Department of Physics, Unit Catalogue 2007/08 |
PH20019 Mathematics for scientists 3 |
| Credits: 6 |
| Level: Intermediate |
| Semester: 1 |
| Assessment: EX 100% |
| Requisites: |
| Before taking this unit you must take PH10008 and take PH10007 |
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Aims: The aim of this unit is to introduce mathematical concepts and techniques required by science students, and to show how these may be used for different applications. It also aims to continue the development of students' problem-solving skills and their understanding of mathematical results.
Learning Outcomes: After taking this unit the student should be able to: * find the eigenvalues and eigenvectors of matrices; * calculate the normal modes of coupled vibrational systems; * define and transform between Cartesian, polar, cylindrical polar and spherical polar coordinates; * evaluate grad, div, curl and ∇2 in the above coordinate systems, and use and interpret vector integral theorems; * perform line, surface and volume integrals; * define scalar, vector and conservative fields. Skills: Numeracy T/F A, Problem Solving T/F A. Content: Eigenvalues and eigenvectors (6 hours): Revision of matrix algebra. Eigenvalues and eigenvectors of symmetric/Hermitian matrices and their properties. Linear transformations. Normal modes of ball and spring systems. Vector analysis (16 hours): Differentiation of vectors. Space curves; parameterisation of curves, tangent vector. Polar coordinates; velocity and acceleration. Introduction to scalar and vector fields. Directional derivative; gradient of a scalar field, ∇ as a vector operator in Cartesian coordinates. Introduction to div and curl in Cartesian coordinates, physical interpretation. Identities involving ∇, definition of ∇2. Tangential line integrals. Classification of fields; conservative fields, potential functions, path independence of line integrals in conservative fields. Orthogonal curvilinear coordinate systems; Cartesian, spherical polar and cylindrical polar coordinates. Surface and volume integrals. Div and curl; definitions as limits of integrals, explicit forms. ∇2 in spherical and cylindrical polar coordinates. Vector integral theorems; divergence, Stoke's and Green's theorems. |