Course Title:                               Multivariable Calculus

Course Code:                             MATH 2270

Level:                                         2

Number of Credits:                     3

Semester:                                  1

Prerequisite(s):                          MATH1142 and MATH1151 (or equivalents)

Course Rationale

In the natural world, whenever variable quantities change in relation to one another or wherever something can be considered as the result of accumulating a large number of small constituents, there one find Calculus. Mathematical models of real life situations in nature are not one-dimensional and it is not possible to study such models with one-variable calculus. Multivariable calculus is therefore used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields and fluid flow.

Course Description

This is a one-semester, three-credit course at the intermediate level in multivariate calculus intended for students who have satisfactorily completed six credits in elementary differential and integral calculus. For this reason, MATH 1142 - Calculus I and MATH 1151 - Calculus II (or their equivalents) are listed as prerequisite courses.

In this course, vector notation is introduced and utilized for modelling and solving problems in multidimensional space. The first section of the course deals with the Calculus of functions of several real variables. The fundamental ideas of limits and continuity are introduced, followed by the technique of partial differentiation via the chain rule and its related applications. One key application covered is the use of the method of Lagrange multipliers for the determination of constrained extrema. This is followed by the calculus of vectors and their description of curves and surfaces in space. Differentiation of vectors is more fully developed, extending elementary notions of differentiation to those involving multiple variables. Integration is developed to encompass double integrals and triple integrals. Finally, line and surface and volume integrals are considered. The Green’s Theorem in a plane, Stokes’ Theorem and the Divergence Theorem are introduced and utilized for the calculation of line, surface and volume integrals.

This course includes proofs and discussions at a level of complexity suitable for those intending to specialize in mathematics, as well as many examples and applications of the theory for those more interested in being able to make use of the theory in their various fields of interest.

Assessment is designed to encourage students to work continuously with the course materials. Active learning will be achieved through problem sheets allowing continuous feedback and guidance on problem solving techniques in tutorials. Assessment  will be based on four marked assignments and in-course tests followed by a final examination based on the whole course.

Content

• Functions of Several Variables
• Limits and continuity.
• Partial Derivatives: Chain rule, Implicit differentiation.
• Applications of Partial Derivatives: Stationary points, Lagrange multipliers.
• Multiple Integration
• Polar, cylindrical and spherical polar coordinate systems, change of variables for multiple integrals.
• Double and triple integrals, calculation of surface area and centre of mass by double integration, calculation of volume by triple integration.
• Vector Calculus
• Basic concepts: Unit vectors, dot and cross product, vector fields, gradient, divergence and curl, directional derivatives, tangent planes, normal to surfaces.
• Line integrals: evaluation of line integrals, line integrals independent of the path taken (for conservative vector fields).
• Surface integrals.
• Green’s Theorem in a plane, Stokes’ Theorem, Divergence Theorem..

Learning Outcomes

By the end of the course, students should be able to:

• Use the definition of the limit of a function of several real variables to prove rigorously that a limit exists at a given point.
•  Investigate the continuity of a function of several real variables at any point or over a specified region.
• Compute partial derivatives of a function of several real variables via the chain rule.
• Find the stationary points of a function of several real variables.
• Utilize the method of Lagrange multipliers to calculate relative extrema subject to given constraints.
• Transform functions from the Cartesian coordinate system to polar, cylindrical and spherical polar coordinate systems.
• Transform multiple integrals into different curvilinear coordinate systems and evaluate the resulting integrals.
• Formulate and evaluate double and triple integrals to calculate surface area, centre of mass and volume.
• Calculate the derivatives and integrals of vector functions.
• Find directional derivatives and gradients of scalar functions.
• Evaluate line integrals through scalar or vector fields and provide physical interpretations of these integrals.
• State and utilize Green’s Theorem in a plane, Stokes’ Theorem, Divergence Theorem.

Teaching Methodology

This course will be delivered through a combination of informative lectures and participative tutorials. The total estimated 39 contact hours are broken down as follows: 29 hours of lectures, 10 tutorials. Supporting course materials will be posted on myeLearning.

Assessment

The course assessment will be broken into two components; a coursework component worth 50% and a final exam worth 50%.

• Two course work exams each worth 15% of the student’s final grade.
• Four assignments (practical questions based on the theory done during lectures) will be given throughout the semester, each worth 5%.
• The final exam will be two hours in length and will be worth 50% of the final grade.

Course Calendar

Week

Lecture Topics

Assignments

Tutorial

1

Course Overview/Introduction

Introduction: Functions of Several Variables: Limits and continuity.

None

Tutorial #1

2

Functions of Several Variables: Partial Derivatives: Chain rule, Implicit differentiation.

Assignment 1 given

Tutorial #2

3

Functions of Several Variables: Applications of Partial Derivatives: Stationary Points.

None

Tutorial #3

4

Functions of Several Variables: Applications of Partial Derivatives: Lagrange multipliers.

Assignment 1 due

Assignment 2 given

Tutorial #4

5

Multiple Integration: Polar, cylindrical and spherical polar coordinate systems, change of variables for multiple integrals.

Coursework Exam #1

(15%)

No tutorial this week

6

Multiple Integration: Change of variables for multiple integrals.

Assignment 2 due

Assignment 3 given

Tutorial #5

7

Multiple Integration: Double and triple integrals: Calculation of surface area and centre of mass by double integration.

None

Tutorial #6

8

Multiple Integration: Double and triple integrals: Calculation of volume by triple integration.

Vector Calculus: Basic concepts: Unit vectors, dot and cross product, vector fields, gradient, divergence and curl.

Assignment 3 due

Assignment 4 given

Tutorial #7

9

Vector Calculus: Directional derivatives, tangent planes, normal to surfaces.  Evaluation of line integrals

None

Tutorial #8

10

Vector Calculus: Line integrals independent of the path taken (for conservative vector fields). Surface Integrals.

Assignment 4 due

Tutorial #9

11

Vector Calculus: Surface integrals. Green’s Theorem in a plane.

Coursework Exam #2

(15%)

No Tutorial this week

12

Vector Calculus: Stokes’ Theorem, Divergence Theorem.

None

Tutorial #10

33

Revision

None

None

REFERENCE MATERIAL:

Books:

Prescribed

STEWART, James. Multivariable Calculus. Brooks Cole; 6th edition (2007): ISBN-13: 978-0495011637

Recommended

MARSDEN, J. & A. Tromba. Vector Calculus, W.H. Freeman & Company, 4th edition      (1996): ISBN-13: 978-0716724322

Highly Recommended

LANG, Serge. Calculus of Several Variables. Springer, 3rd edition (1987): ISBN-13:          978-0387964058

Online Resources: Support material will be made available via myeLearning