# Module MA1S11: Mathematics for Scientists (first semester)

Credit weighting (ECTS)
10 credits
Semester/term taught
Michaelmas term 2013-14
Contact Hours
11 weeks, 6 lectures including tutorials per week
Lecturers
Prof. Tristan McLoughlin, Prof. Vladimir Dotsenko
Learning Outcomes
On successful completion of this module students will be able to
• Manipulate vectors to perform alegebraic operations on them such as dot products and orthogonal projections and apply vector concepts to manipulate lines and planes in space $\mathbb{R}^3$ or in $\mathbb{R}^n$ with $n \geq 4$.
• Use Gaussian elimination techniques to solve systems of linear equations, find inverses of matrices and solve problems which can be reduced to such systems of linear equations.
• Manipulate matrices algebraically and use concepts related to matrices such as invertibility, symmetry, triangularity, nilpotence.
• Manipulate numbers in different bases and explain the usefulness of the ideas in computing.
• Use computer algebra and spreadsheets for elementary applications.
• Explain basic ideas relating to functions of a single variable and their graphs such as limits, continuity, invertibility, even/odd, differentiabilty and solve basic problems involving these concepts.
• Give basic properties and compute with a range of rational and starndard transcendental functions, for instance to find derivatives, antiderivatives, critical points and to identify key features of their graphs.
• Use a range of basic techniques of integration to find definite and indefinite integrals.
• Apply techniques from calculus to a variety of applied problems.
Module Content

The content is divided in two sections, one for each lecturer.

### Calculus with applications for Scientists

The lecturer for this part will be Prof. Vladimir Dotsenko. See http://www.maths.tcd.ie/~vdots/teaching/1S11-1314.html for additional information about this part. The main textbook will be [Anton] and the syllabus will be approximately 7 Chapters of [Anton] (numbered differently depending on the version and edition)

• Before Calculus (9th ed) {was `Functions' in the 8th edition};
• Limits and Continuity;
• The Derivative;
• The Derivative in Graphing and Applications;
• Integration;
• Exponential, Logarithmic and Inverse Trigonometric Functions;

### Discrete Mathematics for Scientists

The order of the topics listed is not necessarily chronological. Some of the topics listed below linear algebra will be interspersed with linear algebra.

• Linear algebra The syllabus for this part will be approximately chapters 1, 3 and parts of 10 from [AntonRorres].
• Vectors, geometric, norm, vector addition, dot product
• Systems of linear equations and Gauss-Jordan elimination;
• Matrices, inverses, diagonal, triangular, symmetric, trace;
• Selected application in different branches of science.
• Computer algebra.

An introduction to the application of computers to mathematical calculation. Exercises could include ideas from calculus (graphing, Newton's method, numerical integration viatrapezoidal rule and Simpsons rule) and linear algebra. We will make use fo the computational software Mathematica which is used in many scientific applications.

• Numbers. An introduction to numbers and number systems e.g. binary, octal and hexadecimal numbers and algorithms for converting between them.

Essential References

[Anton]
Combined edition :
Calculus : late transcendentals : Howard Anton, Irl Bivens, Stephen Davis 10th edition (2013) (Hamilton Library 515 P23*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 512.5 L32*9;-5, S-LEN 512.5 L32*9;6-15]
Module Prerequisite
None
Assessment Detail
This module will be examined in a 3 hour examination in Trinity term. Assignments and tutorial work will count for 20% of the marks. There will be final examination in April/May counting for the remaining 80%. For supplementals, if required, the supplemental exam will count for 100%.