The total derivative is a linear approximation to a map between Euclidean spaces. Its Jacobian records components, and the chain rule describes composition. Directional derivatives alone need not establish total differentiability. The examples make the distinction explicit.

The inverse function theorem supplies a smooth local inverse when the derivative is invertible. The implicit function theorem solves suitable equations for selected variables. Both are local conclusions with regularity assumptions. Test those assumptions in the examples.

A smooth manifold is locally Euclidean with compatible smooth coordinate charts. Transition maps let smoothness be defined independently of a chosen chart. Smooth maps and diffeomorphisms then compare manifolds. The examples explain why one coordinate singularity need not be a geometric singularity.

Tangent vectors admit compatible descriptions using curves and derivations. A smooth map induces a differential on tangent spaces; covectors act linearly on tangent vectors. Tangent and cotangent bundles collect these spaces smoothly over the manifold. Check their coordinate descriptions in the examples.

Tensor fields vary multilinear data smoothly over a manifold. Tensor products and contraction create new tensors, while alternating covariant tensors give differential forms. The wedge product supplies their graded algebra. Use the examples to distinguish a tensor from its coordinate components.

A vector field assigns tangent vectors smoothly. Its local flow solves the associated differential equation, and the flow-box theorem describes nonvanishing fields in adapted coordinates. Lie derivatives compare objects under this motion. Work through the examples before interpreting a flow globally.

A Riemannian metric is a smoothly varying positive-definite inner product. It defines curve lengths and an induced distance whose topology agrees with the manifold topology. Volume and divergence add integration to this geometry. The examples prepare the later comparison with Lorentzian metrics.

A connection differentiates sections and defines parallel transport along curves. Metric compatibility and zero torsion characterize the Levi–Civita connection. The fundamental theorem of Riemannian geometry establishes existence and uniqueness. Check the difference between connection coefficients and geometric objects in the examples.

A geodesic has parallel velocity along itself. Local existence and uniqueness turn initial tangent vectors into curves, and the exponential map records their endpoints. Normal neighborhoods give a local geometric model. Riemannian completeness and Hopf–Rinow have additional hypotheses that must not be transferred uncritically to spacetime. Use the examples to test those distinctions.

Curvature measures a failure of covariant derivatives to commute. Curvature identities and submanifold formulas connect intrinsic and ambient descriptions. Gauss’s Theorema Egregium identifies the intrinsic nature of surface curvature. The examples distinguish a curved surface from a flat coordinate description.

The relativity framework uses a smooth time-oriented Lorentzian four-manifold. Its nondegenerate metric has indefinite signature, introducing causal distinctions absent from positive-definite geometry. Inverse metrics and tensor operations support the physical model’s fields. The examples clarify which statements belong to this framework.

Proper time and free fall connect timelike curves with the metric-selected connection. Geodesic initial data determine local trajectories, and Lorentzian normal-coordinate results describe local geometry. A physics theorem about ideal test bodies also depends on its model premises. Check the examples against those premises.

The curvature tensors organize local spacetime geometry. The Bianchi identity constrains the Einstein tensor, and the Jacobi equation describes relative acceleration in geodesic families. The tidal interpretation uses the free-fall model. Work through the examples to distinguish a coordinate effect from a tidal effect.

Einstein’s equation relates curvature to stress–energy in the adopted classical gravitational model. Variational constructions identify the equation from an action; conservation identities and matter models then constrain its interpretation. The examples keep mathematical deductions distinct from the empirical validation of the physical framework.