| CODE | MAT3221 | ||||||||
| TITLE | Analysis 4 | ||||||||
| UM LEVEL | 03 - Years 2, 3, 4 in Modular Undergraduate Course | ||||||||
| MQF LEVEL | 6 | ||||||||
| ECTS CREDITS | 4 | ||||||||
| DEPARTMENT | Mathematics | ||||||||
| DESCRIPTION | 1. Continuity of functions of several variables (Limits, Various Limits, Continuity, Operations with Continuous Functions); 2. Differentiable Functions (Linear Approximations, Differentials and Partial Derivatives; Derivatives of Composite Functions, Differentiability along Directions, Geometrical Interpretation of Derivatives); 3. Higher Order Derivatives (Partial Derivatives of Higher Order, Mixed Derivatives, Higher Order Differentials); 4. Taylor Formula (Peano Form, Lagrange Form); 5. Local Extrema (Extreme Points, Necessary Condition, Quadratic Forms, Sufficient Condition); 6. Implicit Functions (Functional Equations, System of Functional Equations, Jacobians, Implicit Vector-Valued Functions, Inverse Vector-Valued Functions); 7. Conditional Extremum (Necessary Condition, Lagrange Multipliers); 8. Differentiable Vector-Valued Maps (Jacobian Matrix, Vector-Valued Differential, Differentiable Vector-Valued Functions). Study-unit Aims: This study-unit aims to introduce the students to the differential calculus of functions of several variables. Although there are some similarities with the familiar theory of one real variable, the theory for functions of several variables is far richer. For example, for functions of several variables, all partial derivatives may exist and yet the function may fail to be differentiable. Also, the critical points for functions of several variables might be maxima, minima or saddle points (which are minima in one direction and maxima in another direction). A key idea in the study-unit will be to generalize the definition of the derivative at a point to the the derivative of a map between Euclidean spaces. This is the Fréchet derivative - a linear map that gives the best approximation to the function at the given point. This derivative is used in a number of very elegant and useful results, and is a key notion in the study of the critical points of functions of several variables. Learning Outcomes: 1. Knowledge & Understanding: By the end of the study-unit the student will be able to: - recognise the significance of limits for functions of several variables; - identify the importance of continuity for functions of several variables; - know directional derivatives, partial derivatives and the differentials; also the relationship between them. 2. Skills: By the end of the study-unit the student will be able to: - resolve limits for functions of several variables; - find partial derivatives and use the chain rule; - find the critical points of a function of several variables and determine the nature of these critical points; - apply Lagrange multipliers to simple extremum problems with a constraint. Main Text/s and any supplementary readings: Lecture notes covering all topics. Textbooks: • Apostol, T. (1974) ‘Mathematical Analysis’. Addison-Wesley Publishing Company (Addison-Wesley Series in Mathematics). • Marsden, J.E. and Tromba A.J. (2003, 5th Edition) ‘Vector Calculus’. W.H. Freeman & Company (The University Series in Undergraduate Mathematics). |
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| ADDITIONAL NOTES | Follows from: MAT2213, MAT2212 | ||||||||
| STUDY-UNIT TYPE | Lecture and Independent Study | ||||||||
| METHOD OF ASSESSMENT |
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| LECTURER/S | Emanuel Chetcuti |
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The University makes every effort to ensure that the published Courses Plans, Programmes of Study and Study-Unit information are complete and up-to-date at the time of publication. The University reserves the right to make changes in case errors are detected after publication.
The availability of optional units may be subject to timetabling constraints. Units not attracting a sufficient number of registrations may be withdrawn without notice. It should be noted that all the information in the description above applies to study-units available during the academic year 2024/5. It may be subject to change in subsequent years. |
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