Chapter 2. The Tensor Concept

Scalars: tensors of rank 0. Vectors: tensors of rank 1. Since their specifications in any coordinate system require $3^n$ numbers, called components.

Tensor is more than just a collection of numbers. Key property is the transformation law of its components: The way components in one coordinate system relate to components in another coordinate system. The transformation law is a consequence of the geometric or physical meaning of the tensor.

Properly formulated physical laws are invariant under shifts and rotations of the coordinate system.

2.2 Zero Order Tensors (Scalars)

Scalars are quantities uniquely defined by a single number and invariant under coordinate system changes.

Examples: Let $A$ and $B$ be two points in space, and let $\delta s$ be the distance between them. Then $\delta s$ is a scalar.

For the time being, we confine ourselves to the rectangular coordinate systems.

Tensors written in rectangular coordinates are called Cartesian tensors.

2.3 First Order Tensors (Vectors)

By a vector we mean a quantity specified in any coordinate system by three real numbers (components) which transform under coordinate system changes according to the law:

$$ A'_{i} = \alpha_{i'k} A_{k} \tag{2.2} $$

where $A_{k}$, $A'_{i}$ are the components of the vector in the old and new coordinate systems $K$ and $K'$, respectively, and $\alpha_{i'k}$ are the cosines of the angles between the $i$-th axis of $K'$ and the $k$-th axis of $K$.

This definition is equivalent to the definition of a vector as a directed line segment, but has the advantage of being easily generalizable to arbitrary order tensors.

2.4 Second Order Tensors

By a second order tensor we mean a quantity uniquely specified nine real numbers (components) which transform under coordinate system changes according to the law:

$$ A'_{ik} = \alpha_{i'l} \alpha_{k'm} A_{lm} \tag{2.4} $$

where $A_{lm}$, $A'_{ik}$ are the components of the tensor in the old and new coordinate systems $K$ and $K'$, respectively, and $\alpha_{i'l}$ is the cosine of the angle between the $i$-th axis of $K'$ and the $l$-th axis of $K$.

The components of a second order tensor are often written as a matrix:

$$ \lVert A_{ik} \rVert = \left \lVert \begin{array}{ccc} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{array} \right \rVert $$