MATH 1201 -Introduction to Applied Mathematics I

                                                                                              

Level

Level I undergraduate course, (one semester elective course).

 

Semester

Semester I, 2011/2012.

 

No. of Credits

3 credits.

 

Prerequisites

CAPE Advanced Level Proficiency in Pure Mathematics (Units 1 and 2), GCE A-Level Mathematics or equivalent.

 

 Course Rationale

Introductory courses in Applied Mathematics have been in existence since 1991. The current 6-credit courses are being modified for three main reasons.  First, due to harmonization of courses across the three campuses, the credit rating for this course is being reduced from 6 to 3.  Second, many from the Department of Mathematics and Computer Science agree that the present course’s content is far too specialized for a Level I mathematics course.  This also takes away from presenting the fundamentals of classical applied mathematics in more common areas of interest.  Finally, we wish to align the syllabus more closely with the classical part of CAPE Applied Mathematics.  The reason for this is so that those majoring in mathematics, having done this course and entering into the teaching profession, will be well-prepared to handle this aspect of CAPE Applied Mathematics more confidently.  This new syllabus may also assist those students doing some physics courses, as well as those who may branch across to the field of engineering.  At present, there is no other course on any of UWI’s campuses that handles classical applied mathematics in this manner. Thus there is a great need to expose students to this aspect of the field, which provides the necessary tools, with illustrations, on the usefulness and power of mathematics in working out simplified real world problems

 

Course Description

This course will cover the basic concepts and techniques of vectors and some common topics in statics. It will provide undergraduate students with a good understanding of the fundamental laws and associated applications of vectors, as well the necessary tools used in solving elementary common problems in vectors and statics.    

Prior knowledge of CAPE Advanced Level Proficiency in Mathematics (or its equivalent) will be assumed. 

Assessment is designed to encourage students to work continuously with the course materials. Active learning will be achieved through weekly assignments and problem sheets allowing continuous feedback and guidance on problem solving techniques in tutorials and lectures. Assessment will be based on the weekly assignments and in-course tests (40%)  followed by a final examination  (60%) based on the whole course.

 

Learning Outcomes

 On successful completion of this course, students will be able to:

  • Apply the concept of a vector, characteristics and laws associated with them.
  • Identify properties of vectors such as the dot and cross products, as well as triple products.
  • Apply vectors to establish proofs and to volume and area determination.
  • Use the nabla operator to solve problems involving Gradient, Divergence and Curl.
  • Apply the properties of vectors to problems involving lines, planes and level surfaces.
  • Determine the directional derivative of a vector.
  • Identify certain properties of forces, and use such to determine characteristics of static systems due to the action of coplanar forces.  Light frameworks and heavy jointed rods will also be addressed.
  • Solve problems involving friction and tensions in inextensible strings.
  • Determine the centre of gravity of a body.
  • Solve problems related to moment of inertia of a body.

Content

  • Position vectors, the magnitude of a vector, unit vector, dot & cross products, areas of a triangle & parallelogram.
  • Triple products.
  • Gradient, Divergence & Curl.
  • Vector equation of a line, distance of a point from a line, distance between parallel & skewed lines.
  • Vector & Cartesian equations of planes, line of intersection of planes, angle between planes, distance of a point from a plane, distance between parallel planes.
  • Level Surfaces: unit normal at a point, tangential plane.
  • Directional derivative.
  • Coplanar forces in equilibrium.
  • Friction.
  • Light inextensible strings, (Tension).
  • Moment (Torque) of a force.
  • Couple.
  • Centre of gravity.
  • Light frameworks.
  • Heavy jointed rods.
  • Moment of Inertia.

 

Teaching Methodology

Lectures: Two (2) lectures each week (50 minutes each).

Tutorial: One (1) weekly tutorial session (50 minutes).

 

Assessment

Coursework will be worth 50%.  This will generally be made up of at least two coursework examinations, worth 40% in total, and 10% will be drawn from six of the ten assignments given. The six assignments to be assessed will be determined by the lecturer.

 

Final Examination (one 2-hour written paper) – 50%

 

Course Calendar

 

Week

Lecture subjects

Tutorials / Coursework examinations

1

Introduction/Course Overview

Position vectors, the magnitude of a vector & unit vector.

None

2

Dot & cross products, areas of a triangle & parallelogram.  Triple products.

Assignment # 1

3

Partial derivatives.  Gradient, Divergence & Curl.

Assignment # 2

4

Vector equation of a line, distance of a point from a line, distance between parallel & skewed lines.

 

Assignment # 3

5

Vector & Cartesian equations of planes, line of intersection of planes, angle between planes, distance of a point from a plane, distance between parallel planes. 

Assignment # 4

6

Level Surfaces: unit normal at a point, tangential plane.  Directional derivative.

First coursework examination

7

Coplanar forces in equilibrium.  Friction.  Light inextensible strings, (Tension).

Assignment # 5

8

Moment (Torque) of a force.  Couple. 

Assignment # 6

9

Centre of gravity – Plane laminas and 3D bodies.

Assignment # 7

10

Light frameworks.

Second coursework examination

11

Heavy jointed rods.

Assignment # 8

12

Moment of Inertia.

Assignment # 9

13

Revision

Assignment # 10

 

Required Reading

Reference Texts (No essential textbook. Lecture notes will be prepared and made available for the students).

 

  • Theory and Problems of Vector Analysis – M. Spiegel (Schaum series, Mc Graw Hill).
  • Applied Mathematics (Vols. 1 & 2) – C. Bostock & S. Chandler (Stanley Thomas Ltd.).
  • Mathematics – Mechanics and Probability – L. Bostock, S. Chandler.
  • Further Mechanics and Probability – L. Bostock, S. Chandler.