Unit Description and General Aims

This unit is intended at expanding the scope of engineering mathematics learning further, by introducing the student to the principles and applications of differential and integral calculus including vector calculus and complex analysis. The derivative and integration rules and techniques are brought out clearly, so as to enable the student to solve simple as well as complex engineering problems, using calculus. Further, a detailed overview of the concepts related to analytical geometry, probability and statistics and sets, so that the student will be able to use these mathematical techniques to effectively deal with problems in engineering application areas.

Learning Outcomes

On successful completion of this Unit, students are expected to be able to:

1. Apply the principles of calculus and make use of numerical methods.
   Bloom's Level 3
2. Evaluate the concepts of analytical geometry
   Bloom's Level 4
3. Acquire knowledge of vector spaces and perform vector calculus by applying various theorems.
   Bloom's Level 4
4. Evaluate complex integration
   Bloom's Level 3
5. Apply concepts related to statistics and probability
   Bloom's Level 3

Student assessment

Overall Requirements: Students must achieve a result of 40% or above in the exam itself to pass the exam and must pass the exam to be able to pass the unit. An overall final unit score of 50% or above must be achieved to pass the unit once all assessment, including the exam, has been completed.

Prescribed and Recommended Readings

Suggested Textbooks

• J. Bird, Higher Engineering Mathematics, 9th Edition. Routledge, 2021 - ISBN-13: 978-0367643737

Reference Materials

• P. O'Neil, Advanced Engineering Mathematics, SI Edition, 8th Edition. Cengage, 2018. ISBN 9781337274524
• Kreyszig, E 2011, Advanced Engineering Mathematics, 10th edn, John Wiley & Sons, ISBN-13: 978-0470458365.
• Knovel library: http://app.knovel.com
• IDC Technologies publications
• Other material and online collections as advised during the lectures and in the Reading Guide.

Unit Content

Topic 1

Differentiation

1. Rules of Differentiation/Derivatives Summary
2. Applications of Derivatives
3. Rates of Change
4. Minimum and maximum value problems
5. ODEs
6. Initial Value Problem
7. Application Examples
8. Second order differential equations

Topic 2

Integration 1

1. Integration Rules and Techniques
2. Integration with trigonometric substitutions
3. Integration with partial fractions
4. Integration by parts
5. Double and Triple Integrals

Topic 3

Integration 2

1. Applications of Integration
2. Areas and Arc Length
3. Volumes of Solids of Revolution
4. Centroids
5. Theorem of Pappus
6. Second Moments of Area
7. Parallel Axis Theorem
8. Perpendicular Axis Theorem
9. Additional Applications

Topic 4

Numerical Methods

1. The trapezoidal Rule, Mid-ordinate Rule, and Simpson's Rule
2. Euler’s method, Euler-Cauchy method and the Runge-Kutta method
3. Solution of equations by iteration

Topic 5

Analytical Geometry 1

1. Angles and Lines
2. Triangles
3. Quadrilaterals
4. Polygons
5. Circle Properties
6. Irregular Areas
7. Solid Figures
8. Straight Lines and Equations
9. Circle Equations
10. Parabolas, Ellipses and Hyperbolas

Topic 6

Analytical Geometry 2

1. Planes and spaces
2. Other coordinate systems
3. Parametric equations
4. Spheres
5. Conic sections
6. Transformations in space
7. Geometric intersections
8. Volumes by integration

Topic 7

Vector Spaces

1. Linear combination and spans
2. Linear dependence and independence
3. Subspaces
4. Vector dot and cross product
5. Null space and column space
6. Linear transformations

Topic 8

Vector Differential Calculus

1. Vectors in 2−space and 3−space
2. Velocity, acceleration and Curvature
3. Curves, Arc length
4. Streamlines
5. Gradient of a scalar field and directional derivatives
6. Divergence and curl of a vector field

Topic 9

Vector Integral Calculus

1. Path independence of line integrals
2. Green's Theorem in the plane
3. Independence of path
4. Surface integrals
5. Triple integrals, Divergence theorem of Gauss
6. Stokes' theorem

Topic 10

Complex Integration

1. Line integral in the complex plane
2. Properties of complex integrals
3. Cauchy's integral theorem
4. Consequences of Cauchy’s theorem
5. Deformation theorem
6. Cauchy's integral formula

Topic 11

Statistics and Standard Deviation

1. Data and data averages
2. Mean
3. Variance
4. Elementary probability
5. Laws of probability
6. Standard Deviation
7. Coefficient of Variation

Topic 12

Distributions and Data

1. Normal Distribution and Z-Scores
2. Chebyshev’s Theorem
3. Histograms
4. Correlation and Scatterplots
5. Correlation Coefficient and Regression Equation
6. Utility and Validity

Software/Hardware Used

Software

• Software: N/A
• Version: N/A
• Instructions:  N/A
• Additional resources or files: N/A

Hardware

• N/A

Unit Changes Based on Student Feedback