Module Details
The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
Title MATHEMATICS FOR PHYSICISTS III
Code PHYS207
Coordinator Dr J Alaria
Physics
Jonathan.Alaria@liverpool.ac.uk
Year CATS Level Semester CATS Value
Session 2018-19 Level 5 FHEQ First Semester 15

Aims
  • To re-inforce students'' prior knowledge of mathematical techniques
  • To introduce new mathematical techniques for physics modules
  • To enhance students'' problem-solving abilities through structured application of these techniques in physics

Learning Outcomes

At the end of the module the student should be able to:

  • Have knowledge of a range of mathematical techniques necessary for physics and astrophysics programmes
  • Be able to apply these mathematical techniques in a range of physics and astrophysics programmes

Syllabus
1         Overview

Integral and differential vector calculus:

  • Scalar and vector fields
  • Scalar and vector field functions
  • Polar coordinate systems
  • Derivation of the gradient, divergence and curl  functions
  • Example s of these operations including their physical significance
  • Vector operations in polar coordinate systems
  • Stoke’s theorem with examples
  • Gauss’ theorem with examples
  • Line, surface and volume elements in circular, spherical and cylindrical polar coordinates
  • Line, surface and volume integrals in different coordinate systems - applications
 Vectors and Matrices
  • Real and complex vectors, linear independence, basis, scalar product, orthon ormal basis.
  • Revision of matrices. Sum, product, transposition. Symmetric and antisymmetric matrices. 
  • Trace and determinant of square matrices. Laplace expansion theorem. Row echelon form of a matrix. Rank of a matrix. Application to vectors (coplanarity, collinearity).
  • Systems of linear equations, Gaussian elimination.
  • Inversion of matrices using row operations.
  • Eigenvalues and eigenvectors of matrices. Complex and degenerate eigenvalues.

  • Real symmetric matrices and diagonalisation. Orthogonal transformations and orthogonal matrices. Applications: rotational motion, inertia tensor.

Applications

Application: rotational motion, inertia tensor

Hermitian scalar product of complex vectors. Hermitian matrices and diagonalization. Unitary transformations and unitary matrices.  

 
Application: quantum mechanics.
Revision of Taylor''s theorem, Taylor''s theorem with remainder.
Revision of infinite sums and series. Ratio test. Radius of convergences of power series.
Revision of Taylor series. Generating Taylor series from known Taylor series by substitution and differentiation.

Teaching and Learning Strategies

Lecture - 24 hours of lectures

= 12 x 2 lectures/week

Workshop - To give help with completing work and to give feedback on completed work, and to learn in conversational style with staff.

= 10 x 2-hour workshops


Teaching Schedule
  Lectures Seminars Tutorials Lab Practicals Fieldwork Placement Other TOTAL
Study Hours 24
24 hours of lectures
        20
To give help with completing work and to give feedback on completed work, and to learn in conversational style with staff.
44
Timetable (if known) = 12 x 2 lectures/week
  		        = 10 x 2-hour workshops
  		 
Private Study 106
TOTAL HOURS 150

Assessment
EXAM Duration Timing
(Semester)		 % of
final
mark		 Resit/resubmission
opportunity		 Penalty for late
submission		 Notes
Unseen Written Exam  2 hours  1  70  Yes  Standard UoL penalty applies  Exam Notes (applying to all assessments) Assessment 1: Problem sets in 10 workshops (20%), 2 Homeworks (10%), Assessment 2: Written examination (70%) If the continuous assessment component is failed and a resit is required, the mark for the resit examination will subsume the marks for the continuous assessment component.  
CONTINUOUS Duration Timing
(Semester)		 % of
final
mark		 Resit/resubmission
opportunity		 Penalty for late
submission		 Notes
Coursework  10 x 2 hours worksho  1  30  No reassessment opportunity  Standard UoL penalty applies  Workshop participation/Homework There is no reassessment opportunity, Subsumed by resit exam 

Recommended Texts
Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module.
Explanation of Reading List: