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| Unit name |
Mathematics 1AM |
| Unit code |
MATH10100 |
| Credit points |
40 |
| Level of study |
C/4
|
| Teaching block(s) |
Teaching Block 4 (weeks 1-24)
|
| Unit director |
Professor. Porter |
| Open unit status |
Open |
| Pre-requisites |
A-level in Mathematics or equivalent. |
| Co-requisites |
None |
| School/department |
School of Mathematics |
| Faculty |
Faculty of Science |
Description including Unit Aims
This unit consists of calculus, computational mathematics, linear algebra and multivariable calculus. This differs from MATH10300, Mathematics 1AS by teaching further mathematics instead of statistics. There are regular tutorials, computational practical sessions and assignments.
Aims:
To introduce and develop skill in the mathematics needed to study the sciences at degree level.
Syllabus
Calculus 1, 34 lectures, weeks 1 - 12; Dr R.Porter
The numbers of lectures (shown in brackets) are a rough guide only.
- General introduction, Review of algebra and trigonometry. (2)
- Functions and graphs: important examples, inverse functions. (2)
- Sequences and series; limits of functions; continuous functions (3)
- Exponential function; natural logarithm; hyperbolic functions (2)
- Complex numbers; Argand diagram, polar form, complex exponential, complex roots (4)
- Differential calculus, differentiability, basic methods, higher derivatives, #Leibniz formula; differentiation of inverse functions (3)
- 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)
- Integration: integrals as antiderivatives and as area; standard techniques; infinite integrands; infinite ranges of integration. (4)
- Differential equations: 1st-order separable and first order linear differential equations. (2)
- 2nd order linear differential equations with constant coefficients, homogenous including simple harmonic motion, inhomogeneous including resonance. (4)
- Full-range Fourier series in [-pi, pi] and general intervals. (4)
Linear algebra, 16 lectures, weeks 13 -18; Dr. M. Zaturska
- Matrices and vectors. Definition and motivation. What are they good for?
- Vectors. Addition and scaling, linear independence, bases. Dot product. Orthonormal sets. Cross product.
- Matrices. Basic algebra, inverses. Determinants, geometrical interpretation, calculation of determinants.
- Systems of linear equations. The geometry of solutions.
- 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. M. Zaturska
The numbers of lectures (shown in brackets) are a rough guide only.
- Functions of two variables: contours, sections. Contour surfaces of functions of three variables. (4)
- Partial derivatives: first and second derivatives; directional derivative; chain rule (5)
- Approximations using first order partial derivatives; tangent plane: normal to the surface f = constant. (2)
- Taylor's theorem for two variables: vector form, maxima and minima. (3)
- Implicit functions and derivatives. (1)
- Line integrals over line segments and arcs of circles. (3)
- Double integrals: change of order of integration. (3)
Intended Learning Outcomes
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,
- basic familiarity with vectors and matrices, including eigenvalues and eigenvectors,
- the ability to work with functions of two variables, and their derivatives and integrals.
Transferable Skills:
Mathematical techniques for application in the physical sciences.
Teaching Information
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.
Assessment Information
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 13th January 2012, and the Maths 1AM examination may be on Friday 13th January or Saturday 14th 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.
Reading and References
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.
- 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. There are many similar textbooks in the library.
- 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.
- 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.
- 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.
- 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..
- Jeffrey, A., Essentials of engineering mathematics, Chapman & Hall, London
Similar in style to the previous book, though with slightly less extensive coverage.
- 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)
