Module MA1S12: Mathematics for Scientists (second semester)
 Credit weighting (ECTS)
 10 credits
 Semester/term taught
 Hilary term 201718
 Contact Hours
 11 weeks, 6 lectures including tutorials per week
 Lecturers
 Prof. Sergey Mozgovoy, Prof. Colm Ó Dúnlaing

Calculus with Applications for Scientists
The lecturer for this part will be Prof Sergey Mozgovoy.
 Learning Outcomes
 On successful completion of this module students will be able to
 Apply definite integrals to various geometric problems;
 Apply various methods of integration;
 The concept of a differential equations and methods of their solution;
 The concept of infinite series and their convergence; Taylor series;
 The concepts of parametric curves and polar coordinates,
 Module Content

 Application of definite integrals in geometry (area between curves, volume of a solid, length of a plane curve, area of a surface of revolution).
 Methods of integration (integration by parts, trigonometric substitutions, numerical integration, improper integrals).
 Differential equations (separable DE, first order linear DE, Euler method).
 Infinite series (convergence of sequences, sums of infinite series, convergence tests, absolute convergence, Taylor series).
 Parametric curves and polar coordinates.
Linear Algebera, Probablility & Statistics
The lecturer for this part will be Prof. Colm Ó Dúnlaing
 Learning Outcomes:
 Determinants: define, calculate by cofactor expansion and through upper triangular form.
 Use Cramer's Rule to solve linear equations.
 Use the Adjoint Matrix to invert matrices.
 Construct bases for row space, column space, and nullspace of a matrix.
 Construct orthonormal bases in three dimensions.
 Calculate the matrices of various linear maps.
 Compute linear and quadratic curves matching data through the least squared error criterion.
 Calculate eigenvalues and eigenvectors for 2x2 matrices, with applications to differential equations.
 Probability: derive distributions in simple cases.
 Solve problems involving the Binomial distribution.
 Use the Central Limit Theorem to approximate the binomial distribution for large n.
 Conditional probability: compute P(A_i  D) given P(DA_i).
 Use the Poisson distribution for trafficlight queuing problems.
 Calculate percentage points for continuous distributions: Normal, chisquared, and Student's tdistribution.
 Compute confidence intervals for mean and standard deviation.
 Formulate and decide simple hypotheses.
 Module Content:
 Determinants. Cramer's Rule. Adjoint matrix formula for inverse.
 Row space, column space, and nullspace of a matrix.
 Orthonormal bases in three dimensions.
 Linear maps and matrices.
 Least squared error linear and quadratic estimates.
 Eigenvalues and eigenvectorsfor 2x2 matrices. Systems of linear differential equations.
 Probability: uniform distribution, Binomial distribution, Poisson distribution.
 Conditional probability: compute P(A_i  D) given P(DA_i).
 Poisson distribution and trafficlight queuing problems.
 Normal distribution ; Central Limit Theorem.
 Chisquared, and Student's tdistribution: percentage points.
 Confidence intervals for mean and standard deviation.
 Basic hypothesis testing.
Essential References:
(Anton)
 Combined edition:
 Calculus: late transcendentals: Howard Anton, Irl Bivens, Stephen Davis 10th edition (2013) (Hamilton Library 515P23*9) Or
 Single variable edition.
(AntonRorres)
 Howard Anton & Chris Rorres, Elementary Linear Algebra with supplementary applications. International Student Version (10th edition). Publisher Wiley, c2011. (Hamilton 512.5L32*9;  5, SLEN 512.5 L32*9;615):
Recommended References:
(Kreyszig)
 Erwin Kreyszig, Advanced Engineerin
 Erwin Kreyszig, Advanced Engineering Mathematics (10th edition), (Erwin Kreyszig in collaboration with Herbert Kreyszig, Edward J. Normination), Wiley 2011 (Hamilton 510.24 L21*9)
(Thomas)
 Thomas' Calculus, Author Weir, Maurice D. Edition 11th ed/based on the original work by George B. Thomas, Jr., as revised by Maurice D. Weir, Joel Hass, Frank R. Giordano, Publisher Boston, Mass., London: Pearson/Addison Wesley, c2005. (Hamilton 515.1 K82*10;*)
 Module Prerequisite
 MA1S11 Mathematics for Scientist (First Semester)
 Assessment Detail
 This module will be examined in a 3 hour examination in Trinity term. Continuous assessment in the form of weekly tutorial work will contribute 20% to the final grade at the annual examinations, with the examination counting for the remaining 80%. For supplementals if required, the supplemental exam will count for 100%.