Engineering
MA2101: Mathematics and Applied Mathematics I (5 ECTS)
This module covers topics in both Mathematics and Applied Mathematics for engineering students. The material presented includes: calculus of several variables, multiple integration and integral theorems, coordinate systems, force systems, rigid body motion, Fourier series, and Laplace transforms.
Taught in Semester(s) 1. Examined in Semester(s) 1.
Workload: 123 hours (36 Lecture hours, 12 Tutorial hours, 75 Self study hours).
Module Learning Outcomes.
On successful completion of this module the learner should be able to:
- calculate the partial derivatives of a function of several variables; determine critical points of functions of several variables, including constrained systems using Lagrange multipliers;
- calculate the gradient of scalar fields, calculate the divergence and curl of vector fields;
- integrate functions of several variables, interpret results in terms of areas or volumes, change independent variables to simplify multiple integrals;
- apply the integral theorems of Green, Gauss and Stokes in two- and three-dimensions;
- solve some three-dimensional problems in rigid body statics using vector methods;
- solve some problems for the motion of a rigid body in two and three dimensions;
- calculate the Fourier series associated with a periodic function;
- calculate the Laplace transform and inverse Laplace transform of some simple functions, solve some initial value problems for ordinary differential equations using Laplace transforms.
Indicative Content
This module covers material in both Mathematics and Applied Mathematics for engineering students, and the following topics are considered.
Functions of several variables. Partial derivatives of a function of several variables. Classification of critical points of functions of several variables, constrained systems and Lagrange multipliers. The gradient, the divergence, the curl, the directional derivative, and the Laplacian. Vector identities.
Multiple integrals and integral theorems. Integration in two and three dimensions (areas, volumes), the line integral, surface integrals, Green’s theorem, the divergence theorem, Stoke’s theorem, examples.
Coordinate systems. Definition of cylindrical and spherical coordinate systems, the Jacobian matrix and area/volume elements, coordinate bases and changing vectors from one coordinate system to another, scale factors.
Force systems & rigid body statics. The resultant of a distributed system of forces and couples at a point, mechanical equivalence of force systems, wrench systems. Equations of equilibrium, free body diagrams, supports and hinges, solved problems.
Rigid body dynamics.The equations of motion for a rigid body, motion of the centre of mass, motion about the centre of mass, angular velocity, moments of inertia, solved problems in two and three dimensions.
Fourier Series. Definition of Fourier series, the sine series, the cosine series, examples.
Laplace transforms. Definition and examples, calculating inverse Laplace transforms, Laplace transform of a derivative, solution of some initial value problems for ordinary differential equations.
Module Resources
- Modern Engineering Mathematics, G. James, Prentice Hall
- Advanced Modern Engineering Mathematics, G. James, Prentice Hall
- Advanced Engineering Mathematics, E. Kreyszig, John Wiley & Sons
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