Module Descriptors

Engineering

MM140: Mathematical Methods for Engineers (5 ECTS)

Taught in Semester(s) II. Examined in Semester(s) II.

Workload: 48 hours (36 Lecture hours, 12 Tutorial hours).

Module Learning Outcomes. On successful completion of this module the learner should be able to:

  1. express a problem modelled by a system of linear equations in an appropriate matrix form and solve the resulting system of equations;
  2. use row operations to determine whether or not a system of $m$ linear equations in $n$ unknowns is consistent/has a unique solution /has an infinite number of solutions;
  3. perform elementary calculations involving matrices and determinants;
  4. calculate the characteristic polynomial, eigenvalues and corresponding eigenvectors for a $3\times3$ matrix, and diagonalise such a matrix;
  5. write complex numbers in modulus/argument form, apply de Moivre's theorem, derive expressions for the sin/cosine of multiple angles in terms of powers of sin/cosine x, etc;
  6. factorise real polynomials into irreducible linear and quadratic terms. Determine the nth roots of unity for small values of $n$;
  7. plot direction fields for first order ODEs and solve separable first order ODEs
  8. solve linear first order ODEs by the integrating factor method;
  9. solve linear homogeneous second order ODEs with constant coefficients, solve linear non-homogeneous second order ODEs with constant coefficients by the method of undetermined coefficients and the method of variation of parameters.

Indicative Content

  1. Engineering problems modelled by systems of linear equations, Gaussian elimination, computer demonstrations, vectors, planes and intersections of planes in n-dimensional Euclidean space. Finding the general solutions of a system of $m$ linear equations in $n$ unknowns.
  2. Matrix algebra. Inverse, transpose and adjoint matrices. Linear transformations of the plane. Computer applications to engineering problems such as Kirchoff's Laws, computer graphics, determinants and their efficient computation, volumes of parallelograms/parallelepipeds. Theoretical formula for the inverse of a matrix.
  3. Complex numbers: argument, modulus, Argand diagram, de Moivre's Theorem. Applications to trigonometric functions, complex roots of polynomials, roots of unity, factorization of real polynomials.
  4. Characteristic polynomials, eigenvalues, eigenvectors and diagonalization of matrices, classification of conic sections, orthogonal transformations of the plane, and an application to finding standard forms for conic sections.
  5. First order ordinary differential equations (ODEs): geometrical interpretation, direction fields, plotting direction fields, solving separable first order equations, solution using an integrating factor, Bernoulli equations.
  6. Second order linear homogeneous ODEs with constant coefficients: the three cases that arise and the general solution in each case. The simple harmonic oscillator equation, the damped simple harmonic oscillator, examples of oscillating systems that arise in engineering.
  7. Second order linear non-homogeneous ODEs with constant coefficients: the method of undetermined coefficients, the method of variation of parameters, resonance.

Module Resources

Modern Engineering Mathematics G. James Prentice Hall

Advanced Engineering Mathematics E. Kreyszig Wiley

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