MAU22E02 Engineering mathematics IV

Module Code	MAU22E02
Module Title	Engineering mathematics IV
Semester taught	Semester 2
ECTS Credits	5
Module Lecturer	Prof. Sergey Frolov
Module Prerequisites	MAU22E01 Engineering mathematics III

11 weeks of teaching with 3 lectures and 1 tutorial per week.

On successful completion of this module, students will be able to

Analyse the behaviour of functions of several variables, present the results graphically and calculate partial derivatives of functions of several variables (including those which are defined implicitly).
Obtain equations for tangent lines to plane curves and for tangent planes to space surfaces.
Apply derivative tests to find local and global minima and maxima of functions of several variables.
Calculate multiple integrals in Cartesian, polar, cylindrical and spherical coordinates, and in particular, find areas, volumes, masses and centres of gravity of two- and three-dimensional objects.
Determine whether a vector field is conservative, find a potential function for a conservative field, and use it to calculate line integrals.
Use Green's, Stokes' and the divergence theorems to calculate double, surface and flux integrals.
Solve differential equations using the Laplace transform.

Vector-valued functions: Introduction to vector-valued functions, calculus of vector-valued functions, change of parameter, arc length, unit tangent vector, normal and binormal vectors.
Partial derivatives: Functions of two or more variables, limits and continuity, differentiability, differentials, local linearity, chain rule, directional derivatives and gradients, tangent planes and normal vectors, maxima and minima of functions of two variables.
Multiple integrals: Double integrals over non-rectangular regions, double integrals in polar coordinates, surface area, parametric surfaces, triple integrals in cylindrical and spherical coordinates, centre of gravity, change of variables in multiple integrals, Jacobians.
Topics in vector calculus: vector fields, line integrals, independence of path, conservative vector fields, Green's theorem, applications of surface integrals, flux, divergence theorem, Stokes' theorem.
Laplace transforms: linearity, first shifting theorem, transforms of derivatives, ordinary differential equations, Heaviside function, second shifting theorem, short impulses, Dirac's delta function, convolutions.