Mathematics 2

Module aims

  • To ensure that all students acquire the mathematical knowledge and skills required for the second and later years of their Biomedical Engineering course.

Learning outcomes

Learning Outcomes - Knowledge and Understanding

  • carry out calculations involving the differentiation of functions of two or more variables
  • define the grad, div and curl vector operators; explain their relevance to physical and biological processes; carry out simple calculations involving these operators;
  • evaluate simple line, double and volume integrals; carry out changes of variable in multiple integrals;
  • state Green`s, Gauss` and Stokes` theorems; apply these theorems to biological and engineering problems;
  • describe physical and biological phenomena using simple partial differential equations;
  • carry out calculations involving the Dirac delta function, the Heaviside, square wave, tent, sgn, and auto-correlation functions;
  • calculate Fourier transforms, and find their inverse using the convolution theorem; use the technique of Fourier transforms to solve certain PDEs.
  • calculate Laplace transforms, and find their inverses including the convolution theorem and contour integration; use the technique of Laplace transforms to determine the solution of initial value ODE problems.
  • perform matrix manipulations and compute eigenvalues and eigenvectors

Learning Outcomes - Intellectual Skills

  • Analysis and formulation of engineering problems into mathematical terms
Learning Outcomes - Practical Skills
  • Tools of mathematical analysis
  • Tools to underpin the development of theoretical models in engineering and biology
  • Tools to underpin the understanding of signal analysis, mechanical engineering, fluid dynamics and computation

Module syllabus

Alpha (Dr. Lee)

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.

Beta (Dr. Choi)

PARTIAL DIFFERENTIATION : Differentiation as linearization. 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

Gamma (Dr. Tanaka)

Matrix Algebra: basic matrix operations, eigenvalues and eigenvectors, diagonalisation, Gaussian elimination, linear dependence

Numerical Methods: Euler and Runge-Kutta methods, finite differences

Pre-requisites

BE1-HMATH1 Mathematics 1

Teaching methods

Lectures: 27 hours
Study groups: 15 hours
Labs: 2 hours

Assessments


Written exam: one final exam consisting of 3 questions(1.5 hrs long, 50% of final marks)

Mastery exam: two mastery tests 1) 1-hour combined alpha & beta mastery test and 2) 45-min gamma mastery test (40% of final marks)


Coursework:
●  Item 1:Problem sheet Title:Alpha problem sheet Description: Weighting: 5 %
●  Item 2:Problem sheet Title:Beta problem sheet Description: Weighting: 5 %

Outline answers to past papers will be available

Exam rubric: One final exam consisting of 3 questions (1.5 hrs long, 50% of final marks) Two mastery tests: 1) 1-hour combined alpha & beta mastery test and 2) 45-min gamma mastery test

Feedback : The alpha and beta sub-courses will have 1 marked coursework activity. The coursework will be marked with feedback within 2 weeks (or less) in order to provide useful preparation for the mastery test. Similar activities will be worked on in the study groups. There will be two VLE based mastery test. The first mastery will be scheduled at the beginning of the spring term, and the second test will happen during the spring term. A mock mastery for each sub-course will be provided to assist preparation. Normal mastery test rubrics apply. The result will be revealed within 1 week of the assessment and will be revealed as Pass or Fail.

Reading list

Supplementary