# Instructor guide: Geodesic coordinate-invariance audit

Course: Tensor calculus and differential geometry

Suggested time: 60–75 minutes

## Learning target

Learners separate coordinate descriptions from invariant geometric and numerical statements in a geodesic calculation.

## Prepare

- Review metric compatibility and affine parameters.
- State the chart domain and coordinate singularities.
- Prepare two step sizes for convergence testing.

## Facilitation moves

- Ask whether each reported quantity is a scalar, component, or coordinate.
- Require a convergence check before geometric interpretation.
- Connect Christoffel symbols to derivatives of the metric rather than treating them as tensors.

## Misconception checks

- **A curved coordinate path proves spacetime curvature.** Even flat geometry can have curved coordinate lines; curvature requires invariant tensor information.
- **Christoffel symbols are tensor components.** Their inhomogeneous transformation supplies the coordinate correction needed for covariant differentiation.

## Accessibility and participation

- Pair every trajectory with numeric invariant diagnostics.
- Describe chart changes in words and equations rather than animation alone.
- Provide a symbol glossary for indices and contractions.

## Evidence of learning

- A chart-versus-invariant table
- A step-size convergence result
- A correct coordinate-singularity criterion

## Extension

Compute geodesic deviation for two nearby trajectories and compare separation with a curvature component in an orthonormal frame.

## Evidence boundary

Assess the learner's reasoning only within the declared model and recorded observations. Do not upgrade a simulation result into a claim about an unmodeled physical system.