“God made the integers, all else is the work of man.” – Leopold Kronecker

“Continuum Mechanics may appear as a fortress surrounded by the walls of tensor notation” E. Tadmore, et. al. 1. The word “Tensor” applies to virtually all the quantities encountered in Engineering. Scalars and Vectors are zeroth and first order tensors. However, the word, with no prefix, refers to a second-order tensor. 2. For an object to be proved to be a tensor we need to show that it transforms a vector and its output is also a vector. Secondly, that transformation must be linear. 3. A tensor can be expressed in Component Form. The vector itself is more than its components as components presume reference to particular coordinate system. Outside that the numbers mean nothing. 4. For any tensor, certain scalar-valued functions are characteristic of the tensor, independent of coordinate systems. These are usually the targets of computations of any tensor. They are Principal Invariants. 5. Tensors can be decomposed additively or multiplicatively to simpler tensors. The goal is to make analysis easier and gain valuable insight by removing parts of the tensor not crucial to the problem.

“Continuum Mechanics may appear as a fortress surrounded by the walls of tensor notation” E. Tadmore, et. al. 1. The word “Tensor” applies to virtually all the quantities encountered in Engineering. Scalars and Vectors are zeroth and first order tensors. However, the word, with no prefix, refers to a second-order tensor. 2. For an object to be proved to be a tensor we need to show that it transforms a vector and its output is also a vector. Secondly, that transformation must be linear. 3. A tensor can be expressed in Component Form. The vector itself is more than its components as components presume reference to particular coordinate system. Outside that the numbers mean nothing. 4. For any tensor, certain scalar-valued functions are characteristic of the tensor, independent of coordinate systems. These are usually the targets of computations of any tensor. They are Principal Invariants. 5. Tensors can be decomposed additively or multiplicatively to simpler tensors. The goal is to make analysis easier and gain valuable insight by removing parts of the tensor not crucial to the problem.

“We do not fuss over smoothness assumptions: Functions and boundaries of regions are presumed to have continuity and differentiability properties sufficient to make meaningful underlying analysis...” Morton Gurtin, et al. 1. The first chapter provided the necessary background to study tensors. Chapter 2 explored the properties of tensors. 2. This third chapter begins the applications by introducing the concept of differentiation for objects larger than scalars. 3. The gradient of scalar, vector or tensor is connected to the extension of the concept of directional derivative called the Gateaux differential. 4. The complexity in differentiation of large objects comes, not from the objects themselves, but from the domain of differentiation. Differentiation rules with respect to scalar arguments follows closely the similar laws for scalar fields. 5. Integral theorems of Stokes, Green and Gauss are introduced with computational examples. 6. Orthogonal Curvilinear systems are dealt with in the addendum. Important results and the use of Computer Algebra are given.

“We do not fuss over smoothness assumptions: Functions and boundaries of regions are presumed to have continuity and differentiability properties sufficient to make meaningful underlying analysis...” Morton Gurtin, et al. 1. The first chapter provided the necessary background to study tensors. Chapter 2 explored the properties of tensors. 2. This third chapter begins the applications by introducing the concept of differentiation for objects larger than scalars. 3. The gradient of scalar, vector or tensor is connected to the extension of the concept of directional derivative called the Gateaux differential. 4. The complexity in differentiation of large objects comes, not from the objects themselves, but from the domain of differentiation. Differentiation rules with respect to scalar arguments follows closely the similar laws for scalar fields. 5. Integral theorems of Stokes, Green and Gauss are introduced with computational examples. 6. Orthogonal Curvilinear systems are dealt with in the addendum. Important results and the use of Computer Algebra are given.

The various possible types of motion leaving aside the causes to which the initiation of motion may be ascribed. - E.T. Whittaker