MECH 601: Engineering Analysis
Graduate course, University of Victoria, Department of Mechanical Engineering, 2023
Graduate-level course providing advanced mathematical methods for engineering analysis, focusing on integral transformations, complex analysis, and calculus of variations with applications to engineering research.
Course Information on UVic Website
Course Description
This course seeks to provide a graduate level introduction to advanced mathematical methods used in engineering analysis. The course focuses on the derivation of mathematical concepts and the application of mathematics to engineering.
Core topics include:
- Integral transformations with applications
- Complex analysis
- Calculus of variations
Course Objectives
Provide graduate students with advanced mathematical tools necessary for conducting engineering research and solving complex engineering problems. Students will learn to derive mathematical concepts and apply them directly to engineering applications.
Learning Outcomes
Students who successfully complete this course will be able to use mathematical concepts presented in class in their graduate research, including:
- Apply Laplace and Fourier transformations to solve differential equations arising in engineering systems
- Use complex analysis techniques for solving problems in fluid mechanics, heat transfer, and structural analysis
- Apply calculus of variations to optimization problems in engineering design
- Derive mathematical models for engineering phenomena
- Select and apply appropriate mathematical methods for research problems
Course Topics
1. Integral Transformations (Jan 9 - Feb 17)
Laplace Transformation:
- Definitions and properties
- Applications to ordinary differential equations
- Transfer functions and system response
- Inverse Laplace transforms
Fourier Transformation:
- Fourier series and Fourier integrals
- Frequency domain analysis
- Applications to partial differential equations
- Fast Fourier Transform (FFT)
2. Complex Analysis (Feb 20 - Mar 17)
Analytic Functions:
- Complex variables and functions
- Cauchy-Riemann equations
- Contour integration
Theory of Residues:
- Residue theorem
- Applications to real integrals
- Evaluation of improper integrals
Conformal Mapping:
- Geometric interpretation
- Applications to potential flow and heat conduction
- Schwarz-Christoffel transformation
3. Calculus of Variations (Mar 20 - Apr 6)
Variations:
- Euler-Lagrange equations
- Functionals and variational principles
- Brachistochrone and other classical problems
Constraints:
- Lagrange multipliers in variational problems
- Isoperimetric problems
- Applications to mechanics and optimal control
Assessment
| Component | Weight | Due Dates |
|---|---|---|
| Assignments (6) | 42% | Jan 27, Feb 10, Feb 24, Mar 10, Mar 24, Apr 7 |
| Mid-term Exam | 30% | March 23 (In Class) |
| Final Project | 28% | April 14 |
Assignment Topics
- Assignment 1 (Jan 27): Laplace transformation
- Assignment 2 (Feb 10): Laplace transformation applications
- Assignment 3 (Feb 24): Fourier transformation
- Assignment 4 (Mar 10): Fourier transformation applications
- Assignment 5 (Mar 24): Complex analysis
- Assignment 6 (Apr 7): Calculus of variations
Submission: All assignments administered through Brightspace. Solutions must be scanned and submitted as a single PDF file.
Final Project
The final project requires students to apply advanced mathematical methods to a problem relevant to their graduate research. Students select appropriate mathematical techniques covered in the course and demonstrate their application to solving real engineering problems.
Textbooks and References
Primary Texts:
- Advanced Engineering Mathematics by E. Kreyszig
- Methods of Mathematical Physics by R. Courant and D. Hilbert
Supplementary References:
- Mathematical Techniques by D.W. Jordan and P. Smith
- Advanced Engineering Mathematics by C.R. Wylie and L.C. Barrett
Grading Scale (Graduate Level)
| Grade | Grade Point | Percentage | Description |
|---|---|---|---|
| A+ | 9 | 90-100 | Exceptional work. Technically flawless and original work demonstrating insight, understanding and independent application or extension of course expectations; often publishable. |
| A | 8 | 85-89 | Outstanding work. Demonstrates very high level of integration demonstrating insight, understanding and independent application. |
| A- | 7 | 80-84 | Excellent work. High level of integration, comprehensiveness and complexity; mastery level of relevant techniques/concepts. |
| B+ | 6 | 77-79 | Very good work. Satisfactory level of integration, comprehensiveness and complexity; sound level of analysis. |
| B | 5 | 73-76 | Acceptable work that fulfills course expectations. Satisfactory level of integration of key concepts/procedures. |
| B- | 4 | 70-72 | Acceptable but revealing some deficiencies in knowledge, understanding or techniques. |
Important: Every grade of 4.0 (B-) or lower must be reviewed by the student’s supervisory committee with a recommendation made to the Dean of Graduate Studies.
Instructor Information
Instructor: Prof. Caterina Valeo
Office: EOW 543
Phone: (250) 721-8623
Email: valeo@uvic.ca
Office Hours: TBA
Teaching Assistant
TA: Zhonghao Zhang
Email: zhonghaoz@uvic.ca
Class Schedule
Lectures: Wednesdays and Fridays, 9:00-10:20 AM
Location: HHB 110
Why Take This Course?
This course is essential for graduate students in mechanical engineering and related fields who need advanced mathematical tools for their research. Whether you’re working on:
- Computational Fluid Dynamics (CFD): Complex analysis and conformal mapping
- Heat Transfer: Fourier analysis and partial differential equations
- Structural Mechanics: Variational methods and optimization
- Control Systems: Laplace transforms and transfer functions
- Signal Processing: Fourier analysis and frequency domain methods
The mathematical techniques in this course provide the foundation for rigorous engineering analysis and research.
This graduate course bridges the gap between undergraduate mathematics and research-level engineering analysis, providing students with powerful mathematical tools essential for conducting high-quality graduate research.