Tensor theory is extremely useful in advanced engineering theory. It is used to help describe or model many natural phenomenon such as: physical forces, potential fields, particle or control element motion, wave propagation, etc.
Specific examples are:
aeronautical engineering
Navier-Stokes equations Presented in partial differential equation form.
dynamics of systems of rigid (assumed incompressible) bodies and particles
stress and strain within elastic bodies
electromagnetics Maxwell's Equations
hydrodynamics