Difference between revisions of "Math 251-102 Fall 2022"

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=Course info=
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'''Description.''' Vectors and three-dimensional analytic geometry; curves and surfaces; functions of several variables, calculus of several variables; partial derivatives; gradient and linearization; multiple integrals; volume and surface area.
  
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'''Thursdays at 5pm EDT'''<br />
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'''Instructor.''' Casian Pantea https://math.wvu.edu/∼capantea/
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[https://wvu.zoom.us/j/96317244165?pwd=YVJGWHdxb2lGUHlkZEdRT1NPRFFsUT09 zoom link] (password ''euclid2022'')
 
  
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'''Schedule.''' Mondays, Wednesdays, Fridays 9:50-11AM, Hodges 338
  
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The 2022 WVU Junior Math Club is a mathematical enrichment program open to middle school and high school students in West Virginia and Southeast Pennsylvania. We meet on Zoom every Tuesday in Spring 2022 and we discuss various mathematical topics with special emphasis on competition-type problems. The club is supported by an [https://www.maa.org/programs-and-communities/outreach-initiatives/dolciani-mathematics-enrichment-grants MAA Dolciani grant] and the West Virginia University [https://mathanddata.wvu.edu School of Mathematical and Data Sciences]. If you would like to participate, or for additional info please contact Casian Pantea at [mailto:cpantea@math.wvu.edu cpantea@math.wvu.edu].  
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'''Office hours.''' Mondays 11:10AM-12:10PM and 5PM-6PM, Wednesdays 2:30-3:30 PM in Armstrong Hall 305B; and by appointment on zoom: https://wvu.zoom.us/j/92855687691
  
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= Schedule (updated as we go)=
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'''Textbook.''' Calculus: Early Transcendentals, 8th Edition, by James Stewart; we cover Chapters 12-16.
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{| cellpadding="8"
 
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!align="left" | date 
 
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!align="left" | topic
 
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!align="left" | instructor
 
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!align="left" | materials
 
  
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'''Prerequisite.''' C or better in Math 156 is required;
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=Evaluation=
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'''Grading scheme'''
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*30% Final Exam
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*2x20% Midterms
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*20% Quizzes
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*10% Homework
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Letter grades will be assigned as follows
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{|
 
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|-
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| February 3
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|'''A'''
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| Geometry 1
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|>=90%
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| Pantea
 
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|[[#February 3  | Congruences, similarity and other interesting triangle-related stuff]]
 
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|
 
 
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| February 7
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|'''B'''
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| Algebra 1
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|80-90%
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| Voitiuk
 
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|[[#February 10  | Algebraic inequalities]]
 
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|
 
 
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| February 3
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|'''C'''
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| Combinatorics 1
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|70-80%
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| Goldwasser
 
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|[[#February 17  | Congruences, similarity and other interesting triangle-related stuff]]
 
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|
 
 
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|-
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| February 24
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|'''D'''
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| Combinatorics 1
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|60-70%
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| TBD
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|-
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|[[#February 24  | Counting sets, Pascal triangle, and other combinatorial identities]]
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|'''F'''
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|
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|<60%
 
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|}
  
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=Abstracts=
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=Schedule of topics=
  
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===Veronica Ciocanel===
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{| class="wikitable" style="margin:auto"
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Actin filaments are polymers that interact with myosin motor
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! Date !! Topic !! Section !! Other
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proteins and play important roles in cell motility, shape, and
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development. Depending on its function, this dynamic network of
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| Wed Aug 17 || 3D coordinate system, vectors || 12.1 ||
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interacting proteins reshapes and organizes in a variety of structures,
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including bundles, clusters, and contractile rings. Motivated by
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| Fri Aug 19 || Vectors || 12.2 ||
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observations from the reproductive system of the roundworm C. elegans,
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we use an agent-based modeling framework to simulate interactions
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| Mon Aug 22 || Dot product || 12.3 ||
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between actin filaments and myosin motor proteins inside cells. We also
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develop techniques based on topological data analysis to understand
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| Wed Aug 24 || Cross product || 12.4 ||
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time-series data extracted from these filament network interactions.
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These measures allow us to compare the filament organization resulting
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| Fri Aug 26 || Lines and planes || 12.5 ||
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from myosin motors with different properties. Recently, we have also
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studied how different models of myosin regulation predict actin network
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| Mon Aug 29 || Problem session || 12.1-12.5 ||
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organization during the cell cycle. This work also raises questions
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about how to assess the significance of features in common topological
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| Wed Aug 31 || Quadric surfaces || 12.6 ||
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summaries.
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| Fri Sep 2 ||Quadric surfaces || 12.6 || [[Media:25122Q1Soln.png|'''Quiz 1''']]
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===Radu Precup===
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| Wed Sep 7 || Review || 12.1-12.6 ||
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Harnack type inequalities and localization of solutions
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| Fri Sep 9 || Vector functions and curves || 13.1 || [[Media:25122HW1.png|'''HW 1 due''']]
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First it is explained the problem of localization of solutions for boundary value problems in general. Next, the focus will be on radial solutions for phi-Laplace equations. Numerical solutions are obtained for some concrete equations.
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| Mon Sep 12 || Derivatives and integrals || 13.2 ||
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===Jue Wang===
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| Wed Sep 14 || Arc length and curvature || 13.3 ||
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Locating Objects of Interest from Screening Ultrasound
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| Fri Sep 16 || Motion in space || 13.4 || [[Media:25122Q2Soln.png|'''Quiz 2''']]
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A fast Enclosure Transform is developed to localize complex objects of interest from medical ultrasound images. This approach explores spatial constraints on regional features from a sparse image feature representation. Unrelated, broken ridge features surrounding an object are organized collaboratively, giving rise to the enclosureness of the object. Three enclosure likelihood measures are constructed, consisting of the enclosure force, potential energy, and encloser count. In the transform domain, the local maxima manifest the locations of interest objects, for which only the intrinsic dimension is known a priori. I will demonstrate two medical applications in detecting (1) suspicious breast masses in screening breast ultrasound, and (2) the location of the prostate on trans-abdominal ultrasound for verification of patient positioning in radiotherapy treatment of prostate cancer.
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| Mon Sep 19 ||  ||  ||
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===Matthew Johnston===
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|-
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| Wed Sep 21 || Problem session  ||  ||
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A Mathematical Model of COVID-19 Spread by Vaccination Status
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| Fri Sep 23 || Functions of multiple variables  || 14.1  ||[[Media:25122HW2.png|'''HW 2 due''']]
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The novel coronavirus SARS-CoV-2, and the corresponding illness COVID-19, has afflicted hundreds of millions of people worldwide, killed at least 5 million, and devastated the world economy. The rapid development of effective and safe vaccines, which were authorized in the United States under Emergency Use Authorization in December 2020, have offered some hope that the end of the pandemic may be sight; however, at the same time we have seen the rise of variants, such as Delta, which are much more transmissible than the original strains. In this talk, I will introduce an n-stage vaccination model and corresponding system of differential equations which can simulate a disease outbreak by breaking the population down according to their vaccination status. This allows the mitigation effects of vaccination and accelerating effects of variants such as Delta to be uncoupled from one another, and offers valuable insight for the future course of the COVID-19 pandemic. I fit the model to 2021 data from the Virginia Department of Health.
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| Mon Sep 26 || Review ||  ||
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===Xiang Xu===
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| Wed Sep 28 ||'''Midterm 1''' ||  ||[[Media:25122MT1Soln.png|'''Midterm 1''']]
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Blowup rate estimates of a singular potential in the Landau-de Gennes theory for liquid crystals
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| Fri Sep 30 || Functions of multiple variables  || 14.1 ||
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The Landau-de Gennes theory is a type of continuum theory that describes nematic liquid crystal configurations in the framework of the Q-tensor order parameter. In the free energy, there is a singular bulk potential which is considered as a natural enforcement of a physical constraint on the eigenvalues of symmetric, traceless Q-tensors. In this talk we shall discuss some analytic properties related to this singular potential. More specifically, we provide precise estimates of both this singular potential and its gradient as the Q-tensor approaches its physical boundary.
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| Mon Oct 3 || Limits and continuity || 14.2 ||
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===Lorand Parajdi===
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| Wed Oct 5 || Partial derivatives || 14.3 ||
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On the controllability of some systems modeling cell dynamics related to leukemia
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| Mon Oct 10 || Tangent planes || 14.4 ||
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In this talk, I will present two control problems for a model of cell dynamics related to leukemia. The first control problem is in connection with classical chemotherapy, which indicates that the evolution of the disease under treatment should follow a prescribed trajectory assuming that the drug works by increasing the cell death rates of both malignant and normal cells. In the second control problem, as for targeted therapies, the drug is assumed to work by decreasing the multiplication rate of leukemic cells only, and the control objective is that the disease state reaches a desired endpoint. The solvability of the two problems as well as their stability is proved by using a general method of analysis. Some numerical simulations are included to illustrate the theoretical results and prove their applicability. The results can possibly be used to design therapeutic scenarios such that an expected clinical evolution can be achieved.
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| Wed Oct 12 || Chain rule || 14.5 || [[Media:25122Q3Soln.png|'''Quiz 3''']]
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===Adrian Tudorascu===
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|-
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| Fri Oct 14 || Directional derivatives and gradients || 14.6 || [[Media:25122HW3.png|'''HW 3 due''']]
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On the convexity condition for the semi-geostrophic system
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| Mon Oct 17 || Directional derivatives and gradients || 14.6 ||
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We show that conservative distributional solutions to the Semi-Geostrophic system in a rigid domain are in some well-defined sense critical points of a time-shifted energy functional involving measure-preserving rearrangements of the absolute density and momentum, which arise as one-parameter flow maps of continuously differentiable, compactly supported divergence free vector fields. We also show directly, with no recourse to Monge- Kantorovich theory, that the convexity requirement on the modified pressure potentials arises naturally if these critical points are local minimizers of said energy functional for any admissible vector field.
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| Wed Oct 19 || Maxima and minima || 14.7 ||
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|-
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| Fri Oct 21 || Lagrange multipliers ||14.8  || [[Media:25122Q4Soln.png|'''Quiz 4''']]
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|-
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| Mon Oct 24 || Problem session || ||
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|-
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| Wed Oct 26 || Double integrals over rectangles || 15.1 ||
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|-
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| Fri Oct 28 || Double integrals over general domains || 15.2 || [[Media:25122HW4.png|'''HW 4 due''']]
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|-
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| Mon Oct 31 || Polar coordinates || 15.3 ||
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|-
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| Wed Nov 2 || Polar coordinates || 15.3 ||
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| Fri Nov 4 || Surface area || 15.5 || [[Media:25122Q5Soln.png|'''Quiz 5''']]
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|-
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| Mon Nov 7 || Review ||  ||
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|-
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| Wed Nov 9 ||'''Midterm 2'''||  ||[[Media:25122MT2Soln_part1.png|'''Midterm 2 (file 1)''']]<br> [[Media:25122MT2Soln_part2.png|'''Midterm 2 (file 2)''']]
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|-
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| Fri Nov 11 || Triple integrals || 15.6 || [[Media:25122HW5.png|'''HW 5 due''']]
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| Mon Nov 14 || Cylindrical coordinates || 15.7 ||
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| Wed Nov 16 || Spherical coordinates || 15.8 ||
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| Fri Nov 18 || Vector fields || 16.1 || [[Media:25122Q6Soln.png|'''Quiz 6''']]
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| Mon Nov 28 || Line integrals || 16.2 ||
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| Wed Nov 30 || Fundamental theorem for line integrals || 16.3 ||
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| Fri Dec 2 || Green's theorem || 16.4 || [[Media:25122HW6.png|'''HW 6 due''']]
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|-
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| Mon Dec 5 || Review ||  ||
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| Wed Dec 7 || Review ||  ||
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| Mon Dec 12 ||'''Final Exam'''  ||  || 8-10pm<br /> Hodges 338
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|}

Latest revision as of 13:24, 10 December 2022

Course info

Description. Vectors and three-dimensional analytic geometry; curves and surfaces; functions of several variables, calculus of several variables; partial derivatives; gradient and linearization; multiple integrals; volume and surface area.

Instructor. Casian Pantea https://math.wvu.edu/∼capantea/

Schedule. Mondays, Wednesdays, Fridays 9:50-11AM, Hodges 338

Office hours. Mondays 11:10AM-12:10PM and 5PM-6PM, Wednesdays 2:30-3:30 PM in Armstrong Hall 305B; and by appointment on zoom: https://wvu.zoom.us/j/92855687691

Textbook. Calculus: Early Transcendentals, 8th Edition, by James Stewart; we cover Chapters 12-16.

Prerequisite. C or better in Math 156 is required;

Evaluation

Grading scheme

  • 30% Final Exam
  • 2x20% Midterms
  • 20% Quizzes
  • 10% Homework

Letter grades will be assigned as follows

A >=90%
B 80-90%
C 70-80%
D 60-70%
F <60%

Schedule of topics

Date Topic Section Other
Wed Aug 17 3D coordinate system, vectors 12.1
Fri Aug 19 Vectors 12.2
Mon Aug 22 Dot product 12.3
Wed Aug 24 Cross product 12.4
Fri Aug 26 Lines and planes 12.5
Mon Aug 29 Problem session 12.1-12.5
Wed Aug 31 Quadric surfaces 12.6
Fri Sep 2 Quadric surfaces 12.6 Quiz 1
Wed Sep 7 Review 12.1-12.6
Fri Sep 9 Vector functions and curves 13.1 HW 1 due
Mon Sep 12 Derivatives and integrals 13.2
Wed Sep 14 Arc length and curvature 13.3
Fri Sep 16 Motion in space 13.4 Quiz 2
Mon Sep 19
Wed Sep 21 Problem session
Fri Sep 23 Functions of multiple variables 14.1 HW 2 due
Mon Sep 26 Review
Wed Sep 28 Midterm 1 Midterm 1
Fri Sep 30 Functions of multiple variables 14.1
Mon Oct 3 Limits and continuity 14.2
Wed Oct 5 Partial derivatives 14.3
Mon Oct 10 Tangent planes 14.4
Wed Oct 12 Chain rule 14.5 Quiz 3
Fri Oct 14 Directional derivatives and gradients 14.6 HW 3 due
Mon Oct 17 Directional derivatives and gradients 14.6
Wed Oct 19 Maxima and minima 14.7
Fri Oct 21 Lagrange multipliers 14.8 Quiz 4
Mon Oct 24 Problem session
Wed Oct 26 Double integrals over rectangles 15.1
Fri Oct 28 Double integrals over general domains 15.2 HW 4 due
Mon Oct 31 Polar coordinates 15.3
Wed Nov 2 Polar coordinates 15.3
Fri Nov 4 Surface area 15.5 Quiz 5
Mon Nov 7 Review
Wed Nov 9 Midterm 2 Midterm 2 (file 1)
Midterm 2 (file 2)
Fri Nov 11 Triple integrals 15.6 HW 5 due
Mon Nov 14 Cylindrical coordinates 15.7
Wed Nov 16 Spherical coordinates 15.8
Fri Nov 18 Vector fields 16.1 Quiz 6
Mon Nov 28 Line integrals 16.2
Wed Nov 30 Fundamental theorem for line integrals 16.3
Fri Dec 2 Green's theorem 16.4 HW 6 due
Mon Dec 5 Review
Wed Dec 7 Review
Mon Dec 12 Final Exam 8-10pm
Hodges 338
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