### Aim:

Algebra: In Algebra we'll study systems of linear equations, see how such systems arise in all kinds of engineering problems, and try to get a feel for why "row operations" and "matrix algebra" is so useful in solving these systems. We'll also use the geometric language of vectors, planes (and even hyper-planes) to study some general properties of systems of linear equations.

Calculus: In the Calculus section we'll digest concepts such as "limit", "function", "continuity", "derivative" and illustrate their relevance to engineering problems involving rates of change.

Textbook:

"(Modern) Engineering Mathematics" by Glyn James. There are copies in the bookshop. For just over 80 euros you can buy this book together with the book for Second Year.

### Lectures:

The lecture content won't be too different from that in 2008-2009. A pdf record of the 2008-2009 lectures is below.

Lecture 1: Systems of linear equations and Gaussian elimination. See Examples 5.30, 5.32, 5.33, 5.34 in the book for further related worked examples.

Lecture 2: Matrix multiplication and inversion applied to systems of linear equations. See examples 5.3, 5.4, 5.5, 5.25, 5.29 in the book for  further related examples.

Lecture 3: Using row operations to invert a square matrix. See examples 5.21, 5.22 in the book for an alternative approach to inverting matrices (using a formula). Try these examples with the "row operations method" to make sure you get the same answer. Unfortunately, the book does not cover the row operations approach to inverting matrices.

Lecture 4 and computer output: Electric circuits and systems of linear equations. See exercises 79, 80, 85 in Chapter 5 of the book for further examples of linear systems arising in real-life engineering problems.

Lecture 5: Expressing row operations as matrix multiplication.  Matrix algebra is used to verify that our "row operations method" for inverting a matrix really does always produce the correct answer.

Lecture 6: A taste of geometry. Rotations of the plane and reflections in a line in the plane can be represented by matrix multiplication. Not all transformations of the plane can be represented by matrix multiplication. (Warning: the last line on the last lecture slide is just plain wrong! T is one-one.) See Example 2.40 for an extra worked example on the sine function. See Examples 1.31, 1.32, 1.33, 1.34, 1.35, 1.36 for worked examples on cartesian coordinates in the plane.

Self Assessment: We've covered enough material for you to be able to tackle Question 2(a), Question 2(b) and all of Question 3 on the 2007 summer exam paper II. At this stage you could also have a go at Question 1 from this paper (though future lectures will give you more help here).

Lecture 7: First example of a limit: a derivative. See Examples 8.1(a), (b) and (c) for more worked examples of this kind of limit.

Lecture 8: Second example of a limit: an integral. You'll find more examples like this in Semester II of the course.

Lecture 9: The (rough) definition of a limit, and evaluation of limits. See Example 7.31(a)  for another related worked example.

Lecture 10: Limits of trigonometric functions. See Example 7.31(b) for another related worked example.

Lecture 11: The derivative of a function. Definition and worked examples. See Section 8.2 for more related material and worked examples.

Lecture 12: Limits at infinity, vertical and horizontal asymptotes, and how to sketch the graph of a function. See Examples 2.36 and 2.37 for more worked examples on asymptotes and graph sketching.

Lecture 13: Continuitinuous functions. Read Sections 7.91 and 7.92 for a more detailed account of continuous functions.

Lecture 14: Precise statement of the Intermediate Value Theorem. See Section 7.9.3 for real engineering application of the Intermediate Value Theorem  to locating zeros. (You might find Section 7.9.3 heavy going: in which case just skip it!)

Lecture 15: Three examples of the Intermediate Value Theorem: (i) hill climbing, (ii) atmospheric pressure, (iii) an old exam question.

Lecture 16: Determinant of a 2x2 matrix: definition and four basic properties.

Lecture 17: Determinant of a 3x3 (and nxn) matrix: four basic properties. For further worked examples see Examples 5.13, 5.14, 5.15, 5.16, and 5.17 in the book.

Lecture 18: Adjoint of a 3x3 (and nxn matrix). For further worked examples see Examples 5.18, 5.19, 5.20, 5.22 and 5.23 in the book.

Lecture 19: Review of: (i) transformations of the plane; (ii) lines and vector addition in the plane. (The term "yesterday" on the first slide actually refers to Lecture 6.)

Lecture 20: Lines, planes and hyperplanes in n-dimensional space. See section 4.3 in the book for more examples.

Lecture 21: Solving a system of n equations in m unkowns.

Lecture 22
Lecture 22(b): (Guest lecturer) Techniques of differentiation. See Section 8.3 in the book for more examples

Lecture 23 : (Guest lecturer) Techniques of differentiation. See Section 8.3 in the book for more examples.

Lecture 24 : Applications of differentiation to rates of change problems.

Lecture 25 : Applications of differentiation to optimization problems.

Lecture 26 : Applications of differentiation to curve sketching.

Lecture 27 : Maxima, minima, concavity and points of inflection.

Lecture 28 : Intuitive understanding of logarithms and exponentiation.

Lecture 29 : Rigorous treatment of logarithms.

Lecture 30 : Netwon's law for cooling objects and a differential equation.

Lecture 31 : Radioactive decay. Plus the formal definition of a derivative.

In Lectures 32,33,34 I will go over past examination questions.

In the last two lectures of the Semester - we will have two 50-minute class tests.

Here are some lectures on matrices, eigenvalues and eigenvectors.
Lecture 16

Lecture 17

Lecture 18

Lecture 19

See here for an explanation of how eigenvalues are used to determine the page rank of a web page in the Google search engine.

Lecture 20

Lecture 21

Lecture 22