EE1-10 Mathematics 1 (E-stream and I-stream)
Lecturer(s): Dr Gunnar Pruessner; Dr Henrik JensenAims:
To introduce mathematics as a logical and structured discipline ; to ensure that all students acquire the mathematical knowledge and skills required for their first year electrical engineering courses ; to provide a basis for the more advanced mathematical techniques which are required in later years of the course.
Learning Outcomes:
By the end of the course, students will be able to :
(i) manipulate complex numbers in both standard and polar forms ; apply de Moivre’s theorem;
(ii) apply the techniques of the differential calculus to determine the maxima and minima of functions and to curve sketching ;
(iii) integrate simple functions using the methods of substitution and integration by parts;
(iv) find analytic solutions of certain first order ordinary differential equations ; find the general solution of linear constant coefficient second order ODE’s ;
(v) carry out calculations involving the differentiation of functions of two variables ; determine Taylor’s power series expansion for a function of two variables ; determine stationary points ; sketch contour maps ;
(vi) become familiar with statements using quantifiers, some standard proof techniques and the logical framework ;
(vii) be able to manipulate sets, functions and relations ;
(viii) determine the Maclaurin series expansions of the standard functions and to apply the ratio test to check for convergence;
(ix) carry out calculations involving hyperbolic functions and relate these functions to the trigonometric functions ;
(x) determine the Fourier series expansions of simple functions ;
(xi) carry out simple calculations in vector algebra , vector geometry and matrix algebra ;
(xii) find the inverse of matrices by the Gauss-Jordan method and the solution of systems of linear algebraic equations by Gaussian elimination ;
(xiii) compute eigenvalues and eigenvectors of matrices, and use matrices to solve linear numerical and differential equations.
Syllabus:
Complex Numbers: the complex plane, polar coordinates, polar representation, de Moivre’s theorem, and trigonometric formulae.
Functions of one variable/Differentiation: odd even, composition of functions, inverse functions; Limits, definition, continuous, discontinuous functions, and intermediate values theorem, continuity, differentiability; implicit, derivatives, derivatives of the composition of two functions, logarithmic differentiation, Leibniz’s formula, stationary points, inflection points, curve sketching, Rolle’s and Mean value theorems, and l’Hopital’s rule.
Integration: integrability, fundamental theorem, improper integrals, change of variable, integration by part, and integration of fractions.
Ordinary Differential Equations: first order equations (separable, homogeneous, exact), second order equations with constant coefficients.
Introduction to Partial Differentiation: functions of more than one variable, partial differentiation, total differentials, change of variables, Taylor’s theorem of a function of two variables, introduction to PDEs.
Introduction to discrete mathematics: logical connectives, quantifiers (existential and universal), subsets, Cartesian products, power sets, operations on sets, logical equivalence, converse statement, contrapositive statement, functions, domain, image, injection, surjection, bijection, inverse of a function, relations (transitive, reflexive, symmetric, antisymmetric, orders, functions as relations, graphical representation).
Sequences and Series: recurrence definitions, numerical sequences, limits of sequences, principle of induction, numerical series, Taylor and MacLaurin series, convergence of power series, radius of convergence, alternating series, hyperbolic functions, and trigonometric functions.
Fourier series: Standard formulae, periodic functions, even and odd functions, half- range series; complex form, differentiation and integration of Fourier series, Parseval’s theorem, Introduction to Fourier transforms.
Linear algebra: coordinates, scalar and vector products, applications to geometry, equations of lines and planes, triple products, linear dependence, double suffix notation, sums and products, transpose, inverse, special matrices (symmetric, diagonal, unit, triangular and orthogonal matrices), determinant, Cramer’s rule, linear equations, row and column operations, Gauss-Jordan method, Jordan elimination, characteristic polynomials, eigenvalues and eigenvectors, diagonalisation, application to linear sequences and systems of linear differential equations.
Marked problem sheet questions will contribute to the final assessment.
Assessment:
Two 2-hour examination papers in June
Coursework contribution: 0%
Term: Autumn & Spring
Closed or Open Book (end of year exam): Closed
Coursework Requirement
Non-assessed problem sheets
Oral Exam Required (as final assessment): no
Prerequisite: None required
Course Homepage: unavailable
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