Department: PhysicsModule co-ordinator: Dr. Kate LancasterCredit value: 20 creditsCredit level: CAcademic year of delivery: 2022-23

- See module specification for other years: 2021-22

Occurrence | Teaching period |
---|---|

A | Autumn Term 2022-23 to Summer Term 2022-23 |

Mathematics is fundamental to the study of Physics. This course aims to introduce the concepts of calculus, complex numbers, vectors, linear algebra and statistics. Links will be made to A-Level Mathematics and areas of Physics to be studied.

- Determine local and universal maximum and minimum values of functions
- Differentiate functions using the chain rule, product rule, quotient rule, and Leibnitz's differentiation formula
- Definition and algebraic manipulation of hyperbolic functions
- Integrate functions using standard solutions, substitutions, and integration by parts.
- Define the terms arithmetic and geometric series
- Calculate the limit of an infinite arithmetic and geometric progression
- Use l’Hôpital’s Rule to calculate the limit of a function
- Obtain the Maclaurin expansion of a function by repeated differentiation
- Obtain the Taylor expansion of a function about a given value
- Recognise and manipulate with confidence the three standard forms of a complex number
- Obtain complex solutions of equations, in particular z^n = a where a is complex
- Interpret operations involving complex numbers in the Argand Diagram
- Solve first order differential equations by separation of variables or finding an integrating factor.
- Solve homogeneous and non-homogeneous second order differential equations
- Distinguish between a vector and a scalar
- Algebraic manipulation of vectors including addition, subtraction, and multiplication by a scalar.
- Recall the ideas and use of unit vectors, calculate the magnitude of a vector, and use a position vector to specify a point
- Express a vector in terms of Cartesian components referred to axes defined by unit vectors i, j, k and to relate the components to direction cosines.
- Evaluate the scalar and vector product of two vectors.
- Evaluate scalar triple products and expand the vector triple product ax(bxc)
- Differentiate vector functions with respect to a scalar parameter and describe the application of this in areas of physics such as dynamics
- Derive the vector equations of the straight line and the plane
- Apply the above ideas to the solutions of problems in algebra and geometry and to begin to appreciate the power of vector methods in physical applications.
- Calculate determinants and express vector products in determinant form
- use Cramer's rule to solve linear equations
- Define various types of matrix and be able to multiply, divide, add and subtract matrices.
- Solve linear simultaneous equations using matrix methods.
- Solve sets of homogeneous and non-homogeneous equations.
- Define the rank of a matrix and understand its significance.
- Be able to define linear dependence and independence and be able to test for these conditions.
- Understand the concepts and properties of eigenvalues and eigenvectors.
- Obtain all the partial derivatives of a function f(x,y,z...) up to at least the second order
- Apply the chain rule of partial differentiation
- Change variables in an expression containing partial derivatives
- Expand a function of one or two variables as a power series about any given point
- Evaluate a double integral analytically by straight- forward integration
- Evaluate a double integral analytically by changing the order of integration
- Evaluate a double integral analytically by changing the variables
- Use cylindrical polar coordinates or spherical polar coordinates in appropriate circumstances
- Express and evaluate a triple integral in terms of the Cartesian, cylindrical polar and spherical polar coordinate systems.

**Syllabus**

*Differentiation:* Function of a function; product rule; maxima and minima (local and global); expressions involving hyperbolic functions.

*Sequences and series:* Arithmetic, geometric, series; infinite series; ratio test for the infinite series; interval of convergence techniques for summing series; Maclaurin and Taylor expansion of well-known functions; nature of Taylor approximations and their use.

*Hyperbolic functions:* Definitions; relationship with circular functions; hyperbolic form.

*Integration:* Partial fractions; by parts; substitution (algebraic, trig and hyperbolic); completing the square; integration of powers and products of trig. functions; standard integrals; reduction formulae; substitution of hyperbolic functions; integrals as mean value and rms value.

*Limits:* Taking limits at x = 0 and x expansions Hopitals rule.

*Complex numbers:* Argand diagram; real and imaginary parts; modulus and argument; complex exponentials, de Moivres theorem, complex conjugate; nth roots; division by a complex number (rationalisation); cos and sin as complex exponentials.

*First order differential equations:* Separation of variables; change of variable; integrating factor; examples: charging and discharging a capacitor, radioactive decay.

*Second order differential equations:* Solution of homogeneous equation, e.g. exponential decay and SHM; complementary function; particular integral; examples: driven, damped, harmonic oscillator, resonance.

*Vector algebra:* Vectors in three dimensions; Cartesian components; unit vectors; dot and cross products; scalar and vector triple products; linear dependence; normal vector; lines and planes; differentiation of vectors.

*Linear algebra:* Matrices, Determinants, cross product, triple products. Matrix operations, Linear simultaneous equations, Cramers rule

*Matrices:* basic operations, addition, multiplication, division. Determination of the inverse matrix via Gauss elimination and calculation of the cofactor matrix. Determinants of a 2x2 and 3x3 linear matrix, the rank of a matrix, linear dependence.

*Matrix applications:* linear simultaneous equations. Eigenvalues and eigenvectors of a matrix.

*Differentiation of functions of several variables:* Partial differentiation; geometric interpretation; higher and mixed derivatives; total differential; function of a function; chain rule; differential operators; change of variables; Taylor series for functions of two variables.

*Integration of functions of several variables:* Double integrals; geometric interpretation; change of order of integration; change of variables between Cartesian and polar; multiple integrals; physical dimensions in integrals.

*3D coordinate systems:* Cartesian, cylindrical polar and spherical polar coordinate systems.

Task | Length | % of module mark |
---|---|---|

Closed/in-person Exam (Centrally scheduled)Maths I spring exam |
1.5 hours | 40 |

Closed/in-person Exam (Centrally scheduled)Maths I summer exam |
1.5 hours | 40 |

Essay/courseworkMaths I assignment T1 |
N/A | 10 |

Essay/courseworkMaths I assignment T2 |
N/A | 10 |

None

Task | Length | % of module mark |
---|---|---|

Closed/in-person Exam (Centrally scheduled)Maths I spring exam |
1.5 hours | 50 |

Closed/in-person Exam (Centrally scheduled)Maths I summer exam |
1.5 hours | 50 |

Our policy on how you receive feedback for formative and summative purposes is contained in our Department Handbook.

Stroud K A: *Engineering Mathematics*, 5th Ed 2001 (Palgrave) ****

James G: Modern *Engineering Mathematics*, 4th Ed 2010 (Pearson) ****

Mary L. Boas: *Mathematical Methods in the Physical Sciences*, Wiley ***

K F Riley & M P Hobson: *Foundation Mathematics*, Cambridge University Press 2011 **

Jenny Olive: *Maths; A Students Survival Guide*, Cambridge University Press **

**Preparatory Reading and Lecture Notes: Term 1 **

The term 1 course is very well covered in the book by Stroud. It will assume that you know the Foundation Topics in Part I. Chapters F11 and F12 on Differentiation and Integration are especially important.

Almost all the course is at the level of Part II of Stroud s book, and covers the following Chapters: 1 and 2 (Complex numbers), 3 (Hyperbolic functions), 6 (Vectors), 7 and 9 (Differentiation), 13 and 14 (Series), 15 and 17 (Integration), 24 and 25 (Differential equations).

The book by James also covers the course well, and comes with the online course MyMathLab if you buy it new. Pearson offers a discount on this book if you also buy Young and Freedman s University Physics.

The book by Boas is more advanced - it is a recommended text for Year 2 Mathematics, and has some sections that are used in Year 1.