Search Rutgers Search Physics Physics 464/511 Fall 2016 Supplementary Notes

 Different books use different conventions, and sometimes even my notes may differ. Here is a possibly useful table.
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I have written up some notes which may be helpful as a background in mathematical techniques, and some that give more technical details of things presented in the lecture notes without all the details.

These are notes on mathematical techniques which should be part of your background. Some are not directly relevant to this course.

• Notes on using indices correctly: “On Indices and Arguments”: (view) or (print). Some cautionary notes on how indices are used and how to avoid making nonsense when evaluating expressions with dummy indices. This seems trivial, but I have seen many students have difficulties.

• Using ε's and determinants
• εijk and cross products in 3-D Euclidean space: Notes on totally antisymmetric tensors, or Levi-Civita symbols, (view), (print)
• εμνρ... in higher dimensional Euclidean or Minkowski space: (view), (print), including their use with matrices and determinants. The Levi-Civita symbol is also essential in curved spaces, but that is not covered in this note.
Also, on ε and determinants, "Properties of Determinants": (view), (print)
• Differentials
• The gradient operator view or print.
• On differential forms. If you have not seen differential forms before, you might want to look at my notes from 507, pages 153-4 on 1-forms and pp 167-175 on higher k-forms, or Zapolsky's notes. They are discussed in advanced calculus texts, e.g. Buck, "Advanced Calculus".
• Wronskian: view or print.
• Vector Identities from cover of Jackson, or all on one sheet version.
• The beta function B(x,y) and Γ(ε) for ε &≅ 0: view or print.
• Note on the Surface "Area" of a D-dimensional ball, and on the Euler Γ function: “Γ(N/2) and the Volume of SD-1”: ( view or print).
• Rodrigues' Formula and Orthogonal Polynomials (view or print).
• Notes on Bessel functions ( view or print), giving in particular the orthonormality properties.
• Notes on Lie Algebras and Groups: You really ought to take a course on group theory, but if you haven't, here are
• “ Lightning review of groups”: (view or print). Their definition and representations, the connection of continuous groups to Lie algebras, Killing forms and Casimir operators.
• “Note on Representations, the Adjoint rep, the Killing form, and antisymmetry of cijk”: (view or print).
• Schwinger trick and Feynman Parameters: view or print.
• Group Invariant Metric, which discusses a metric gμν on the group with the property that lengths are preserved under multiplication by a common group element. As a consequence g1/2 is the Haar measure.
• Quantum Mechanics on a Riemannian Space view or print.

Joel Shapiro (shapiro@physics.rutgers.edu)