAAgACIaA0BA0DQNNJpADZ3unxdyRThQkJHChj4%3d\" target=\"_blank\" rel=\"noopener noreferrer\" data-auth=\"NotApplicable\" data-linkindex=\"11\">(213) 740-7166<\/a> for assistance.<\/p>\n<p>Ggrades are due 96 hours after the university-scheduled final examination day and time. Therefore, it might not be possible to accommodate late student requests for an alternate, makeup final examination after the published examination period.<\/p>\n<p><strong> Academic Support <\/strong> <a href=\"https:\/\/kortschakcenter.usc.edu\/\" target=\"_blank\" rel=\"noopener\"> The Kortschak Center for Learning and Creativity <\/a><\/p>\n<p><strong> Supplementary materials <\/strong><\/p>\n<p><strong>Sample exams<\/strong><\/p>\n<p><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Math445-MT1.pdf\" target=\"_blank\" rel=\"noopener\">Earlier MT1s<\/a>\u00a0 \u00a0\u00a0 <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/M445-MT1-sol.png\" target=\"_blank\" rel=\"noopener\">MT1-S2022<\/a> \u00a0<a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2024\/12\/Math445-F2022-MT1.pdf\" target=\"_blank\" rel=\"noopener\"> MT1-F2022<\/a> \u00a0<a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2024\/10\/Math445-F2024-MT1-Sol.pdf\" target=\"_blank\" rel=\"noopener\">MT1-F2024<\/a> <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2026\/03\/Math445-S2026-MT1-sol.pdf\" target=\"_blank\" rel=\"noopener\">MT1-S2026<\/a><\/p>\n<p><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Math445-MT2.pdf\" target=\"_blank\" rel=\"noopener\">Earlier MT2s<\/a>\u00a0 \u00a0 <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Math445-S2022-MT2-sol.pdf\" target=\"_blank\" rel=\"noopener\">MT2-S2022<\/a> \u00a0\u00a0<a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Math445-F2022-MT2-Prep.pdf\" target=\"_blank\" rel=\"noopener\">MT2-F2022<\/a>\u00a0 <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2024\/12\/Math445-F2024-MT2-Sol.pdf\" target=\"_blank\" rel=\"noopener\">MT2-F2024<\/a><\/p>\n<p><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Math445-Final.pdf\" target=\"_blank\" rel=\"noopener\">Earlier Finals<\/a> \u00a0\u00a0 <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Math445-S2022-Final-sol.pdf\" target=\"_self\">Final-S2022<\/a>\u00a0 <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Math445-F2022-Final-Answ.pdf\" target=\"_blank\" rel=\"noopener\">Final-F2022<\/a> <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2024\/12\/Math445-F2024-Final-sol.pdf\" target=\"_blank\" rel=\"noopener\">Final-F2024<\/a><\/p>\n<p><strong> Other materials <\/strong><\/p>\n<p><strong> My notes <\/strong><\/p>\n<ul>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2024\/08\/GeneralIntro-Ugrad-CalcIIIAndUp-Beamer.pdf\" target=\"_blank\" rel=\"noopener\">Introduction<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Vectors.png\" target=\"_blank\" rel=\"noopener\">Vectors<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/VectorCalc.pdf\" target=\"_blank\" rel=\"noopener\">Some names and faces behind some formulas<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/CentralField.pdf\" target=\"_blank\" rel=\"noopener\">Motion in the central field<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/CurvLinCoords-1.pdf\" target=\"_blank\" rel=\"noopener\"> Grad, Div, Laplacian, and Curl in non-cartesian coordinates <\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Fluids.pdf\" target=\"_blank\" rel=\"noopener\">PDEs describing fluids<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/FTEtAl-Examples.pdf\" target=\"_blank\" rel=\"noopener\">Fourier and Laplace Transforms<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Fourier-Applications.pdf\" target=\"_blank\" rel=\"noopener\">Applications of Fourier analysis<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/PDEsOnR.pdf\" target=\"_blank\" rel=\"noopener\">PDEs (Transport, Heat, Wave, Laplace) in the whole space<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/BoundedDomain.pdf\" target=\"_blank\" rel=\"noopener\">PDEs in a bounded domain<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/TelegraphEq.pdf\" target=\"_blank\" rel=\"noopener\">Telegraph equation<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Inhomogenous.pdf\" target=\"_blank\" rel=\"noopener\">Inhomogeneous equations and inhomogeneous boundary conditions<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/SolvingPDEs.pdf\" target=\"_blank\" rel=\"noopener\">Solving PDEs: Separation of variables and the method of characteristics <\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2026\/01\/SturmLiouville.pdf\" target=\"_blank\" rel=\"noopener\">Sturm-Liouville theory<\/a><\/li>\n<li>On Music<br \/>\n<a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/MusicMath-1.pdf\" target=\"_blank\" rel=\"noopener\">Basic<\/a><br \/>\n<a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/MathSoundsOfMusic.png\" target=\"_blank\" rel=\"noopener\">PDEs: a summary <\/a><br \/>\n<a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/PDEsAndMusic.pdf\" target=\"_blank\" rel=\"noopener\">PDEs: details<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/WeierApprTh-2.pdf\" target=\"_blank\" rel=\"noopener\">The Weierstrass Approximation Theorem<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/QuantumMechanics.pdf\" target=\"_blank\" rel=\"noopener\">Elementary Quantum Mechanics<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/08\/GammaBetaFunctions.pdf\" target=\"_blank\" rel=\"noopener\">Gamma and Beta functions<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/LadderTheorem.png\" target=\"_blank\" rel=\"noopener\">The ladder theorem and beyond<\/a><\/li>\n<\/ul>\n<p><strong> Other Notes and Illustrations<br \/>\n<\/strong><\/p>\n<ul>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/TumblingBox.pdf\" target=\"_blank\" rel=\"noopener\"> An article about the mathematics of the tumbling box (and also a tennis racquet) <\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Rotation1.gif\" target=\"_blank\" rel=\"noopener\">Rotation in space: a .gif video<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/DoubleIntegralFourWays.png\" target=\"_blank\" rel=\"noopener\">A double integral four different ways.<\/a> \u00a0 <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Wisdom1.jpg\" target=\"_blank\" rel=\"noopener\">Why?<\/a><\/li>\n<li>The &#8220;complex origins&#8221; of the HP corporation\n<ul>\n<li>The first product: <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2024\/08\/Hp200a-front-panel.jpg\" target=\"_blank\" rel=\"noopener\">front<\/a> and <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2024\/08\/Hp200a-insides-crop.jpg\" target=\"_blank\" rel=\"noopener\">inside<\/a><\/li>\n<li>The <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2024\/08\/HPOscillator.png\" target=\"_blank\" rel=\"noopener\">patent<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2024\/08\/12AE10CPentodeByGE.jpg\" target=\"_blank\" rel=\"noopener\">Pentode<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2024\/08\/Complex-origins-of-HP-Company.png\" target=\"_blank\" rel=\"noopener\">Simplified analysis<\/a>, following Section 5.6 of the book &#8220;An Imaginary Tale&#8221; by Paul Nahin (Princeton University Press, 1998)<\/li>\n<\/ul>\n<\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/hanging-chain.pdf\" target=\"_blank\" rel=\"noopener\">Oscillations of a heavy chain: the equation<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Musical-string-inharmonicity-Chris-Murray.pdf\" target=\"_blank\" rel=\"noopener\">Advanced music theory<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/GreekLetters-3.pdf\" target=\"_blank\" rel=\"noopener\">How to write Greek letters <\/a> (by Olga Korosteleva, CSULB)<\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Compass_and_Straightedge_Construction.pdf\" target=\"_blank\" rel=\"noopener\">Basic exercises on ruler-compass construction <\/a>(by Olga Korosteleva, CSULB)<\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Sampling-AMSNotices-1.pdf\" target=\"_blank\" rel=\"noopener\"> An article about the sampling theorem <\/a><\/li>\n<li>Laplacian in higher dimensions: <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2025\/08\/LaplEigf-2D.pdf\" target=\"_blank\" rel=\"noopener\">lecture notes<\/a>, a <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2025\/08\/Laplace-Beltrami-1.pdf\" target=\"_blank\" rel=\"noopener\">research paper<\/a>, and some <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2025\/08\/DirichletLapl-disk.pdf\" target=\"_blank\" rel=\"noopener\">numerical examples<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2026\/01\/SturmLiouville-1.pdf\" target=\"_blank\" rel=\"noopener\">Notes on Sturm-Liouville theory<\/a><\/li>\n<li>The book <strong> Fourier Analysis <\/strong> by T. Korner (Cambridge University Press, 1988) is a great reference for many topics in this class. Here are some examples from the book:<br \/>\n<a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/NowhereDiffFunction.pdf\" target=\"_blank\" rel=\"noopener\"> A nowhere differentiable function <\/a><br \/>\n<a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/GibbsPhenom.pdf\" target=\"_blank\" rel=\"noopener\"> Analysis of the Gibbs phenomenon <\/a><br \/>\n<a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/NonUniqueness-HeatEq.pdf\" target=\"_blank\" rel=\"noopener\"> Non-uniqueness for the heat equation <\/a><br \/>\n<a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/HeatEq-HalfLine.pdf\" target=\"_blank\" rel=\"noopener\">Heat equation on the half-line<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2025\/12\/Ostrogradsky-s-Instability-ThirdOrder.pdf\" target=\"_blank\" rel=\"noopener\">Why no higher-order equations in physics<\/a>?<\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2025\/12\/SpaceTimeDimensions.pdf\" target=\"_blank\" rel=\"noopener\">Why is space 3d and time 1d<\/a>?<\/li>\n<\/ul>\n<p><strong>More<\/strong><\/p>\n<ul>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/OnHandwriting-2.pdf\" target=\"_blank\" rel=\"noopener\">On handwriting<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Motivation-1.pdf\" target=\"_blank\" rel=\"noopener\">What makes an expert<\/a>?<\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/OptionalMaterial-2.jpg\" target=\"_blank\" rel=\"noopener\">No optional material<\/a><\/li>\n<li>Real-life math: <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Math-for-dummies.jpg\" target=\"_blank\" rel=\"noopener\">addition<\/a>, <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/RoundOffError.jpg\" target=\"_blank\" rel=\"noopener\">construction<\/a>, <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/MathInRealLife.jpg\" target=\"_blank\" rel=\"noopener\">cartoonSB<\/a>, <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/Math.jpg\" target=\"_blank\" rel=\"noopener\">work<\/a>, <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/MathJoke1.jpg\" target=\"_blank\" rel=\"noopener\">treasure hunt<\/a>, <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/MathJoke2.jpg\" target=\"_blank\" rel=\"noopener\">cartoonS<\/a>, <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/MoreAlgebra.jpg\" target=\"_blank\" rel=\"noopener\">algebra<\/a>, <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/ElectricalEng.jpg\" target=\"_blank\" rel=\"noopener\">electricity<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2024\/07\/AllSchrodinger.pdf\" target=\"_blank\" rel=\"noopener\">About Schrodinger and his cat<\/a><\/li>\n<li>(Non-quantum) Uncertainty Principle in real life: <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/SchrodingerPlates.jpg\" target=\"_blank\" rel=\"noopener\">Plates<\/a>\u00a0 <a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/UncertainlySign.png\" target=\"_blank\" rel=\"noopener\">Directions<\/a><\/li>\n<li><a href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/wp-content\/uploads\/sites\/211\/2023\/06\/PrimeNumbersFormula-1.png\" target=\"_blank\" rel=\"noopener\">When a closed-form explicit formula does not help that much!<\/a><\/li>\n<\/ul>\n<hr \/>\n<p align=\"center\">USC Math Department <a href=\"https:\/\/dornsife.usc.edu\/mathematics\/\" target=\"_blank\" rel=\"noopener\">Homepage<\/a><\/p>\n<hr \/>\n\n\n\n<\/div>\n\n\n  <\/div><\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":282,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":""},"class_list":["post-968","page","type-page","status-publish","hentry"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.1.1 - 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