MAST20009
Vector Calculus

Semester II 2018

Lecturer: Arun Ram, 174 Richard Berry, email: aram@unimelb.edu.au

Time and Location:
Lecture: Tuesday 11:00-12:00, Thursday 13:00-14:00, Friday 12:00-13:00 Physics South Laby Theatre L108
Practice class: One per week, see the timetable.

Pre-Exam consultation hours:

Andrei Ratiu Monday 5 November 10-12 and 1-2pm in Room G14.

Ana Mamatelashvili Thursday 1 November 11:30-1 and 2:15-3:15 in Room G14

Arun Ram, Wednesday 31 October 2-5pm in Room 174.

Ana Mamatelashvili Tuesday 30 October 11:30-1 and 2:15-3:15 in Room G50

See https://ms.unimelb.edu.au/study/mslc/consultation

The student representatives are
Dinglin Bu email: dinglinb@student.unimelb.edu.au and
Stella Ulm email: sulm@student.unimelb.edu.au

Announcements

Prof. Ram reads email but generally does not respond by email.
The lectures will be recorded and made available in the LMS. The lecture recordings are NOT a complete resource and are usually a very incomplete record of the lecture. It is unreasonable to expect that you can do well on the assessment without more comprehensive and thorough utilisation of all resources.
Plagiarism declaration: It is required to complete the plagiarism declaration in LMS, by clicking on the Plagiarism declaration link for this subject and completing the submission of the Plagiarism declaration through LMS for this subject.
Academic Misconduct The start of semester pack includes a statement about Academic Misconduct (pdf file), Further information regarding academic misconduct and plagiarism is available at http://academichonesty.unimelb.edu.au/
Generic Skills statement: In addition to learning specific technical skills that will assist you in your future careers in science, engineering, commerce, education or elsewhere, you will have the opportunity to develop, in this subject, generic skills that will assist you in whatever your future career path:
- You will develop problem-solving skills (especially through tutorial exercises) including engaging with unfamiliar problems, and identifying relevant strategies.
- You will develop analytical skills - the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of the analysis.
- Through tutorials and other interactions with fellow students, you will develop the ability to work in a team. The school distinguishes between ethical collaboration, which is strongly encouraged, and plagiarism, which is prohibited.
Printing arrangements for Peter Hall building Students must use UNICARD to print documents. The UNICARD printer is located near the G70 computer lab. Please note: there is not a money uploader in the Peter Hall Building. For more information about printing at the University and for locations of UNICARD uploaders direct students to Student IT Support: http://studentit.unimelb.edu.au/
Housekeeping: The start of semester pack includes: Housekeeping (pdf file), SSLC responsibilities (pdf file).
Stop1 Quick Reference Guide pdf file contains information on Requisite waivers, Enrolment after week 2, Electives, subject and major selection, study abroad, credit exemptions, academic adjustments, special consideration, exam reviews, Stop1 services, Healthand well being, further information.
it is required to complete the plagiarism declaration in LMS, by clicking on the Plagiarism declaration link for this subject and completing the submission of the Plagiarism declaration through LMS for this subject.
It is University Policy that:
“a further component of assessment, oral, written or practical, may be administered by the examiners in any subject at short notice and before the publication of results. Students must therefore ensure that they are able to be in Melbourne at short notice, at any time before the publication of results” (Source: Student Diary).
Students who make arrangements that make them unavailable for examination or further assessment, as outlined above, are therefore not entitled to an alternative opportunity to present for the assessment concerned (i.e. a ‘make-up’ examination).

Subject Outline

The handbook entry for this course is at https://handbook.unimelb.edu.au/view/2018/MAST20009.

This subject studies the fundamental concepts of functions of several variables and vector calculus. It develops the manipulation of partial derivatives and vector differential operators. The gradient vector is used to obtain constrained extrema of functions of several variables. Line, surface and volume integrals are evaluated and related by various integral theorems. Vector differential operators are also studied using curvilinear coordinates.

Functions of several variables topics include limits, continuity, differentiability, the chain rule, Jacobian, Taylor polynomials and Lagrange multipliers. Vector calculus topics include vector fields, flow lines, curvature, torsion, gradient, divergence, curl and Laplacian. Integrals over paths and surfaces topics include line, surface and volume integrals; change of variables; applications including averages, moments of inertia, centre of mass; Green's theorem, Divergence theorem in the plane, Gauss' divergence theorem, Stokes' theorem; and curvilinear coordinates.

Assessment

There will be one three hour examination at the end of the semester, and four written assignments during semester. For your final mark, the exam counts for 80% and the assignments count for a total of 20% (5% each). Note that each piece of assessment is compulsory.

Assignments

Assignments will be due by 11am on the following dates:

Tuesday, 14 August: Assignment 1
Tuesday, 4 September: Assignment 2
Tuesday, 2 October: Assignment 3
Tuesday, 16 October: Assignment 4

Each assignment is made available in the UniMelb LMS site for this course, two weeks before it is due.

These assignments must be your own work. While students are encouraged to discuss their coursework and problems with one another, assignments must be written up independently. It is University policy that students submit a signed plagiarism sheet at the start of each semester. If you do not submit this sheet your assignments will be given a mark of zero.

it is required to complete the plagiarism declaration in LMS, by clicking on the Plagiarism declaration link for this subject and completing the submission of the Plagiarism declaration through LMS for this subject.

Students who are unable to submit an assignment on time and qualify for special consideration should contact the lecturer as soon as possible after the due date.

Prerequisites

The prerequisites are listed in the handbook entry for this course at https://handbook.unimelb.edu.au/view/2018/MAST20009.

Lecture notes

Lecture notes and the Problem booklet (together in a single shrink wrapped set) will be available for sale at the University Coop. These contain information on lecture schedule, supplementary references, and course procedures.

Lectures this semester

I like to prepare by writing my own handwritten lecture notes (even though, in class, I usually follow the Melbourne University tradition and annotate the course note book on the document projector. I will post my handwritten lecture notes below.

Week 1: Functions of several variables

24 July 2018 Lecture 1: Limits and continuity; Intuitive idea of limits of functions of several variables
handwritten lecture notes (pdf file)
27 July 2018 Lecture 2: Limits and continuity of functions of several variables; handwritten lecture notes (pdf file).
28 July 2018 Lecture 3: Partial differentiation revision; Differentiability of functions of several variables and C^N; handwritten lecture notes (pdf file).

Week 2: Functions of several variables

31 July 2018 Lecture 4: Matrix version of chain rule, Jacobian handwritten lecture notes (pdf file).
2 August 2018 Lecture 5: Taylor polynomials for functions of several variables, extrema handwritten lecture notes (pdf file).
3 August 2018 Lecture 6: Constrained extrema, Lagrange multipliers handwritten lecture notes (pdf file).

Week 3: Space curves and vector fields

7 August 2018 Lecture 7: Functions of several variables: Lagrange multipliers; handwritten lecture notes (pdf file).
9 August 2018 Lecture 8: Parametric paths, velocity, acceleration, differentiating dot and cross products; handwritten lecture notes (pdf file).
11 August 2016 Lecture 9: Arc length, tangent and normal vectors, curvature torsion; handwritten lecture notes (pdf file).

Week 4: Space curves and vector fields

14 August 2018 Lecture 10: vector fields, flow lines;
16 August 2018 Lecture 11: Differential operators: grad, div, curl, Laplacian; handwritten lecture notes (pdf file).
17 August 2018 Lecture 12: Laplacian vector analysis identities, scalar potentials; handwritten lecture notes (pdf file).

Week 5: Double and triple integrals

21 August 2018 Lecture 13: Using the vector analysis identities; handwritten lecture notes (pdf file).
23 August 2018 Lecture 14: Double integrals over rectangular and general domains; handwritten lecture notes (pdf file).
24 August 2018 Lecture 15: Area using double integrals, changing order of integration; handwritten lecture notes (pdf file).

Week 6:

28 August 2018 Lecture 16: Triple integrals over rectangular box domains, volume using triple integrals; handwritten lecture notes (pdf file).
30 August 2018 Lecture 17: Triple integrals over general domains; handwritten lecture notes (pdf file).
31 August 2018 Lecture 18: Polar coordinates, cylindrical coordinates; handwritten lecture notes (pdf file).

Week 7: Double and triple integrals

4 September 2018 Lecture 19: Spherical coordinates, change of variables for multiple integrals; handwritten lecture notes (pdf file).
6 September 2018 Lecture 20: Change of variables for multiple integrals; handwritten lecture notes (pdf file).
7 September 2018 Lecture 21: Applications: averages, centre of mass, moment of inertia; handwritten lecture notes (pdf file).

Week 8: Integrals over paths and surfaces;
- 11 September 2018 Lecture 22: Parametrization of paths, path integrals; handwritten lecture notes (pdf file).
- 13 September 2018 Lecture 23: Line integrals, parametrization of surfaces; handwritten lecture notes (pdf file).
- 14 September 2018 Lecture 24: Normals, tangent planes to parametrized surfaces, surface area; handwritten lecture notes (pdf file).
Week 9: Integrals over paths and surfaces

18 September 2018 Lecture 25: Integrals of scalar functions over surfaces; handwritten lecture notes (pdf file).
20 September 2018 Lecture 26: Integrals of scalar and vector functions over surfaces; handwritten lecture notes (pdf file).
21 September 2018 Lecture 27: Integrals of vector functions over surfaces; handwritten lecture notes (pdf file).

Week 10: Integral theorems

2 October 2018 Lecture 28: Green's theorem in the plane; handwritten lecture notes (pdf file).
4 October 2018 Lecture 29: Area via line integrals, divergence theorem in the plane; handwritten lecture notes (pdf file).
5 October 2018 Lecture 30: Stokes' theorem handwritten lecture notes (pdf file).

Week 11: Integral theorems
- 9 October 2018 Lecture 31: Stokes' theorem conservative fields; handwritten lecture notes (pdf file).
- 11 October 2018 Lecture 32: Conservative fields, Gauss' divergence theorem; handwritten lecture notes (pdf file).
- 12 October 2018 Lecture 33: Gauss' divergence theorem; handwritten lecture notes (pdf file).
Week 12: Integral theorems and general curvilinear coordinates

16 October 2018 Lecture 34: Applications of integral theorems to physics and engineering; handwritten lecture notes (pdf file).
18 October 2018 Lecture 35: General orthogonal curvilinear coordinates, surface area and volume element; handwritten lecture notes (pdf file).
19 October 2018 Lecture 36: Differential operators, simple PDEs in spherical and cylindrical coordinates; handwritten lecture notes (pdf file).