Mathematics and Engineering 2

Module aims

The aims of this modules are to:

Ensure that all students acquire the mathematical knowledge and skills required for the second and later years of their Molecular Engineering programme.

Introduce the principles of signals and control as required for further courses in the Molecular Bioengineering degree.

Provide practice in solving mathematical, signals and control problems posed in a Bioengineering context.

Learning outcomes

Upon successful completion of this module, you will be able to: 

1) Differentiate functions of two or more variables, evaluate simple line, double and volume integrals and carry out changes of variable in multiple integrals.

2) Define and calculate the vector operators grad, div and curl, and state Green's, Gauss' and Stokes' theorems involving those operators.

3) Derive and classify partial differential equations, recognizing the importance of boundary conditions and apply standard methods to solve and verify the solutions of such equations

4)  Explain and use concepts related to matrices including: linear transformations performed by a matrix, the significance of eigenvectors and eigenvalues of matrices, eigendecomposition and diagonalisation

5) Describe and apply simple methods for numerical integration.

6) Explain applications of signal processing and control systems in biomedical engineering - this includes in imaging techniques, and describing physiological signals. 

7) Model or represent a signal and/or control system using either mathematical expressions or simulations in Matlab, or an equivalent programming language.

8) Analyse signals quantitatively and with the use of computational methods.

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; 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 (PDEs): Derivation from microscopic considerations. Classification of PDEs and boundary conditions. Examples such as the wave equation, the diffusion equation, Laplace's equation. Verification of solutions by substitution, solution of linear equations via separation of variables and similarity solutions, principle of superposition. The importance of eigenfunctions and orthogonality in the solution of linear PDEs. 

MATRIX ALGEBRA: basic transformation, cofactor expansion, rank, eigenvalues and eigenvectors, diagonalisation 

NUMERICAL METHODS: Newton's method, Euler and Runge-Kutta methods. 

SIGNALS AND CONTROL: Definition of Signals & Systems with examples in bioengineering. Fourier Representations and the use of Fourier analysis in bioengineering. Sampling and digital signals Control including PID control and Frequency domain analysis 

Assessments

Mathematics

  • Alpha/beta progress test - 10%
  • Gamma progress test - 5%
  • Main exam - 35%

Signals and Control

  • Signals progress test - 10%
  • Control progress test - 5%
  • Main exam - 35%