## Trinity College Dublin, The University of Dublin

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# Module MAU11S02: Mathematics for Scientists (second semester)

Credit weighting (ECTS)
10 credits
Semester/term taught
Hilary term 2021-22
Contact Hours
11 weeks, 6 lectures and two tutorials per week

Lecturers
Prof. Colm Ó Dúnlaing Prof Miriam Logan

### Calculus with Applications for Scientists

The lecturer for this part will be Prof Miriam Logan

Learning Outcomes
On successful completion of this module students will be able to
• Solve definite integrals using a variety of methods of integration;
• Use integration to solve geometrical problems, such as finding volumes, areas and lengths;
• Evaluate improper integrals
• Formulate and solve first order differential equations;
• Determine if an infinite sequence converges or not;
• Test a series for convergence;
• Approximate a function by polynomials using Taylor and Maclaurin series;
Module Content
• Application of definite integrals in geometry (area between curves, volumne 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 fo sequences, sums of infinite series, convergence tests, absolute convergence, Taylor series).
• Parametric curves and polar coordinates.

• Single Variable Calculus 7th ed. Early Transcendentals by James Stewart.
• Calculus, 9th edition, Early Transcendentals by H. Anton, I. Bivens, S. Davis.

### Further linear algebra and statistics for Scientists

The lecturer for this part will be Prof. Colm Ó Dúnlaing

Module Content:
• Linear Algebra - This reference for this part of the course will be (AntonRorres). The syllabus will be approximately chapters 2, 5, section 4.2 and a selection of application topics from chapter 11 of (AntonRorres).
• Determinants, Evaluation by Row Operations and Laplace Expansion, Properties, Vector Cross Products, Eigenvalues and Eigenvectors;
• Introduction to Vector Spaces and Linear Transformations. Least Squares Fit via Linear Algebra;
• Differential Equations, System of First Order Linear Equations;
• Selected Application in Different Branches of Science;
• Probability - Basic Concepts of Probability; Sample Means; Expectation and Standard Deviation for Discrete Random Variables; Continuous Random Variables; Examples of Common Probability Distributions (binomial, Poisson, normal) (sections 24.1 - 24.3, 24.5 - 24.8 of (Kreyszig).

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-LE N 512.5 L32*9;6-15):

Recommended References:

(Kreyszig)