# 0.1 Cartesian vectors and tensors: their algebra  (Page 5/5)

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If a and b are two vectors, the set of nine products, a i b j =A ij , is a second order tensor, for

$\begin{array}{ccc}{\overline{A}}_{pq}& =& {\overline{a}}_{p}{\overline{b}}_{q}={l}_{ip}{a}_{i}{l}_{jq}{b}_{j}={l}_{ip}{l}_{jq}\left({a}_{i}{b}_{j}\right)\\ & =& {l}_{ip}{l}_{jq}{A}_{ij}\end{array}$

An important example of this is the momentum flux tensor. If ρ is the density and v is the velocity, ρ v i is the i th component in the direction Oi . The rate at which this momentum crosses a unit area normal to Oj is the tensor, ρ v i v j .

If α is a scalar and A a second order tensor, the scalar product of α and A is a tensor α A each of whose components is α times the corresponding component of A .

The sum of two second order tensors is a second order tensor each of whose components is the sum of the corresponding components of the two tensors. Thus the ij th component of A + B is A ij + B ij . Notice that the tensors must be of the same order to be added; a vector can not be added to a second order tensor. A linear combination of tensors results from using both scalar multiplication and addition. α A + ß B is the tensor whose ij th component is α A ij + ß B ij . Subtraction may therefore be defined by putting $\alpha =1$ , $ß=-1$ .

Any second order tensor can be represented as the sum of a symmetric part and an antisymmetric part. For

${A}_{ij}=\frac{1}{2}\left({A}_{ij}+{A}_{ji}\right)+\frac{1}{2}\left({A}_{ij}-{A}_{ji}\right)$

and changing i and j in the first factor leaves it unchanged but changes the sign of the second. Thus,

$\mathbf{A}=\frac{1}{2}\left(\mathbf{A}+\mathbf{A}\text{'}\right)+\frac{1}{2}\left(\mathbf{A}-\mathbf{A}\text{'}\right)$

represents A as the sum of a symmetric tensor and antisymmetric tensor.

## Contraction and multiplication

As in vector operations, summation over repeated indices is understood with tensor operations. The operation of identifying two indices of a tensor and so summing on them is known as contraction . A ii is the only contraction of A ij ,

${A}_{ii}={A}_{11}+{A}_{22}+{A}_{33}$

and this is no longer a tensor of the second order but a scalar, or a tensor of order zero. The scalar A ii is known as the trace of the second order tensor A . The notation tr A is sometimes used. The contraction operation in computing the trace of a tensor A is analogous to the operation in the calculation the magnitude of a vector a , i.e., $|\mathbf{a}{|}^{2}=\mathbf{a}\mathbf{\cdot }\mathbf{a}={a}_{1}{a}_{1}+{a}_{2}{a}_{2}+{a}_{3}{a}_{3}$

If A and B are two second order tensors, we can form 81 numbers from the products of the 9 components of each. The full set of these products is a fourth order tensor. Contracted products result in second order or zero order tensors. We will not have an occasion to use products of tensors in our course.

The product A ij a j of a tensor A and a vector a is a vector whose i th component is A ij a j . Another possible product of these two is A ij a I . These may be written A a and a A , respectively. For example, the diffusive flux of a quantity is computed as the contracted product of the transport coefficient tensor and the potential gradient vector, e.g., $\mathbf{q}=-\mathbf{k}\mathbf{\cdot }\mathbf{\nabla }T$

## The vector of an antisymmetric tensor

We showed earlier that a second order tensor can be represented as the sum of a symmetric part and an antisymmetric part. Also, an antisymmetric tensor is characterized by three numbers. We will later show that the antisymmetric part of the velocity gradient tensor represents the local rotation of the fluid or body. Here, we will develop the relation between the angular velocity vector, ω , introduced earlier and the corresponding antisymmetric tensor.

Recall that the relative velocity between a pair of points in a rigid body was described as follows.

$\Delta \mathbf{v}=\omega ×\Delta \mathbf{x}$

We wish to define a tensor Ω that also can determine the relative velocity.

$\begin{array}{ccc}\mathbf{\Delta }\mathbf{v}& =& \omega ×\mathbf{\Delta }\mathbf{x}\\ & =& \mathbf{\Delta }\mathbf{x}\mathbf{•}\Omega \end{array}$

The following relation between the components satisfies this relation.

$\begin{array}{l}{\Omega }_{ij}={\epsilon }_{ijk}{\omega }_{k}\\ {\omega }_{k}=\frac{1}{2}{\epsilon }_{ijk}{\Omega }_{ij}\end{array}$

Written in matrix notation these are as follows.

$\omega =\left[\begin{array}{c}{\omega }_{1}\\ {\omega }_{2}\\ {\omega }_{3}\end{array}\right],\Omega =\left[\begin{array}{ccc}0& {\omega }_{3}& -{\omega }_{2}\\ -{\omega }_{3}& 0& {\omega }_{1}\\ {\omega }_{2}& -{\omega }_{1}& 0\end{array}\right]$

The notation vec Ω is sometimes used for ω . In summary, an antisymmetric tensor is completely characterized by the vector, vec Ω .

## Canonical form of a symmetric tensor

We showed earlier that any second order tensor can be represented as a sum of a symmetric part and an antisymmetric part. The symmetric part is determined by 6 numbers. We now seek the properties of the symmetric part. A theorem in linear algebra states that a symmetric matrix with real elements can be transformed by its eigenvectors to a diagonal matrix with real elements corresponding the eigenvalues. (see Appendix A of Aris.) If the eigenvalues are distinct, then the eigenvector directions are orthogonal. The eigenvectors determine a coordinate system such that the contracted product of the tensor with unit vectors along the coordinate axis is a parallel vector with a magnitude equal to the corresponding eigenvalue. The surface described by the contracted product of all unit vectors in this transformed coordinate system is an ellipsoid with axes corresponding to the coordinate directions.

The eigenvalues and the scalar invariants of a second order tensor can be determined from the characteristic equation.

$\begin{array}{l}\mathrm{det}\left({A}_{ij}-\lambda {\delta }_{ij}\right)=\psi -\lambda \Phi +{\lambda }^{2}\Theta -{\lambda }^{3}\\ \text{where}\\ \Theta ={A}_{11}+{A}_{22}+{A}_{33}=tr\mathbf{A}\\ \Phi ={A}_{22}{A}_{33}-{A}_{23}{A}_{32}+{A}_{33}{A}_{11}-{A}_{31}{A}_{13}+{A}_{11}{A}_{22}-{A}_{12}{A}_{21}\\ \psi =\mathrm{det}\mathbf{A}\end{array}$

## Assignment 2.1

1. Relative velocity of points in a rigid body. If x and y are two points inside a rigid body that is translating and rotating, determine the relation between the relative velocity of these two points as a function of their relative positions. If x and y are points on a line parallel to the axis of rotation, what is their relative velocity? If x and y are points on opposite sides of the axis of rotation but with equal distance, r, what is their relative velocity? Draw diagrams.
2. Prove that: a •( b × c ) = ( a × b )• c
3. Show a •( b × c ) vanishes identically if two of the three vectors are proportional of one another.
4. Show that if a is coplanar with b and c , then a •( b × c ) is zero.
5. Prove that: $\mathbf{a}×\left(\mathbf{b}×\mathbf{c}\right)=\left(\mathbf{a}•\mathbf{c}\right)\mathbf{b}-\left(\mathbf{a}•\mathbf{b}\right)\mathbf{c}$
6. Prove that the contracted product of a tensor A and a vector a , A a , transforms under a rotation of coordinates as a vector.
7. Show that you get the same result for relative velocity whether you use ω or Ω for the rotation of a rigid body.

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