tensor-calculus-differential-forms

Tensor Calculus and Differential Forms

ID: math.tensor_calculus
Domain: Mathematics → Geometry
Difficulty: 5/5

Prerequisites

• math.vector_calculus
• math.linear_algebra

Core Concepts

• tensors
• index notation
• covariant/contravariant
• metric tensor
• connection

Key Equations

$$ T^{\mu\nu} = \Lambda^\mu_{\ \alpha}\Lambda^\nu_{\ \beta}T^{\alpha\beta} $$ $$ \Gamma^\lambda_{\mu\nu} = \tfrac12 g^{\lambda\sigma}(\partial_\mu g_{\sigma\nu} + \partial_\nu g_{\sigma\mu} - \partial_\sigma g_{\mu\nu}) $$

Methods

• Manipulate indices carefully
• Raise/lower with the metric
• Compute covariant derivatives

Typical Problem Types

• Stress-energy tensor
• Curvature computation
• Relativistic kinematics

Common Pitfalls

• Summing over the wrong pair of indices
• Mixing up upper/lower indices

Related Skills

• rel.general.geodesics
• rel.special.four_vectors

References

• Schutz Geometrical Methods
• Misner Thorne Wheeler Ch.3