REVIEW OF DIFFERENTIAL FORMS
MTH 437/537

The treatment of general relativity used in this course is based on differential forms. This page first summarizes the basic skills that should be regarded as prerequisites for the course, then provides links to resources that should assist you in acquiring (or reviewing) them, then lists some sample problems.

Basic Skills

Manipulation of differential forms:

Use ∧, ∗, and d

Vector-valued forms:

Understand relation between ds², dr, and bases

Understand difference between orthonormal basis and coordinate basis

Calculation of Curvature:

Determine connection 1-forms

Compute curvature 2-forms

Resources

• Formula sheet
• Quick review
• More detailed review
• Online notes on differential forms, especially
  §2.1–§2.2, §2.7, §2.10, §3.1–§3.4, §3.6, §3.7, §3.13, §4.1–§4.2, §4.5–§4.6, §6.1–§6.6, §7.4
• Older notes on Hodge duality, connections, and curvature

Sample Problems

• Hodge dual: MTH 434 HW #4
  (Question 2b should be treated as optional.)
• Connection & curvature: MTH 434 HW #6 and MTH 434 HW #7
  (You may use any method you wish. One possibility is to use the structure equations, and not work with vectors at all.)