[NOTES/ME-14005]-Vectors and Tensors

1.Vectors

A vector can be defined as an object with 3 components \(\vec{A}=(A_1,A_2,A_3)\) which under rotation of axes transform as \begin{equation} A_i^{'} = R_{ij} A_j. \end{equation} 

An example As an example, consider two vectors \(\vec{A}= (A_1,A_2,A_3)\) and \(\vec{B}=(B_1,B_2,B_3)\) Let us define a nine component object \(X_{ij}\) by \(X_{ij}=A_iB_j\). Knowing the transformation properties of vectors under rotations, we can work out the components of \(X\) in rotated frame of reference. For the two vectors \(\mathbf A, {\mathbf B}\) we will have \begin{equation} A_j{'} = R_{j\ell}A_\ell, \qquad B_k{'} = R_{km}B_m. \end{equation} Using these two equations we will get \begin{eqnarray}\nonumber X{'}_{jk} \equiv A_j {'} B_k{'} &=& (R_{j\ell} A_\ell)(R_{km} B_m) \\\nonumber &=& R_{j\ell} R_{km} (A_\ell\, B_m)\\ &=& R_{j\ell} R_{km} X_{\ell m}\label{EQ09} \end{eqnarray} Thus the object \(X\) has the property that under a rotation its nine components w.r.t. the rotated axes are linear combinations of components w.r.t. the first set of axes and the two sets are related by \eqRef{EQ09}.

2.Tensors of rank two 

Definition 1 An object \(\mathbf T \) having nine components, \(T_{ij}\) will be called a {\it second rank tensor} if the components transform as \begin{equation} T_{ij}{'}= R_{im}R_{jn}T_{mn}. \end{equation} In matrix notation, the above property takes the form \begin{equation} \underline{\mathbf T}{'} = \underline{R}^T \, \underline{\mathbf T}\, \underline{R} \end{equation}

3.Tensors of higher rank

 A tensor of rank \(N\) has \(N\) indices \(T_{ijk..}\) and its components transform as \begin{equation} (T^{'})_{ijk ...} = \big(R_{i\ell} R_{jm} R_{kn} \big) T_{\ell m n ...}. \end{equation} An important example of higher rank (pseudo) tensor is Levi-Civita symbol. More about will be discussed separately.\cite{eps}

Definition 2 A tensor \(S_{ijk...}\) is called a pseudo tensor, if its components transform as \begin{equation} (S^{'})_{ijk ...} = (\det R)\big(R_{i\ell} R_{jm} R_{kn} \big) T_{\ell m n ...}. \end{equation} 

Examples

• All objects, such as scalar product of vectors which remain unchanged under rotations, may be regarded as a {\it zero rank tensor}.
• As an example, all vectors may be regarded as tensors of rank one.
• If \({\mathbf A},{\mathbf B}\) are two vectors, their cross product \({\mathbf C=}{\mathbf A}\times{\mathbf B} \) is an example of pseudovector, also called {\it axial vector}.