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Programme & Unit Catalogues

## PH20019: Mathematics for scientists 3

Credits: 6
Level: Intermediate (FHEQ level 5)
Period: Semester 1
Assessment: EX 100%
Supplementary Assessment: PH20019 - Mandatory Extra Work (where allowed by programme regulations)
Requisites: Before taking this unit you must take PH10008 and take PH10007
Description: Aims:
The aim of this unit is to introduce mathematical concepts and techniques required by science students, and to show how these may be used for different applications. It also aims to continue the development of students' problem-solving skills and their understanding of mathematical results.

Learning Outcomes:
After taking this unit the student should be able to:
* find the eigenvalues and eigenvectors of matrices;
* calculate the normal modes of coupled vibrational systems;
* calculate and interpret derivatives of vector functions of 1 variable;
* parameterise curves;
* define and transform between Cartesian, plane polar, cylindrical polar and spherical polar coordinates;
* visualise points, lines, planes and volumes in these coordinates;
* define scalar, vector and conservative fields;
* 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;
* evaluate and interpret line, surface and volume integrals in the above coordinate systems;
* identify conservative fields and find their potential functions;
* use and interpret vector integral theorems.

Skills:
Numeracy T/F A, Problem Solving T/F A.

Content:
Eigenvalues and eigenvectors (6 hours): Revision of matrix algebra. Eigenvalues and eigenvectors of symmetric/Hermitian matrices and their properties. Linear transformations. Normal modes of ball and spring systems.
Vector calculus (16 hours): Differentiation of vectors. Space curves; parameterisation of curves, unit tangent vector. 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, 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 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.
Programme availability:

#### PH20019 is Compulsory on the following programmes:

Department of Physics
• USPH-AFB01 : BSc (hons) Physics (Full-time) - Year 2
• USPH-AFB05 : BSc (hons) Physics with Computing (Full-time) - Year 2
• USPH-AKB06 : BSc (hons) Physics with Computing (with Placement) (Full-time with Thick Sandwich Placement) - Year 2
• USPH-AAB06 : BSc (hons) Physics with Computing with Year Abroad (Full-time with Study Year Abroad) - Year 2
• USPH-AKB02 : BSc (hons) Physics (with Placement) (Full-time with Thick Sandwich Placement) - Year 2
• USPH-AAB02 : BSc (hons) Physics with Year Abroad (Full-time with Study Year Abroad) - Year 2
• USPH-AFM02 : MPhys Physics (Full-time) - Year 2
• USPH-AFM04 : MPhys Physics with Research Placement (Full-time) - Year 2
• USPH-AAM03 : MPhys Physics with Year Abroad (Full-time with Study Year Abroad) - Year 2

#### PH20019 is Optional on the following programmes:

Programmes in Natural Sciences
• UXXX-AFB01 : BSc (hons) Natural Sciences (Full-time) - Year 2
• UXXX-AKB02 : BSc (hons) Natural Sciences with Industrial Placement (Full-time with Thick Sandwich Placement) - Year 2
• UXXX-AAB02 : BSc (hons) Natural Sciences with Study Year Abroad (Full-time with Study Year Abroad) - Year 2
• UXXX-AFM01 : MSci (hons) Natural Sciences (Full-time) - Year 2
• UXXX-AKM02 : MSci (hons) Natural Sciences with Professional Placement (Full-time with Thick Sandwich Placement) - Year 2
• UXXX-AAM02 : MSci (hons) Natural Sciences with Study Year Abroad (Full-time with Study Year Abroad) - Year 2
Department of Physics
• USPH-AFB09 : BSc Physics (Full-time) - Year 3
• USPH-AKB09 : BSc Physics (with Placement) (Full-time with Thick Sandwich Placement) - Year 4

NB. Programmes and units are subject to change at any time, in accordance with normal University procedures.