|Unit level:||Level 2|
|Teaching period(s):||Semester 1|
|Offered by||School of Mathematics|
|Available as a free choice unit?:||N
The course unit aims to provide a second year, first semester course in calculus and algebra to students in school of Electrical and Electronic Engineering.
On completion of this unit (2E1) successful students will be able to:
1. (Laplace Transforms) Calculate the Laplace transforms of simple functions using the definition and integrating from -8 to 8. Calculate inverse transforms using tables. Calculate transforms of first and second derivatives and of indefinite integrals. Solve ordinary differential and integro-differential equations, in particular the equations of an RLC circuit, using the Laplace Transform. Extend this to situations in which the current and/or voltage are suddenly switched on or off using, where necessary, the time-delay and s-shift theorems
2. (Vector Calculus) Calculate the gradient of a scalar field and the divergence and curl of a vector field in 3D and use these in a physical context. Calculate second derivatives of scalar and vector fields using the Laplacian operator. Verify solutions of the wave equation and the heat (or diffusion) equation. Express the equations of curves and surfaces in 3D using parameters. Calculate the tangent to a curve and the normal to a surface. Calculate the line integral of a scalar or vector field along a curve. Determine whether or not a vector field is conservative. Calculate the surface integral of a vector field over a given bounded surface. Change the volume integral of the divergence of a vector field to the surface integral over the surface of the volume using the Divergence theorem. Change the surface integral of the curl of a vector field over a given bounded surface into a line integral of the vector field around the boundary. Change from the integral form to the differential form and vice versa of three fundamental theorems of electricity and magnetism: Ampère’s Law, Gauss’s Law and the Maxwell-Faraday equation.
3. (Linear Algebra) Give information about matrices e.g. number of rows, columns etc. and use this information to determine whether particular matrix operations can be carried out. Add, subtract and multiply matrices and multiply a matrix by a scalar. Where they exist, find the determinant and inverse of a matrix. Solve linear systems of equations (including systems with solutions involving a parameter) by inverse matrix, factorisation or elimination methods or determine that no solution exists. Find the eigenvalues and eigenvectors of a given matrix and hence diagonalise a matrix. Use diagonalisation of matrices to solve linear systems of differential euqations.
- Other - 20%
- Written exam - 80%
Assessment Further Information
Coursework 1 (week 7) Weighting within unit 10%. Written or computerised Assignment
Coursework 2 (week 11) Weighting within unit 10%. Written or computerised Assignment
2 hour (semester 1) examination Weighting within unit 80%
5 lectures: Laplace Transforms : Definitions : Transforms of simple functions (integration from -infinity to infinity). Inverse Transforms. Transforms of first and second derivatives. Delta and Step Functions. Shift Theorems. Solution of Ordinary Differential Equatios by Laplace Transforms. Application to LRC circuits. Transfer Functions.
8 lectures : Vector Calculus : Scalar and Vector fields. Gradient, divergence and curl. Laplacian. Identities. Line, surface and volume integrals Divergence Theorem. Stokes Theorem. Green's Theorem in the Plane.
9 lectures : Linear Algebra : Matrices, matrix algebra and inverse matrices. Soluton of Linear Equations, Gaussian Elimination. Linear Spaces, transformations and Projections. Eigenvalues and Eigenvectors. Diagonalisation of Square Matrices. Solution of sytems of linear ODEs Note :- Some terminology consistent with Matlab should be used.
- Assessment written exam - 2 hours
- Lectures - 24 hours
- Tutorials - 11 hours
- Independent study hours - 63 hours