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Mathematics 1AM (MATH 10100)

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Contents of this document:

Administrative Information

  1. Unit number and title: MATH 10100 Mathematics 1AM
  2. Level: C/4 (Open)
  3. Credit point value: 40 credit points
  4. Year: 10/11
  5. First Given in this form: 1989/90 (broadly in its current form, but under the name Mathem
  6. Unit Organiser: Richard Porter
  7. Lecturer: Dr. R Porter (Calculus 1, weeks 1 - 12), Dr. J. Elmer (Linear Algebra, Calculus 2, weeks 12 - 23)
  8. Teaching block: 1 and 2
  9. Prerequisites: A-level Mathematics, or equivalent.

Unit aims

To introduce and develop skill in the mathematics needed to study the sciences at degree level.

General Description of the Unit

The unit begins with some basic ideas revising and extending school-level calculus, and then goes on to a thorough treatment of the calculus from the point of view of scientific applications. The subject is developed as far as differential equations and Fourier series. The mathematics is treated with enough logical precision to enable correct calculations and correct deductions to be made.

The linear algebra section deals with vectors, matrices and eigenvalues (which are fundamental to atomic physics, molecular chemistry, computational graphics, and many other branches of science and engineering).

The Calculus 2 section picks up and develops some ideas from earlier in the unit, and then proceeds to extend the ideas and methods of calculus to functions of more than one variable. (This material is essential for physics and chemistry: electromagnetic fields and quantum wave functions are functions of several variables.)

Teaching Methods

The unit is based on lectures supported by problems classes and tutorials on how to apply the techniques in solving problems.

The lecturers will distribute problems sheets based on the work done in lectures, and they will set specific problems which you will be required to hand in. During the first few weeks  problems classes will be held and work handed in centrally. Later the problems classes will be replaced by weekly tutorials and work will be habded in to tutors for marking.

Experience shows that progress in mathematics depends crucially on regular work at examples. For this reason you are REQUIRED to attend tutorials and to hand in the set work. See the section Formal Requirements of the Unit below.

Tutorials

Weekly tutorials will be held after the first few weeks. You will be given the time of your tutorial .

 

 

Learning Objectives

After taking this unit, students should have:

  • a good understanding of single-variable calculus, as far as Taylor series,
  • techniques for solving simple differential equations and working with Fourier series,
  • a basic familiarity with vectors and matrices, including eigenvalues and eigenvectors,
  • the ability to work with functions of two variables, and their derivatives and integrals.

Assessment Methods

The final mark for Mathematics 1AM is made up as follows:

  • 10% from a 1 1/2 hour examination in Calculus 1 in January,
  • 40% from a 3-hour examination in Calculus 1 in May/June,
  • 50% from a 3-hour examination in Calculus 2 and Linear Algebra in May/June.

More information is given below.

Use of Calculators and Notes

Candidates may bring into the examination room a calculator of the approved type (non-programmable, no text facility).

Candidates may bring into the examination room one double-sided, A4-sized sheet of notes.

Details of the Summer Examination

Paper 1 (3 hours) is on the Calculus 1 material. The paper is in two sections.

  • Section A has 10 short questions, ALL of which should be answered; it carries 40% of the marks for this paper.
  • Section B has 6 longer questions, of which you should do FOUR. If you do more than four, your best four answers from this section will be used for assessment. Section B carries 60% of the marks for this paper.

Paper 2 (3 hours) is in two sections.

  • Section A has 10 short questions, 5 on Linear Algebra and 5 on Calculus 2. ALL of these questions should be answered. Section A carries 40% of the marks for this paper.
  • Section B has 3 longer questions on Linear Algebra. You should do TWO questions from section B; if you do more than two, your best two answers will be used for assessment. Section B carries 30% of the marks for this paper.
  • Section C has 3 longer questions on Calculus 2. You should do TWO questions from section C; if you do more than two, your best two answers will be used for assessment. Section C carries 30% of the marks for this paper 

January examinations

The January examinations are right at the start of the second term. This term begins on Friday 14th January 2010, and the Maths 1AM examination may be on Friday 14th January or Saturday 15th January. IT IS YOUR RESPONSIBILITY to ensure that you are in Bristol to sit the examination; otherwise your mark will be zero (unless you have a certified illness or other special circumstances of which the department has been notified). You will be notified of the date, time and place of the January examination before the end of the first term.

The January examination paper (1 1/2 hours) contributes 10% to your overall mark and consists of two sections.

  • Section A has 5 short questions, ALL of these questions should be answered. Section A carries 40% of the marks for this paper.
  • Section B has 3 longer questions. You should do TWO questions from section B; if you do more than two, your best two answers will be used for assessment. Section B carries 60% of the marks for this paper.

September examinations

If you fail this unit in June, you may (depending on which Faculty you are in and how you have done in your other units) be allowed to resit it in September. The September examination papers have the same structure as in June. If you are offered a resit, you must take the resit examination.

Award of Credit Points

Formal requirements of the unit

You must gain 120 credit points each year in order to be allowed to continue on your degree programme. There are also conditions on how much you must pass: details will be available from your Faculty Handbook or Departmental Handbook. We explain below what you must do  (1) to pass this unit, and (2) to gain the 40 credit points for this unit.

Note: we will make allowances for illness and other such good reasons, PROVIDED that you inform the unit organiser as soon as possible AND that you provide proper written documentation of your case (e.g. a doctor's certificate, specifying the date(s) you were unable to undertake academic work).

(1) Passing the unit

You will pass if your assessment mark is 40 or more. The assessment mark is calculated as described in the Assessment section above.

(2) Gaining credit points for the unit

You will be awarded the 40 credit points for the unit if

  • either you pass the unit with a mark 40 or over,
  • or you score a mark between 30 and 39 inclusive, and you
  • attended at least 75% of the tutorials;
  • made a serious attempt in at least 75% of the weekly homework assignments.

Transferable Skills

Mathematical techniques for application in the physical sciences.

Texts

Recommended Texts:

The following book is recommended, but it is not essential.

Jordan, D.W. & Smith, P. Mathematical Techniques: An introduction for the engineering, physical, and mathematical sciences (4th edition), Oxford University Press, Oxford, 2008.

Supplementary Booklist

These are alternative texts. They should be available in the library, and you may find them useful in different ways, as discussed below.

  1. Stewart, J., Calculus - Early Transcendentals, Brooks/Cole
    A very clearly written and comprehensive introduction to calculus, going beyond the Maths 1AM course. Includes vectors but not matrices. Recommended - if you can afford it. There are many similar textbooks in the library.
  2. Gilbert, J. and Jordan, C., Guide to Mathematical Methods, Palgrave (Macmillan) 2002.
    Introduces topics in a fairly elementary way, but does not cover all the material.
  3. Berry, J., Northcliffe, A., & Humble, S., Introductory mathematics through science applications, Cambridge University Press, Cambridge.
    Introduces topics in a fairly elementary way. May be useful if you feel you need to strengthen your basic skills.
  4. Boas, M.L., Mathematical methods in the physical sciences, Wiley,
    Useful for the second-year physics course: you may find it too demanding at the beginning of the 1AM course.
  5. Jeffrey, A., Mathematics for engineers and scientists, Chapman & Hall, London
    Covers most of the syllabus, and a good deal more besides, in a terse style..
  6. Jeffrey, A., Essentials of engineering mathematics, Chapman & Hall, London
    Similar in style to the previous book, though with slightly less extensive coverage.
  7. Spiegel, M.R., Schaum's outline of theory and problems of vector analysis, McGraw-Hill, New York.
    Brief but lucid explanations of the theory, with many worked examples and problems. (American)

Syllabus

Calculus 1, 34 lectures, weeks 1 - 12; Dr R.Porter

The numbers of lectures (shown in brackets) are a rough guide only.

  1. General introduction, Review of algebra and trigonometry. (2)
  2. Functions and graphs: important examples, inverse functions. (2)
  3. Sequences and series; limits of functions; continuous functions (3)
  4. Exponential function; natural logarithm; hyperbolic functions (2)
  5. Complex numbers; Argand diagram, polar form, complex exponential, complex roots (4)
  6. Differential calculus, differentiability, basic methods, higher derivatives, Leibniz formula; differentiation of inverse functions (3)
  7. Taylor approximations; Taylor series; convergence of the series; ratio test for power series; applications of Taylor series: maxima and minima; l'Hospital's rule for limits (4)
  8. Integration: integrals as antiderivatives and as area; standard techniques; infinite integrands; infinite ranges of integration. (4)
  9. Differential equations: 1st-order separable and first order linear differential equations. (2)
  10. 2nd order linear differential equations with constant coefficients, homogenous including simple harmonic motion, inhomogeneous including resonance. (4)
  11. Full-range Fourier series in [-pi, pi] and general intervals. (4)

Linear algebra, 16 lectures, weeks 13 -18; Dr. J. Elme

  1. Matrices and vectors. Definition and motivation. What are they good for?
  2. Vectors. Addition and scaling, linear independence, bases. Dot product. Orthonormal sets. Cross product.
  3. Matrices. Basic algebra, inverses. Determinants, geometrical interpretation, calculation of determinants.
  4. Systems of linear equations. The geometry of solutions.
  5. Eigenvalues, calculation for 2 x 2 and 3 x 3 case by the characteristic equation. Completeness of eigenvectors. Eigenvalues and eigenvectors of symmetric matrices. Applications.

Calculus 2, 20 lectures, weeks 19 - 23; Dr. J. Elmer

The numbers of lectures (shown in brackets) are a rough guide only.

  1. Functions of two variables: contours, sections. Contour surfaces of functions of three variables. (4)
  2. Partial derivatives: first and second derivatives; directional derivative; chain rule (5)
  3. Approximations using first order partial derivatives; tangent plane: normal to the surface f = constant. (2)
  4. Taylor's theorem for two variables: vector form, maxima and minima. (3)
  5. Implicit functions and derivatives. (1)
  6. Line integrals over line segments and arcs of circles. (3)
  7. Double integrals: change of order of integration. (3)

Advice for Students

This is a 40 credit-point unit, which means that you should spend a third of your working time on it. The university expects students to work roughly 40 hours a week, which means 13 or 14 hours per week throughout the academic year spent on mathematics. Some students may need to spend more time than this in order to master the subject, some may need less.

Remember that you may not be allowed to remain in the University unless you work well enough to obtain 40 credit points from this course; see the section Formal Requirements of the Unit below for more details.

Studying mathematics is different from many other subjects. Mathematicians use words in a way different from everyday life: mathematical terms have very precisely defined meanings, which may sometimes take a bit of work to understand fully. You must read your texts and lecture notes very carefully, thinking about the meaning of every word and every symbol until you have it all clear in your mind. If you just skim instead of reading carefully, the subject will soon become a vague blur in your mind, and you will not make much progress.

Another feature of mathematics is the way that each topic builds on knowledge of the previous material. If you don't have a really firm grasp of earlier material, you will not be able to grasp later parts of the course.

It is essential to practice doing mathematics, in order to build speed and confidence in mathematical techniques; that is an important part of this unit. Think of it as like training for a sport: if you don't do training and practice, you can't expect to perform well. We provide you with exercises to do; you should work them carefully. If you get stuck with a question, it is helpful to go back to your lecture notes and textbook: make sure that you understand the mathematical ideas behind the question, and then look for worked examples that might help.

We encourage you to discuss the exercises with other students - working in small groups is fun, and is a good way of learning. But the work that you hand in should be your own write-up, even if some of the ideas were generated in group discussion.

Tutorials and, to a lesser extent, problems classes give you a chance to discuss detailed difficulties you have in working the exercises or understanding the material from the lectures. You can also discuss more general issues of how to approach studying mathematics: the kind of issue discussed in the previous few paragraphs. Tutorials are an essential part of our teaching - make the most of them, by bringing up points for discussion which will help you in mastering the course.

Calculators

Don't feel that you have to buy a graphics calculator - they are not essential, and graphic calculators may not be used in the examination. An ordinary scientific calculator (cost from £6) is sufficient, but you should check that it satisfies our requirements on calculators used in examinations.