This module will impart the essential foundational elements of linear algebra in a matrix representation. The learner will carry out the basic calculations of matrix algebra, while interpreting these concepts geometrically. They will then expand their knowledge of calculus to include more methods of integration and to the solution of number of classes of ordinary differential equations. Finally they will apply their knowledge of calculus to power series and Fourier series representations of functions.
Linear algebra: The definition of a matrix. Matrix algebra, including the addition and multiplication of matrices and multiplication by scalars. The representation of vectors as matrices. The definition of the determinant and the inverse for square matrices. Methods for calculating the inverse of a matrix, including the cofactor method and Gauss-Jordan elimination. Solving systems of linear equations using the inverse. Eigenvalues and eigenvectors.
Power Series: Arithmetic and geometric sequences and series, their properties and applications. The Binomial expansion and approximations of functions using the binomial series. The definition of McLaurin and Taylor series and their tests for convergence. Calculating these series for a range of common functions. The link with even and odd functions. Basic tests for convergence for series, including power series.
Integration: Integration by Parts and integration by substitution, for common functions. The definition of piecewise continuous functions and the calculation of their indefinite and definite integrals.
Introduction to Fourier Series: The concept and definition of the Fourier coefficients and the Fourier series. Harmonics and their interpretation. Calculating the Fourier series of standard periodic functions, including square waves, saw-tooth functions, piecewise linear functions. The link with even and odd functions and the use of the half-wave sine and cosine series expansions. The Gibbs phenomenon and the accuracy of Fourier series.
Ordinary Differential Equations: The Methods of solution of Ordinary Differential Equations; first order linear equations and the integrating factor, separable first order equations, second order differential equations with constant coefficients and the characteristic equation. The concept of an engineering system and its modelling as a differential equation. The treatment of linear Ordinary Differential equations as systems solvable by the use of Phasors. Producing empirical approximate solutions for ordinary differential equations using the McLaurin series.
Linear algebra
The definition of a matrix. Matrix algebra, including the addition and multiplication of matrices and multiplication by scalars. The representation of vectors as matrices. The definition of the determinant and the inverse for square matrices. Methods for calculating the inverse of a matrix, including the cofactor method and Gauss-Jordan elimination. Solving systems of linear equations using the inverse. Eigenvalues and eigenvectors.
Power Series
Arithmetic and geometric sequences and series, their properties and applications. The Binomial expansion and approximations of functions using the binomial series. The definition of McLaurin and Taylor series and their tests for convergence. Calculating these series for a range of common functions. The link with even and odd functions. Basic tests for convergence for series, including power series.
Integration
Integration by Parts and integration by substitution, for common functions. The definition of piecewise continuous functions and the calculation of their indefinite and definite integrals.
Introduction to Fourier Series
The concept and definition of the Fourier coefficients and the Fourier series. Harmonics and their interpretation. Calculating the Fourier series of standard periodic functions, including square waves, saw-tooth functions, piecewise linear functions. The link with even and odd functions and the use of the half-wave sine and cosine series expansions. The Gibbs phenomenon and the accuracy of Fourier series.
Ordinary Differential Equations
The Methods of solution of Ordinary Differential Equations; first order linear equations and the integrating factor, separable first order equations, second order differential equations with constant coefficients and the characteristic equation. The concept of an engineering system and its modelling as a differential equation. The treatment of linear Ordinary Differential equations as systems solvable by the use of Phasors. Producing empirical approximate solutions for ordinary differential equations using the McLaurin series.
The calculus element of this module is best thought by introducing the methods of solution for ODEs, Series, Transforms and so on in lectures, explaining how and why they work and then practising as many examples as feasibly possible in the time allowed. The Matrices content is also best taught with examples provided via worksheets.
| Module Content & Assessment | |
|---|---|
| Assessment Breakdown | % |
| Formal Examination | 70 |
| Other Assessment(s) | 30 |