Calculus and Applications A (Physics)

MATH11222 Calculus and Applications A (Physics) 10 Semester 2 Undergraduate Level 1 Matthias Heil Unit coordinator 1 FHEQ level (Framework for Higher Education Qualifications) ' First part HE study/Bachelors ' European Credit Transfer & Accumulation System Rating : 5.0 The unit provides a basic introduction to ordinary differential equations (ODEs) and classical mechanics. The course will discuss both the methods and theory associated with general first and second order ODEs. A brief introduction to the concepts of scaling, non-dimensionalisation and regular perturbation methods will be given. The unit provides a basic introduction to ordinary differential equations (ODEs) and classical mechanics. The course will discuss both the methods and theory associated with general first and second order ODEs. A brief introduction to the concepts of scaling, non-dimensionalisation and regular perturbation methods will be given. 1.Provide a classification of ODEs2.Provide methods of solving both first and second-order ODEs3.Introduce the concepts of scaling and non-dimensionalisation.4.Introduce the concept of a regular perturbation expansion. On successful completion of this unit students will be able to: <ul> <li> Classify ordinary equations (in terms of order, linear/nonlinear, autonomous/non-autonomous) and assess the existence and uniqueness of their solutions.</li> <li> Use graphical and analytical methods to obtain solutions to first-order ODEs.</li> <li> Assess the existence and uniqueness of solutions to linear second-order ODEs and state the general structure of these solutions. Find these solutions for the case of constant-coefficient ODEs.</li> <li> Solve certain second-order nonlinear ODEs that have a special form.</li> <li> Use perturbation methods to find approximate solutions to ODEs containing small parameters.</li> </ul>   1.General introduction. Notation. What are ODEs? Implicit versus explicit form. Classification: order, linearity, autonomous ODEs. Boundary and initial conditions. Boundary and initial value problems. Existence and uniqueness for linear and nonlinear ODEs. [3] 2.First-order ODEs. Graphical methods; separable ODEs, ODEs of homogeneous type; integrating factor. [4] 3.Second-order ODEs. Existence and uniqueness. Linear ODEs: superposition of solutions, fundamental solutions and the general solution for homogeneous ODEs. The general solution of constant-coefficient ODEs; particular solutions for specific RHS; the method of undetermined coefficients. [If time permits (probably not): Power series expansions about regular points.] Some nonlinear ODEs with special properties (autonomous ODEs and ODEs that do not contain the dependent variable). [8] 4.Mechanics applications of second-order ODEs Damped harmonic motions of mechanical oscillators; harmonic forcing and resonance. [3] 5.Non-dimensionalisation and scaling. Exploiting small parameters in an ODE: perturbation methods. Motivation via the roots of quadratic polynomials (singular perturbations only mentioned); applications to selected (regularly perturbed) ODEs. [4] 0 Other 10% 1 Written exam 90% <ul> <li>Submission of homework for supervisions: weighting 10%</li> <li>End of semester examination: weighting 90%</li></ul> Feedback supervisions will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour. MATH10121 Calculus and Vectors A Pre-Requisite Compulsory N Lectures 22 Tutorials 11 0 67 This course unit detail provides the framework for delivery in 20/21 and may be subject to change due to any additional Covid-19 impact.  Please see Blackboard / course unit related emails for any further updates