Application of tensor theory in engineering

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.

Constructions notes:

A^i'j'_k' = x^i'_i x^j'_j y^k_k' A^ij_k

Specific examples are:

aeronautical engineering

 Navier-Stokes equations  Presented in partial differential equation form.

Vorticity is an important quantity in various research, modeling and design calculations regarding lift, drag, and propulsion. It is a tensor quantity defined as: insert gif here when available.

Continuum mechanics

dynamics of systems of rigid (assumed incompressible) bodies and particles

stress and strain within elastic bodies

electromagnetics Maxwell's Equations

Hydrodynamics

Tensor equations to model fluid flow can be derived as follows:

Assume the fluid consists of particles which can be individually tracked as they move in relation to Euclidean 3-space. Thus an individual particle can be tracked as it moves.

We shall use rectangular cartesian coordinates to describe our Euclidean 3 space .... z_r

In the Lagrangian method, all particles are then described by:

Equation (1) z_r=z_r(a,t) where a stands for the set of 3 labels representing the 3 dimensions or axis of Euclidean space ... x_i,x_j,x_k.