Course Description
Multi-variable functions: continuity, differentiability, partial derivatives, Jacobi matrices, chain rule. Inversion theorem and theorem of implicit functions. Vector differential calculus: vector fields, differential operators, orthogonal curvilinear coordinates. Vector analysis and applications: theorems of Green, Gauss and Stokes. Differential forms: degree of differential forms, exact and closed differential forms, exterior differential of differential forms, vector fields and differential forms and integrals of differential forms.
Course Objectives & Outcomes
Objectives:
- Equip the student with necessary knowledge and skills to enable him to discuss the continuity and the differentiability of multi-variable functions.
- Develop skills in applying the inversion theorem and the theorem of implicit functions.
- Enable the student to determine differential operators (gradient, divergence, curl and Laplacian operators) in orthogonal curvilinear coordinates.
- Develop skills in applying Green, Gauss and Stokes theorems.
- Enable the student to calculate all operations on differential forms such as addition, exterior multiplication, exterior differential and integrals of differential forms.
Outcomes: Upon successful completion of this course, the student will be able to:
- Explain the continuity and the differentiability of functions in several variables,
- Evaluate partial derivatives, directional derivatives, chain rules and Jacobian matrix,
- Discuss the inversion theorem and the theorem of implicit functions,
- Identify differential operators and mention their formulas in orthogonal coordinates,
- Evaluate line, surface and volume integrals,
- Describe differential forms and outline exterior differentiation and integration of differential forms,
- Discuss vector analysis theorems: Green, Gauss and Stokes theorems.
References
1. Colley, S.J. (2012) Vector Calculus, 4th edition, Pearson, ISBN-13: 978-0321780652, ISBN-10: 0321780655.
2. Weintraub, S.H. (¬1996) Differential Forms: a Complement to Vector Calculus, Academic Press, ISBN-13: 978-0127425108, ISBN-10: 0127425101.
3. Edwards, C.H. (1995) Advanced Calculus, Dover Publications, ISBN-13: 978-0486683362
ISBN-10: 0486683362.
4. Kaplan, W. (1993) Advanced Calculus, Addison publishing, ISBN 0-201-57888-3.
5. BRESSOUD, D.M. (2001) Second Year Calculus, Springer-Verlag, ISBN 978-1-4612-0959-1.
6. Hubbard, J.H. and Hubbard, B.B. (2002) Vector Calculus, Linear Algebra and Differential Forms: a Unified Approach, Prentice Hall, ISBN-13: 978-0136574460, ISBN-10: 0136574467.
Course ID: MATH 405
Credit hours | Theory | Practical | Laboratory | Lecture | Studio | Contact hours | Pre-requisite | 4 | 4 | 4 | MATH 301 - MATH 303 |
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