Mathematics 2

Module aims

 The aims of this module are to: Ensure that all students acquire the mathematical knowledge and skills required for the second and later years of their Biomedical Engineering programme. This module will build on the content of Mathematics 1 and extend subjects introduced there.

Learning outcomes

 Upon successful completion of this module you will be able to: Carry out calculations involving the differentiation of functions of two or more variables Define and use the grad, div and curl vector operators and explain their relevance to physical and biological processes Evaluate simple line, double and volume integrals and carry out changes of variable in multiple integrals State Green`s, Gauss` and Stokes` theorems and be able to apply these theorems to biological and engineering problems; Carry out calculations involving the Dirac delta function, the Heaviside, square wave, tent, sgn, and auto-correlation functions; Perform matrix manipulations and compute eigenvalues and eigenvectors. Apply the previously described mathematical methods, tools and notations in the analysis and solution of mathematical problems described in a biomedical context.

Module syllabus

 In this module you will cover the following topics: VECTOR CALCULUS: parameterised curves; scalar and vector fields; grad, div and curl; arc length; line integrals; conservative fields; double and triple integrals; Jacobians; Green`s theorem in the plane; surface integration; Gauss` and Stokes` theorems. PARTIAL DIFFERENTIATION : Differentiation as linearisation. Functions of more than one variable: partial differentiation, Jacobian; total differentials, chain rule, changes of variable. Taylor`s theorem for a function of two variables; stationary values; contours. PARTIAL DIFFERENTIAL EQUATIONS: application to the description of biological and engineering problems; classification; wave equation; characteristics. Diffusion equation; similarity solutions. Laplace`s equation. Separation of variables. TRANSFORMS: Fourier transforms; definition, inverse and properties. Fourier convolution theorem. Application to the solutions of PDE`s. Laplace transforms: definition, inverse and properties. Laplace convolution theorem MATRIX ALGEBRA: basic matrix operations, eigenvalues and eigenvectors, diagonalisation, Gaussian elimination, linear dependence NUMERICAL METHODS: Euler and Runge-Kutta methods, finite differences

Teaching methods

Lecture 27 hours

Study groups 12 hours

Assessments

  • Mastery exam 40%
  • Non-Mastery exam 60%

Reading list