# KC5028 - Advanced Mathematics for Physics

## What will I learn on this module?

The module is designed to provide you with the advanced mathematical and statistical techniques required to underpin study of physics at level 5 and beyond. Techniques covered will include Matrices, Fourier Series and Fourier and Laplace Transforms, Probability distributions, and an introduction to vector calculus (including div, grad and curl).

Students will develop skills in the use of advanced mathematical and statistical techniques, applying suitable mathematical calculations over a range of key topics, including explaining how a periodic waveform can be represented as an infinite series of sinusoids, and applying Fourier Transforms. The concepts of the eigenvalue and eigenvectors of a matrix, and how these can be found by algebraic means will also be covered. Finally, students will be introduced to vector calculus and vector operators, including div, grad and curl, and the Kronecker delta and Levi-Civita epsilon.

Linear Algebra
Algebraic evaluation of the eigenvalues and eigenvectors of a matrix (i.e. Matrices to the level of eigenvalues and eigenvectors). Application to the solution of a system of linear ordinary differential equations.

Vector Calculus
Coordinate systems; line, surface and volume integrals; Vector operators Grad, Div and Curl; Gauss’ (Divergence) Theorem, Stokes’ Theorem; Introduction to Cartesian tensors. Applications of vector calculus.

Fourier Series and Fourier and Laplace Transforms
Fourier series and periodic functions. Full-range and half-range series. Even and odd functions. Coefficients in complex form. Application to the solution of partial differential equations by the method of separation of variables. Fourier Transforms. Laplace Transforms. The convolution theorem. An introduction to the solution of partial differential equations.

Probability Distributions
Sample space, types of events, definition of probability, addition and multiplication laws, conditional probability. Discrete probability distributions including Binomial, Poisson. Continuous probability distributions including the Normal distribution.

### How will I learn on this module?

A wide range of learning and teaching approaches are used in this module. The module will be delivered using a combination of lectures, set work and skills periods in which you will be able to obtain help with problems associated with the module. Lectures allow you to experience and understand the formalism of advanced mathematical techniques and include relevant examples. You will have an opportunity to enhance your understanding of the subject through seminars which promote independent learning and tackle relevant problems. You will be provided with formative feedback on problems in seminars and have the opportunity to problem solve within peer groups. The mathematical rigour associated with this module naturally increases your potential for employability and is a highly transferable skill.

The module will be assessed by coursework (30%) and formal examination (70%). Exam feedback will provided individually and also generically to indicate where the cohort has a strong or a weaker answer to examination questions. Written feedback will be provided on the coursework. Formative feedback will be provided during the seminars.

Formative feedback will be provided on seminar work which will include problems designed to aid your learning and understanding.

Written feedback will be provided on the coursework. Exam feedback will provided individually and also generically to indicate where the cohort has a strong or a weaker answer to examination questions.

### How will I be supported academically on this module?

In addition to direct contact with the module team during lectures and seminars, you will be encouraged to develop your curiosity by making direct contact with the module team either via email or the open door policy operated throughout the programme. You will also be regularly referred to supporting resources including relevant texts and relevant multimedia materials.

References to these resources will be made available through the e-learning portal and in lectures and seminars.

### What will I be expected to read on this module?

All modules at Northumbria include a range of reading materials that students are expected to engage with. The reading list for this module can be found at: http://readinglists.northumbria.ac.uk

### What will I be expected to achieve?

Knowledge & Understanding:
1. Determine the eigenvalues and eigenvectors of a matrix and use them to solve a system of linear ordinary differential equations.
2. Determine the Fourier Series of a periodic waveform, and apply and obtain Fourier Transforms.
3. Apply vector calculus and manipulate Cartesian tensors.
4. Apply appropriate statistical methods to statistics problems.

Intellectual / Professional skills & abilities:

Personal Values Attributes (Global / Cultural awareness, Ethics, Curiosity) (PVA):

### How will I be assessed?

SUMMATIVE
1. Coursework (30%) – 1, 2, 3, 4
2. Examination (70%) – 1, 2, 3, 4

FORMATIVE
1. Seminars – 1, 2, 3, 4

Students will be assessed by coursework (30%) and a formal examination (70%). The coursework will provide an opportunity for the student to demonstrate knowledge of the Fourier and Laplace Transforms topics. The examination will cover all topics from the module.

Formative feedback will be provided on seminar work which will include problems designed to aid student understanding. Written feedback will be provided on the coursework.

None

None

### Module abstract

The module is designed to provide you with the advanced mathematical and statistical techniques required to underpin study of physics at level 5 and beyond. You will develop your abilities in the utilisation of advanced mathematical and statistical techniques, applying suitable mathematical calculations over a range of key topics, including explaining how a periodic waveform can be represented as an infinite series of sinusoids, and applying Fourier Transforms. We will cover the concepts of the eigenvalue and eigenvectors of a matrix, and how these can be found by algebraic means. Finally, you will be introduced to vector calculus and vector operators, including div, grad and curl, and the Kronecker delta and Levi-Civita epsilon. Assessment of the module is by one class test (30%) and one formal examination (70%) and the module will be delivered using a combination of lectures and seminars.

### Course info

UCAS Code F3F5

Credits 20

Mode of Study 3 years full-time or 4 years with a placement (sandwich)/study abroad

Department Mathematics, Physics and Electrical Engineering

Location City Campus, Northumbria University

City Newcastle

Start September 2024 or September 2025

## Physics with Astrophysics BSc (Hons)

All information is accurate at the time of sharing.

Full time Courses starting in 2023 are primarily delivered via on-campus face to face learning but may include elements of online learning. We continue to monitor government and local authority guidance in relation to Covid-19 and we are ready and able to flex accordingly to ensure the health and safety of our students and staff.

Contact time is subject to increase or decrease in line with additional restrictions imposed by the government or the University in the interest of maintaining the health and safety and wellbeing of students, staff, and visitors, potentially to a full online offer, should further restrictions be deemed necessary in future. Our online activity will be delivered through Blackboard Ultra, enabling collaboration, connection and engagement with materials and people.

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