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Module MA1S12: Mathematics for Scientists (second semester)

Credit weighting (ECTS)
10 credits
Semester/term taught
Hilary term 2017-18
Contact Hours
11 weeks, 6 lectures including tutorials per week
Lecturers
Prof. Colm Ó Dúnlaing
 

Calculus with Applications for Scientists

The lecturer for this part will be

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(D|A_i).
  • Use the Poisson distribution for traffic-light queuing problems.
  • Calculate percentage points for continuous distributions: Normal, chi-squared, and Student's t-distribution.
  • 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(D|A_i).
  • Poisson distribution and traffic-light queuing problems.
  • Normal distribution ; Central Limit Theorem.
  • Chi-squared, and Student's t-distribution: 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, S-LEN 512.5 L32*9;6-15):

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%.