Level 4 · Advanced undergraduate teaching kit · Third- and fourth-year university
Tensor calculus and differential geometry
Use the learner record during the live investigation, then use the instructor guide to facilitate comparison, address misconceptions, and assess evidence-bounded reasoning.
Learner lab record
Geodesic coordinate-invariance audit
Which features of a computed trajectory reflect geometry, and which reflect the chosen coordinate chart?
Setup
Use the geodesic laboratory. Integrate one physical initial condition in the baseline chart, change a coordinate parameterization, and compare invariant and coordinate-dependent diagnostics.
Predict first
- 1. Predict which plotted coordinates may change after a chart change.
- 2. Predict which tangent norm should remain invariant along an affinely parameterized geodesic.
| Variable | Role | Unit |
|---|---|---|
| Metric parameters | geometric inputs | declared model units |
| Initial position and tangent | initial data | chart coordinates |
| Step size | numerical control | affine parameter |
| Coordinate path and norm residual | dependent diagnostics | coordinates and dimensionless |
Observation columns
Analyze
- 1. Which differences are coordinate artifacts?
- 2. Does the tangent-norm residual converge with step size?
- 3. Where do connection coefficients enter the trajectory?
- 4. Which curvature invariant would distinguish a true singularity from a chart singularity?
Conclusion frame
Changing from chart ___ to ___ changed coordinate path ___ while invariant diagnostic ___ remained ___; the numerical residual scaled ___.
Instructor guide · 60–75 minutes
Teach the investigation, not the interface
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.
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
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.
Extension
Compute geodesic deviation for two nearby trajectories and compare separation with a curvature component in an orthonormal frame.