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Determinants: basic properties; Cramer’s rule.• Elementary row operations; linear dependence and rank; Gauss-Jordan method; Gaussian elimination; LU factorisation.• Eigenvalues and eigenvectors: diagonalizable matrices; Hermitian matrices.Ordinary Differential Equations (15 lectures)• First order equations: separable, homogeneous, exact, linear.

Basic elements from: Numbers and Arithmetics; Algebra; Combinatorics – Binomial Theorem; Functions and Graphs; Cartesian Geometry; Trigonometry; Differential Calculus; Integral Calculus; Vectors and Mechanics Lectures are complemented by weekly tutorials. Students are required to start preparing and revising their basic mathematics prior to arriving at IC.

A diagnostic test takes place in in the first week of term to test basic mathematics skills.

Students who do not do well at the test will be pointed towards appropriate revision material and self-study.

The course is 100% assessed through one 3-hour examination in the summer.

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This course introduces mathematics as a logical and structured discipline.

It aims to ensure that all students acquire the mathematical knowledge and skills required for their first year chemical engineering courses.

The course provides a basis for the more advanced mathematical techniques required in later years of the course.

By the end of the course, student should be able to:(i) apply the techniques of the differential calculus to determine the maxima and minima of functions and to curve sketching;(ii) determine the Maclaurin series expansions of the standard functions and to apply the ratio test to check for convergence;(iii) manipulate complex numbers in both standard and polar forms ; apply de Moivre’s theorem; (iv) integrate simple functions using the methods of substitution and integration by parts;(v) carry out simple calculations in vector algebra , vector geometry and matrix algebra;(vi) find the inverse of matrices by the Gauss-Jordan method and the solution of systems of linear algebraic equations by Gaussian elimination and by LU factorisation;(vii) find the eigenvalues and eigenvectors of matrices;(viii) find analytic solutions of certain first order ordinary differential equations; find the general solution of linear constant coefficient second order ODEs; find the general solution of certain linear second order ODEs with variable coefficients;(ix) find qualitative solutions of autonomous first order non-linear equations by means of the phase plane. Convergence of power series; ratio test; radius of convergence.

The course consists of the following topics: Autumn term Analysis (15 lectures)• Functions of one variable: odd, even, inverse functions. • Differentiation: continuity and differentiability; implicit and logarithmic differentiation; Leibniz’s formula; stationary points and points of inflection; curve sketching; polar coordinates. Complex Numbers (5 lectures)• The complex plane; polar representation; de Moivre’s theorem; ln z and exp z.

Integration (5 lectures)• Definite and indefinite integrals• The fundamental theorem • Improper integrals; integration by substitution and by parts; partial fractions• Applications to path lengths Spring term Linear Algebra (15 lectures)• Vector algebra: basic rules; Cartesian coordinates; scalar and vector products; applications to geometry; equations of lines and planes; triple products; linear dependence.

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