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Week | Topics | Problem types (examples) ---|---:|--- Week 1 — Foundations | Scalars, vectors, coordinate transforms, index notation | Convert vector ops between component and index forms; raise/lower indices; prove transformation rules Week 2 — Tensor Algebra | Tensor product, contraction, symmetrization, alternating tensor | Prove uniqueness of decomposition into symmetric/antisymmetric parts; compute tensor products and contractions Week 3 — Metrics & Duals | Metric tensor, inverse metric, dual vectors, orthonormal bases | Show g_ij transforms as tensor; compute components in polar/spherical; Gram–Schmidt examples Week 4 — Covariant Derivative | Connection coefficients, parallel transport, geodesics | Derive Christoffel symbols for given metrics; solve simple geodesic ODEs Week 5 — Curvature | Riemann, Ricci, scalar curvature, Bianchi identities | Compute Riemann for 2D surfaces (sphere, cone); verify symmetries and Bianchi identity Week 6 — Differential Forms & Hodge | Exterior derivative, Lie derivative, Hodge star | Compute forms on R^3, prove d^2=0, apply Stokes' theorem examples Week 7 — Applications I | Continuum mechanics: stress, strain, index form of PDEs | Write Cauchy momentum in index form; compute small-strain tensor examples Week 8 — Applications II | General relativity basics, Einstein eqns linearized gravity | Linearize metric perturbations; compute Einstein tensor for simple metrics tensor analysis problems and solutions pdf free
For Schwarzschild metric, identify ( g_tt ) and ( g_rr ). Many physics and math students upload their solved
Solution:
( ds^2 = -(1 - \frac2GMr) dt^2 + (1 - \frac2GMr)^-1 dr^2 + r^2 d\Omega^2 )
Thus ( g_tt = -(1 - \frac2GMr),\quad g_rr = (1 - \frac2GMr)^-1 ) compute components in polar/spherical
Compute ( \varepsilon_ijk \varepsilon_ajk )
Solution:
Use identity: ( \varepsilon_ijk\varepsilon_ajk = 2\delta_ia ).