| CODE | MAT1091 | ||||||||
| TITLE | Mathematical Methods | ||||||||
| UM LEVEL | 01 - Year 1 in Modular Undergraduate Course | ||||||||
| MQF LEVEL | 5 | ||||||||
| ECTS CREDITS | 4 | ||||||||
| DEPARTMENT | Mathematics | ||||||||
| DESCRIPTION | • Matrices: • Determinant, rank, trace and inverse of a matrix; • Solution of linear equations; • Eigenvalues and diagonalisation; • Applications. • Ordinary differential equations: • Ordinary differential equations of the first order, Separation of variables; • Ordinary differential equations of the second order with constant coefficients; • Transformation of equations: change of variables, homogeneous functions, Euler’s equation, Bernoulli’s equation; • Partial differentiation and exact differential equations; • Fourier series (including complex form); • Cosine and sine series, odd and even functions. Aims: The aim of this study-unit is to provide an in-depth understanding of a wide range of mathematical methods used for: solving algebraic systems of equations; performing operations with matrices; evaluating determinants; finding eigenvalues and eigenvectors and diagonalising square matrices; partial differentiation; solving first-order differential equations and second order differential equations with constant coefficients; Fourier series expansion of periodic functions. Learning Outcomes: 1. Knowledge and understanding By the end of the study-unit the student will be able to: - Know what is meant by a system of linear equations and its solution set; - Write down the coefficient matrix and augmented matrix of a linear system; - Carry out elementary row operations and understand the step-by-step procedures to reduce any matrix to row-echelon and reduced row-echelon form; - Understand the mathematical operations that can be applied to modify matrices such as matrix addition, subtraction, scalar multiplication, transposition and raising a matrix to a power; - Know what is meant by an invertible matrix and understand when it exists; - Evaluate the determinants of square matrices by expansion, row-reduction and by making use of their properties; - Understand the concepts of eigenvalues, eigenvectors and diagonalisation; - Identify the first-order differential equations as linear, separable, exact, homogeneous, Bernoulli, linear coefficients, Euler; - Solve first-order differential equations by the method of integrating factors and separation of variables; - Know the substitutions and transformations used to solve homogeneous, Bernoulli and linear coefficient equations; - Identify the domain and range of functions of two variables, and know how to compute their first and second partial derivatives; - Understand the periodicity of a function and know how to determine whether a function is even or odd. 2. Skills By the end of the study-unit the student will be able to: - Apply elementary row operations to reduce matrices to echelon forms, and make use of echelon forms in finding the solution sets of linear systems; - Add, subtract and multiply matrices, raise a matrix to a power, and take the transpose of a matrix; - Select and apply appropriate mathematical methods for inverting matrices and evaluating determinants; - Find eigenvalues and eigenvectors of square matrices; - Diagonalise square matrices, whenever possible; - Select and apply appropriate mathematical methods for solving first-order differential equations (linear, separable, exact, homogeneous, Bernoulli, linear coefficients, Euler) and second-order differential equations with constant coefficients; - Find the Fourier series expansions of periodic functions; - Use algebraic tools to create well developed accurate solutions; - Demonstrate enhanced transferable/professional skills such as independent critical thinking and problem solving. Main Text/s and any supplementary readings: • Andrilli S. and Hecker D., Elementary Linear Algebra, Elsevier Academic Press, 4th Edition, 2010. • Nagle R.K., Saff E.B., and Snider A.D., Fundamentals of Differential Equations and Boundary Value Problems, Addison-Wesley, 5th Edition, 2008. • Kreyszig E., Advanced Engineering Mathematics, John Wiley & Sons, 9th Edition, 2006. • Derrick W. and Grossman S., Elementary Differential Equations with Boundary Value Problems, Addison-Wesley, 4th Edition, 1996. • Folland G.B., Fourier Analysis and its Applications, AMS, 1992. • Korner T.W., Fourier Analysis, C.U.P., 1988. |
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| ADDITIONAL NOTES | Pre-requisite qualification: Advanced Level in Pure Mathematics Leads to: MAT1115, MAT1116 and MAT3711. Cannot be taken by students pursuing the Bachelor of Engineering (Honours) course. |
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| STUDY-UNIT TYPE | Lecture and Tutorial | ||||||||
| METHOD OF ASSESSMENT |
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| LECTURER/S | Cristiana Sebu |
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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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