### Contents

### Course Description

Math 210 is the third and the final part of our standard three-semester calculus sequence. The distinct feature of this part of the course is its focus on the multi-dimensional analysis, as opposed to one-dimensional analysis that you learned in Math 180 (Calculus I) and Math 181 (Calculus II). This semester you will get familiar with such important concepts as a vector, a vector field, a function of several variables, partial derivative, a line-integral and multi-variable integrals. You will see that these concepts, as scary as they may sound, are actually a natural generalization of the things you already know from calc I and II. This is how the tree of mathematics is built - going from simple to more complicated. The ideas of the vector calculus apply to numerous areas of human knowledge such as engineering, physics, pure mathematics, biology, and many others. Some of them we will see in the course, some will surface later in your future special courses, yet some may wait until you become a professional.

Students enter Math 210 from a variety of backgrounds: many of you have taken Calculus I and II at UIC, some have transferred from other schools, or placed directly into Calculus III following your calculus studies in other schools. Regardless of your background coming in, our goal is to provide instructorship and all the resources necessary for every one of you succeed, and enjoy yourselves as much as possible in the process! In spite of this, you may find vector calculus very challenging. Like in Math 180 and 181 your success in Math 210 requires a lot of hard work, hours of study and problem solving, and your active involvement in learning, both in and outside of the classroom. Our course is designed with the aim of helping you stay constantly connected with the course and the material, and within easy reach of some of your best resources: your instructor, your teaching assistants, and your colleagues!

### Textbook

Calculus, Early Transcendentals, by W. Briggs and L. Cochran, second edition (the bundle version, Chapters 1-14) OR Multivariable Calculus, by W. Briggs and L. Cochran (Chapters 8-14). We will only go through Chapters 11-14. This textbook has been in our use since 2011. Your instructor is not required to follow the text line-by-line or to use the same problems, so please take notes in class and use them as your primary source.

### Course Structure

The class involves three hours of lectures on MWF, and one hour on T or Th of problem solving session. Please see your class schedule for specific time and classroom. In addition, your instructor and TA will be available during their office hours. They can be found in Sections.

### Prerequisites

Grade of C or better in MATH 181. The prerequisite is enforced throughout all sections of the course without exceptions. Students that have not met the prerequisite will not be allowed to take the course.

### Syllabus

WEEK | CONTENT | TOPICS |
---|---|---|

1 | 11.1, 11.2 | Discussion of course policies; proficiency exam on Wednesday; vectors on plane; vectors in space*. |

2 | Labor Day, 11.2, 11.3, 11.4 | Distance, sphere, dot product, work of force, cross product, torque. |

3 | 11.5, 11.6, 11.7 | Vector-valued functions; calculus operations on them*; parametric equation of a line; curves; physical concepts of motion (velocity, acceleration, speed) using vector calculus. |

4 | 11.7, 11.8, 11.9 | Motion in a gravitational field; motion in space*; arc length, arc length parametrization*; definition of curvature. |

5 | 11.9; review; 1st midterm on 11.1 - 11.9; 12.1 | Alternative curvature formulae; principal normal; components of acceleration* on Monday; Review on Wednesday; 1st midterm on Thursday; Equation of a plane, cylinders on Friday. |

6 | 12.1, 12.2, 12.3 | Quadratic surfaces; functions of 2 variables, graphs, level curves; functions of 3 variables*; calculus of multivariable functions, limits, two-path test. |

7 | 12.4, 12.5, 12.6 | Partial first and higher order derivatives, Clairaut Theorem, differentiability; the Chain Rule; implicit differentiation; concept of gradient, directional derivative; applications. |

8 | 12.7, 12.8 | Tangent plane; linear approximation, concept of the differential; Local extrema, critical points, 2nd derivative test; Absolute extrema -- strategies on closed sets, opens sets and unbounded regions. |

9 | 12.9; review; 2nd midterm on 12.1 - 12.9; 13.1 | The method of Lagrange Multipliers, application in economics, optimization problems, finding extreme distances; Double integral over rectangles, iterated integrals, integral as a volume. |

10 | 13.1; 13.2; 13.3 | Integrals over non-rec standard regions, changing the order of integration, volumes of regions between 2 surfaces, area of a plane region using double integrals; Integrals in polar coordinates, polar rectangles, polar non-rec standard regions; averages*. |

11 | 13.4, 13.5, 13.6* | Triple integration over boxes, regions between surfaces; computation of mass and volume using triple integral, averages*; Cylindrical and spherical coordinates, changing of variables, (lots) of examples; Center of mass formulae*. |

12 | 13.7, 14.1, 14.2 | Plane transformations, Jacobian, general change of variable formula; Linear planar changes, changes determined by integrand, by region; Vector fields, radial, gradient, potential; Line integrals of scalar functions. |

13 | 14.2, 14.3; 14.4 | Integrals of fields, lots of physical interpretations - circulation, flux, work of force; Conservative fields, finding potentials, independence of path, FTC for those fields; Green's Theorem in circulation and flux forms, finding areas using GT, explanation of why GT works on rectangles. |

14 | 14.5, 14.6 | Div and Curl in 3D, and as a generalization of 2D wrt GT; Surface integrals of scalar functions, surface area element; shortcut formulae for spherical, cylindrical, and graph cases; Flux of a vector field through a surface, physical picture. |

15 | 14.7*, 14.8*, Review for final | Stokes' and Divergence Theorems explained as 3D analogues to 2D Green's Theorems in circulation and flux forms, respectively; Review for the final exam |

16 | Final Exam | Cumulative Final is given on the date to be announced. |

A topic marked by * may be covered briefly for one or more of the following reasons: it is similar to another one covered previously; it is of less importance for future development of the course material; it is relatively simple and may be given as a reading assignment; it is too advanced at the first reading. Please follow instructions in your class pertaining to these topics.

### COURSE POLICIES

### Diagnostic Test

Given the variety of students taking the course, it is important to ensure that every one of you has the necessary mathematical background which allows you to fully focus on the wealth of new material which you must learn in Math 210. That is why your instructor will administer a 50-minute diagnostic test on Wednesday of the first week of classes. This test will consist of up to ten problems based on topics from basic algebra to the material of Math 180 and Math 181 which is needed for Math 210. It will be graded based on a simple Satisfactory/Unsatisfactory system. The results of the diagnostic test will not effect or in any way be counted towards your final grade for the course. The grade of U means that your current skills may not be sufficient to continue in Math 210 without substantial difficulties and danger to fail the course, unless you take steps to improve. Since the test is intended for your own evaluation, you are not required to do anything in case of a U grade. However, you are encouraged to talk to your instructor/advisor to discuss possible options. Those may include (re)taking Math 181, enrolling in the 4 week review session that runs weeks 2 through 5, enrolling in additional ESP-sections, using Math Learning Center, seeking tutoring help, etc. All these options are subject to availability.

### Attendance

As explained in the course description, your active involvement in learning is essential in order to successfully complete the course! A basic requirement of the course is therefore a serious commitment on your part to attend both the lectures and the problem sections. Attendance in the course will be taken as follows.

In lectures: Attendance will be taken at the beginning of randomly chosen lectures, by means of an attendance sheet listing all the students registered in the class. The sheet will be circulated in the classroom, and every student present will be required to sign the rubric corresponding to her or his name. The attendance sheet will be returned to the instructor 15 minutes after the beginning of class. In addition, a minimum of 14 short quizzes will be given during the semester. The quizzes will be unannounced, and given at the end of a lecture on randomly chosen days. On the day of a quiz, submitted quiz sheets will be used for attendance check. Submission of a quiz sheet on behalf of another student or signing the rubric under the name of another student on any attendance sheet will be considered a serious violation of course policies. On a first such occurrence, the student will be called to meet with the instructor, and a warning will be issued, while a second such occurrence will lead to a grade of F for the course. It is mandatory to attend at least 80% of lectures. Failure to do so without official excuse will lead to a grade of F for the course.

In problem sessions: The TAs will take attendance in each problem session. 80% attendance is mandatory, and failure to achieve this without official excuse will lead to a grade of F for the course.

Excused Absence Policy: In order to be excused from attendance, students must inform the instructor and/or TA (as appropriate) in advance (except in cases of emergency), and must provide documentation (for example, a letter from a doctor).

### Methods of evaluation and grading policies

Your final grade in Math 181 will be determined by the number of points you earn on the following scale:

Points | Grade |
---|---|

83 - 100 | A |

65 - 82 | B |

50 - 64 | C |

35 - 49 | D |

0 - 34 | F |

There will be no curve for the final grade. You can earn points as follows:

Up to 20 | Midterm 1 |

Up to 20 | Midterm 2 |

Up to 30 | Final exam |

Up to 20 | Homework |

Up to 5 | Quizzes |

Up to 5 | Attendance and participation in problem sessions |

Midterm grades: Although it is not MSCS policy to assign midterm grades to 200-level courses we will do our best to ensure that you receive a feedback of your performance before October 22nd. The midterm grades will follow the same cut-offs as for the final course grades, but with the following contributions:

40% | Homework |

10% | Quizzes (lowest 10% of quiz grades, or at least 1, dropped) |

10% | Attendance and participation in problem sessions |

40% | Midterm 1 |

Tips on interpreting your midterm grade can be found at http://tigger.uic.edu/depts/oaa/advising/student_midterm.html

### Quizzes, homework, exams

Quizzes: The quizzes will be given during your regular lecture time on randomly chosen days. They will typically consist of one or two questions bases on recent material with the purpose of keeping you involved and active in the lectures and letting you know if you are following the concepts. Grading scheme of a quiz is based on 0, 1, 2 points for each problem. It will be graded by the instructor, and returned in lecture or your problem session. There will be no make-up quizzes given, but only the highest 80% of quiz grades will be considered when computing the points corresponding to the quizzes on the final grade. Remember that quizzes will also be used for your attendance check.

Homework: Homework for the course is assigned every week by the course coordinator, and is the same for all sections. Assignments will be posted on Blackboard, on this page under Homework link, and distributed by instructors via e-mail. You are very strongly encouraged to work together with a group of colleagues on the homework problems, but you must write up the solutions by yourself! The homework will be due on the days announced by your instructor. One or two problems chosen at random from each homework will be graded by the TAs. It is very important to note that the solutions to the problems will be graded in full, and just an answer will not earn any credit. You should pay a lot of attention to the comments made by your TA in each graded homework, since the midterms and the final exam will be graded in a very similar way. Shortly after the due date solution keys will be posted as well. In addition to the mandatory assignments, optional sets of problems can be found under Homework link. The same list will be available for use in you MyMathLab account. These problems are designed to build your basic problem solving skills and solidify understanding of the core course material. We recommend that you do them before taking on the more involved required problems. Late homework can be submitted only with a written excuse document, for example a note from doctor, and no homework will be accepted more than 2 days after the deadline. One worst homework score will be dropped at the end of the semester.

Exams: Two midterms will be given on **Thursdays of weeks 5 and 9** of the semester, and one final on the week following
the last week of classes. Midterm 1 will include Sections 11.1 - 11.9, Midterm 2 will include 12.1 - 12.9. The final exam is cumulative
and includes material from the entire course. Updates on time schedules, room assignments and preparation materials can be found here.
Make-ups can be given to students that comply with the Excused Absence Policy above for the day of the exam. Schedule of make-ups will be announced
at least one week before the corresponding exam.

### Calculators

The use of any electronic devices with computing capabilities is prohibited during exams and quizzes.

### Academic Integrity Policy

As an academic community, UIC is committed to providing an environment in which research, learning, and scholarship can flourish and in which all endeavors are guided by academic and professional integrity. All members of the campus community - students, staff, faculty, and administrators - share the responsibility of insuring that these standards are upheld so that such an environment exists. Instances of academic misconduct by students will be handled pursuant to the Student Disciplinary Policy: http://www.uic.edu/depts/dos/docs/Student%20Disciplinary%20Policy.pdf

### Academic Deadlines

Current academic calendar and the list of deadlines can be found here.

### Disability Policy

The University of Illinois at Chicago is committed to maintaining a barrier-free environment so that students with disabilities can fully access programs, courses, services, and activities at UIC. Students with disabilities who require accommodations for access to and/or participation in this course are welcome, but must be registered with the Disability Resource Center (DRC). You may contact DRC at 312-413-2183 (v) or 312-413-0123 (TTY) and consult the following: http://www.uic.edu/depts/oaa/disability_resources/faq/accommodations.html.

### Religious Holidays

Students who wish to observe their religious holidays shall notify the faculty member by the tenth day of the semester of the date when they will be absent unless the religious holiday is observed on or before the tenth day of the semester. In such cases, the student shall notify the faculty member at least five days in advance of the date when he/she will be absent. The faculty member shall make every reasonable effort to honor the request, not penalize the student for missing the class, and if an examination or project is due during the absence, give the student an exam or assignment equivalent to the one completed by those students in attendance. If the student feels aggrieved, he/she may request remedy through the campus grievance procedure. http://www.uic.edu/depts/oae/docs/ReligiousHolidaysFY20122014.pdf

### Grievance Procedures

UIC is committed to the most fundamental principles of academic freedom, equality of opportunity, and human dignity involving students and employees. Freedom from discrimination is a foundation for all decision making at UIC. Students are encouraged to study the University's "Nondiscrimination Statement". Students are also urged to read the document "Public Formal Grievance Procedures". Information on these policies and procedures is available on the University web pages of the Office of Access and Equity: www.uic.edu/depts/oae.