PH22010: Mathematics for physics 2
[Page last updated: 03 June 2024]
| Academic Year: | 2024/25 |
| Owning Department/School: | Department of Physics |
| Credits: | 10 [equivalent to 20 CATS credits] |
| Notional Study Hours: | 200 |
| Level: | Intermediate (FHEQ level 5) |
| Period: |
|
| Assessment Summary: | EXCB 50%, EXOB 50% |
| Assessment Detail: |
|
| Supplementary Assessment: |
|
| Requisites: |
Before taking this module you must take PH12003 OR take PH12062
While taking this module you must take PH22006 |
| Learning Outcomes: |
After taking this unit the student should be able to:
* define and transform between Cartesian, plane polar, cylindrical polar and spherical polar coordinates; * evaluate and interpret grad, ��2 and directional derivatives of scalar fields in the above coordinate systems; * evaluate and interpret div and curl of vector fields in the above coordinate systems; * use and interpret vector integral theorems; * evaluate Fourier series and transforms, and use their properties to solve problems; * apply Fourier techniques to problems in the physical sciences; * recognise and solve some of the key equations which arise in the natural sciences; * apply the separation of variables method to linear partial differential equations, and solve the resulting ordinary differential equations by series solution; * outline key properties of finite- and infinite-dimensional vector spaces. |
| Synopsis: | Vector calculus, Fourier analysis and methods to solve linear partial differential equations are important parts of the physicists mathematical toolbox. You will learn about these more advanced mathematical concepts and techniques, that will be applied in second year and higher-level Physics units, and further develop your problem-solving skills and understanding of mathematical results. |
| Content: | Vector calculus: Scalar and vector fields in Cartesian coordinates. Gradient and directional derivative of a scalar field, �� as a vector operator. Div and curl in Cartesian coordinates, their physical interpretation. Identities involving ��, definition of ��2. Tangential line integrals. Conservative fields and potential functions. Surface, flux and volume integrals in Cartesian coordinates. Orthogonal curvilinear coordinate systems. Plane polar coordinates; velocity and acceleration, equations of motion. Spherical polar and cylindrical polar coordinates. Line, surface and volume integrals in curvilinear coordinates. Grad, div, curl and ��2 in curvilinear coordinates; div and curl as limits of integrals. Meaning and uses of ��2. Vector integral theorems; Divergence, Stoke's and Green's theorems. Use of vector integral theorems.
Fourier series: Periodic functions. Harmonic synthesis. Representation as a Fourier series, Fourier components. Expansion of finite range functions. Applications of Fourier series. Complex form of Fourier series and coefficients. Discrete amplitude spectra.
Transition to aperiodic functions: The Fourier transform. Integral definition and properties of the Fourier transform. Use of tables in evaluating transforms. Dirac delta function. Convolution, sampling theorem. Uses and applications of Fourier techniques in the physical sciences.
Linear equations of science: Linear operators and linear PDEs. Classification of PDEs, examples of linear PDEs in the physical sciences. Superposition, boundary conditions. Solution by separation of variables in Cartesian, cylindrical and spherical coordinate systems. Fourier transform method for solving ODES/PDEs. Series solution of ODEs, recurrence relation. Convergence of infinite series solutions, and relevance to PDEs describing a physical situation. Ordinary and singular points. Bessel functions, Legendre polynomials and associated Legendre polynomials. Laplace's equation in spherical polar co-ordinates, spherical harmonics, and their use for representing functions defined on a sphere. Solving the Schroedinger equation for the hydrogen atom.
Matrices and vector spaces: Basis vectors, definition of a vector space, complex N-dimensional vector spaces. Inner product, norm. Linear operators acting on vectors. Symmetric, Hermitian, orthogonal, and unitary matrices. Eigenvectors and eigenvalues. Commutation. Infinite dimensional vector spaces, Hilbert space. Basis functions, Gram-Schmidt orthogonalization. Hermitian operators. Applications to quantum mechanics, Dirac notation revisited. |
| Course availability: |
PH22010 is a Must Pass Unit on the following courses:Department of Physics
PH22010 is Compulsory on the following courses:Department of Chemistry
PH22010 is Optional on the following courses:Department of Chemistry
|
Notes:
|