Samer Adeeb

Linear Maps between vector spaces: Einstein Summation Convention

Einstein summation convention is a notational convention in Mathematics that is commonly used in the applications of linear algebra in continuum mechanics. The purpose is to achieve notational brevity. According to Einstein summation convention, when an index appears twice in a single term it implies summation of that term over all the values of the index which are almost always the values of \{1, 2, 3\} since the underlying space is \mathbb{R}^3. For example, if B=\{e_1,e_2,e_3\} is a basis set for \mathbb{R}^3, and u\in\mathbb{R}^3, then applying Einstein summation convention implies the following equality:

    \[ u=u_1e_1+u_2e_2+u_3e_3=\sum_{i=1}^3u_ie_i=u_ie_i \]

The following equalities relating the Kronecker delta, the alternator, and the vectors of the basis set B are very useful when dealing with continuum mechanics and with Einstein summation convention:

    \[ \begin{split} \delta_{ij}&=e_i\cdot e_j\\ \varepsilon_{ijk}&=e_i\cdot(e_j\times e_k) \end{split} \]

These, and using Einstein summation convention can be used to show the following identity:

    \[ u\cdot e_i=u_je_j\cdot e_i=u_j\delta_{ji}=u_i \]

Notice that the expression u_j\delta_{ji} is actually summed over the values of j=1 to 3. However, \delta_{ij} is 0 except when i=j and therefore, we are left with only one component u_i.
Similarly, if M\in\mathbb{M}^3,

    \[ e_i\cdot Me_j=e_i\cdot M_{kj}e_{k}=M_{kj}(e_i\cdot e_k)=M_{kj}\delta_{ki}=M_{ij} \]

The cross product can be simplified using the alternator and the Einstein summation convention as follows. If u,v\in\mathbb{R}^3, then the i^{th} component of the vector u\times v has the form:

    \[ (u\times v)_i=\varepsilon_{ijk}u_jv_k \]

For example, the convention can also be used for the following component forms. Let M,N,K\in\mathbb{M}^3, and u,v\in\mathbb{R}^3 then:

    \[\begin{split} & Mu\in\mathbb{R}^3\Rightarrow (Mu)_i=\sum_{j=1}^3M_{ij}u_j=M_{ij}u_j\\ & MN\in\mathbb{M}^3\Rightarrow (MN)_{ij}=\sum_{k=1}^3M_{ik}N_{kj}=M_{ik}N_{kj}\\ & MN^T\in\mathbb{M}^3\Rightarrow (MN^T)_{ij}=\sum_{k=1}^3M_{ik}N_{jk}=M_{ik}N_{jk}\\ & MNK\in\mathbb{M}^3\Rightarrow (MNK)_{ij}=\sum_{k,l=1}^3M_{ik}N_{kl}K_{lj}=M_{ik}N_{kl}K_{lj}\\ & MN^TK\in\mathbb{M}^3\Rightarrow (MN^TK)_{ij}=\sum_{k,l=1}^3M_{ik}N_{lk}K_{lj}=M_{ik}N_{lk}K_{lj}\\ & v\cdot Mu\in\mathbb{R}\Rightarrow v\cdot Mu=\sum_{i,j,k=1}^3v_ie_i\cdot M_{kj}u_je_k=v_iM_{kj}u_j\delta_{ki}=v_iM_{ij}u_j\\ & u\cdot (v\times w)\in\mathbb{R}\Rightarrow u\cdot (v\times w)=\sum_{i=1}^3u_ie_i\cdot\sum_{j,k,l=1}^3\varepsilon_{ljk}v_jw_ke_l=\sum_{i,j,k,l=1}^3\varepsilon_{ljk}\delta_{il}u_iv_jw_k=\varepsilon_{ijk}u_iv_jw_k \end{split} \]

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