Vetorivit

Vetorivit is a hypothetical mathematical construct used to describe a structured field that combines a vector field with a scalar field. Concretely, on a smooth manifold M, a vetorivit is defined as a section of the product bundle TM × R, which assigns to each point p a pair (v_p, s_p) with v_p in the tangent space T_pM and s_p in R. The two components are required to vary smoothly with p. This framework extends ordinary vector fields by incorporating a pointwise scalar weight alongside the direction field.

Notes on notation and operations: If X denotes a vector field and f a scalar function on M, one can view the vetorivit (X, f) as a unified object. Addition and scalar multiplication are performed componentwise: (X1, f1) + (X2, f2) = (X1+X2, f1+f2) and a·(X, f) = (aX, af). Under a diffeomorphism φ: M → N, a vetorivit transforms by the natural actions on TM and on R, preserving its product structure.

Applications: In differential geometry, vetorivit provides a compact way to couple directional flows with local densities, enabling weighted transport, interpolation, and analysis of anisotropic processes. In computer graphics and physical simulations, vetorivit-like objects can model directional quantities that also carry a magnitude or cost. The term is not part of standard textbooks and is used here as a conceptual illustration.

See also: vector field, tangent bundle, product bundle, weighted vector field.