Tensor Calculus

Mathematical Principles & Theorems

Investigates tensor algebra on smooth manifolds. A type \((r, s)\) tensor transforms multilinearly under coordinate charts \(x^\mu \to \bar{x}^\mu\). The Metric Tensor \(g_{\mu\nu}\) defines Riemannian inner products \(ds^2 = g_{\mu\nu}dx^\mu dx^\nu\). Christoffel symbols of the second kind \(\Gamma^\sigma_{\mu\nu} = \frac{1}{2}g^{\sigma\lambda}(\partial_\mu g_{\nu\lambda} + \partial_\nu g_{\mu\lambda} - \partial_\lambda g_{\mu\nu})\) define Levi-Civita covariant differentiation \(\nabla_\mu V^\nu = \partial_\mu V^\nu + \Gamma^\nu_{\mu\lambda}V^\lambda\). Riemann Curvature Tensor \(R^\rho_{\sigma\mu\nu}\) quantifies manifold curvature.

Operating Instructions

• Select coordinate system: Cartesian 3D, Cylindrical, Spherical, or Toroidal coordinates.
• Click Compute Metric to derive the covariant metric tensor matrix \(g_{ij}\) and inverse contravariant metric \(g^{ij}\).
• Inspect the computed non-zero Christoffel connection symbols \(\Gamma^k_{ij}\).
• Review coordinate transformations and differential line element formulas \(ds^2\).