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 phenomena 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

• electromagnetism 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.