covariant derivative
Curvature-aware vector calculus
The covariant derivative is a mathematical operation that generalizes ordinary differentiation to curved spaces and spaces with additional geometric structure.
In flat space, taking a derivative is straightforward, but on curved surfaces or in spaces where coordinates twist and turn, the covariant derivative accounts for how the coordinate system itself changes from point to point.
It is essential in differential geometry and general relativity, where it describes how vectors and tensors change as they move through curved spacetime.
The covariant derivative of the metric tensor vanishes in Riemannian geometry.
- connectionA joining or relation
- parallel transportVector movement preserving covariant derivative
- curvatureDeviation from flatness
- tensorMultilinear algebraic object
- manifoldMany and various
geodesic · Christoffel symbols · gauge · affine · tangent · vector field · torsion · metric · gradient · divergence · section · infinitesimal · transformation · index notation · contraction · commutator · spacetime · general relativity
- Gauge TheoryYang-Mills theory, connection, parallel transport
- differential operatorsdel, gradient, divergence, curl, Hessian
- Relativistic Quantumantimatter, antiparticle, antiparticles, positron, chirality
- spacetimespacetime, space-time, four-dimensional, tesseract, hypercube
- Spacetime Geometryspacetime, curvature, geodesics, general relativity, manifolds
- connectionslink, links, connect, connection, tie
covariant derivativepartial derivative
Differentiation fixing other variables
covariance, variant, variants, variation, variations, varies, varied, varying, variable