

Different books use different conventions, and sometimes even my notes
may differ. Here is a
possibly useful table.





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
 ε_{ijk} and cross products
in 3D Euclidean space:
Notes on totally antisymmetric tensors, or LeviCivita symbols,
(view),
(print)
 ε_{μνρ...} in higher dimensional
Euclidean or Minkowski space:
(view),
(print),
including their use with matrices and determinants. The
LeviCivita 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 1534 on 1forms and pp 167175 on
higher kforms, 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 Ddimensional ball, and on the Euler
Γ function: “Γ(N/2) and the Volume
of S^{D1}”: (
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 c_{ij}^{k}”:
(view or
print).
 Schwinger trick and Feynman Parameters:
view or
print.
 Group Invariant Metric,
which discusses
a metric g_{μν}
on the group with the property that
lengths are preserved under multiplication by a common group element.
As a consequence g^{1/2} is the Haar measure.
 Quantum Mechanics on a Riemannian Space
view or
print.

Joel Shapiro
(shapiro@physics.rutgers.edu)
Last modified: Tue Nov 1 14:38:17 2016