Welcome to Engineering Mathematics 2

1. Welcome to Engineering Mathematics 2 SET Feedback Positives Negatives Talk more in seminars Make the content more relevant to engineering More interesting questions Improve the PPT slides Do revision at the start of class Provide more exercise questions Explain more Speak more slowly Draw attention to important points Seminar is boring More classes Clear Example sheets and handouts Patient and friendly PPT is easy to understand and the handout supports it Explains questions clearly Clear slides Patient Logical PPT Well organised Distributes handout before lecture

2. Exam Feedback Question a) Calculate the binomial expression of b) Using the expansion from part (a) and de Moivre’s representation Show that

3. Exam Feedback Answer a) 2 marks Therefore 2 marks

4. Exam Feedback Answer b) Therefore Let a = eiθ Hence 2 marks 2 marks 2 marks

5. Welcome to Engineering Mathematics 2 These are the topics we will cover this semester Determinants Evaluation of 2nd and 3rd order determinants. Properties. Vectors Vector Algebra: addition, subtraction, unit vectors, resolution into components, scalar product, vector product, scalar triple product. Forces, moments, work done. Applications to three dimensional problems in geometry. Differentiation of vectors. Definition of the vector differential operators grad, div and curl.

6. Welcome to Engineering Mathematics 2 These are the topics we will cover this semester Matrices and systems of equations Matrix algebra. Zero and identity matrices. Inverse matrix. Systems of linear equations. Solution by Gauss elimination Solution by row reduction methods and Jacobi iteration. Inversion matrices by Gauss-Jordan method. Eigenvalues and eigenvectors

7. Definitions A scalar quantity has a magnitude and satisfies the usual rules of arithmetic. Mass, temperature, volume, speed are examples of scalars. A vector quantity has magnitude and direction and requires special rules called Vector Algebra. Force, acceleration & velocity are examples of vectors. A free vector may be anywhere in space e.g. the wind, the tide. A bound vector has a point of application e.g. a force applied to a particular point on a body. A vector is uniquely defined by either (a) its three components with respect to an orthogonal set of axes, or (b) its magnitude and its direction.

8. Vector Notation A vector is written a or . The vector along the line joining the point A to point B with magnitude AB is written . Let us assign an (O, x, y, z) reference frame. Then there are two alternative choices for constructing a set of three numbers that specify the same vector. Let the vector be denoted by r. Where, the subscripts 1, 2 & 3 correspond to the x, y & z directions respectively. r1 = a, r2 = b, r3 = c

9. Vector Notation Secondly, we can describe the vector in terms of its direction cosines. We write Where and i.e.

10. Vector Revision Direction Cosines

11. Vector Revision The length of the vector is easily described in terms of a, b & c The result is that we can write a vector in the following ways Thus we can easily deduce that Which are all equivalent Hence, if we know two direction cosines then we can easily deduce the third

12. Conclusion Essential reading for next week HELM Workbook 9.1 Basic Concepts of Vectors HELM Workbook 9.2 Cartesian Components of Vectors We have covered 4topics today 1. Exam Feedback2. Vector Revision 3. Definitions 4. Vector Notation