# 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)