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
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,
- εμνρ... in higher dimensional
Euclidean or Minkowski space:
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":
- The gradient operator
- 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
They are discussed in advanced calculus texts, e.g.
Buck, "Advanced Calculus".
- Wronskian: view or
- Vector Identities
from cover of
Jackson, or all on one
- The beta function B(x,y) and Γ(ε) for ε &≅ 0:
- 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”: (
- Rodrigues' Formula and Orthogonal Polynomials
- 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”:
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”:
- Schwinger trick and Feynman Parameters:
- Group Invariant Metric,
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
Last modified: Tue Nov 1 14:38:17 2016