PMAT 22293: Functions of Several Variables

Course Code            : PMAT 22293

Title                          : Functions of Several Variables

Pre-requisites          : PMAT 21263

 Learning Outcomes:

At the end of this course, the student should be able to:

1. appraise the geometrical aspects of functions of several variables in different coordinate systems
2. parametrize curves and surfaces
3. give an account of the concepts of limit, continuity, partial derivative, gradient and differentiability for
functions of several variables
4. prove and apply Young’s Theorem, Schwarz’s Theorem
5. classify local and global extremes of functions of two variables
6. determine local extremes under constrains using Lagrange Multiplier Method
7. outline the definition of the multiple integral, compute multiple integrals and use multiple integrals to
compute volumes in different coordinate system
8. make use of change of variables to evaluate double integrals.

 Course Contents:

Geometrical Aspects: 

• Domain and range of functions of several variables, Level curves, Parametric surfaces; Some special surfaces; planes, spheres, cylinders and cones; Surface area.

Analytical Aspects: 

• Domain of a function of two variables; Neighborhoods in the Plane, Limits and Continuity; Partial derivatives; Clairaut’s Theorem (Young’s Theorem), Schwarz Theorem, Differentials, Differentiability; Tangent planes and linear approximations; Chain rules; Gradient of a Function, Directional Derivative, Tangent Planes and Normal lines, Maxima and Minima; Critical Points and Second Partial Test, Lagrange multipliers.

Coordinate Systems: 

• Cartesian, Polar, Spherical and Cylindrical Coordinate Systems Multiple Integrals: Double integrals and Volume; Iterated integrals; Fubini’s Theorem, Double integrals over general regions, Double integrals in Polar coordinates, Change of variables in double integrals, Triple integrals in Cartesian, Cylindrical and Spherical coordinates.

 Method of Teaching and Learning : A combination of lectures and tutorial discussions.

 Assessment : Based on tutorials, tests and end of course examination.

 Recommended Reading:

1. Stewart, J. (2020). Calculus Early Transcendental, Cengage Learning, Inc.
2. Larson, R. & Edwards, B.H. (2018). Calculus, Brooks/Cole, Cengage Learning
3. Ross, K.S. (2015). Elementary Analysis: The Theory of Calculus, Springer.
4. Malik, S.C. & Arora, S. (2017). Mathematical Analysis, New Age International.