Einstein's summation convention

The dot product between two vectors \(\textbf{A} = (A_{x},A_{y},A_{z})\) and \(\textbf{B} = (B_{x},B_{y},B_{z})\) yields \(\textbf{A} \cdot \textbf{B} = A_{x}B_{x}+A_{y}B_{y}+A_{z}B_{z}\). Here, the subscripts \(x,y\) and \(z\) denote components of vectors \(A\) and \(B\) along \((x,y\) and \(z\) axes respectively. In cases where the number of dimensions exceed 3, it is more convenient to label dimensions as {\(1,2,3,...\)} rather than letters {\(x,y,z,...\)}. With this change in labeling method, the vectors \(\textbf{A}\) and \(\textbf{B}\) can be rewritten as \(\textbf{A} = (A_{1},A_{2},A_{3})\) and \(\textbf{B} = (B_{1},B_{2},B_{3})\); and the dot product of the two vectors is \(\textbf{A} \cdot \textbf{B} = A_{1}B_{1}+A_{2}B_{2}+A_{3}B_{3}\). Thus, we label or index the three dimensions as {\(1,2,3\)} - a set of three numbers. We can rewrite the dot product as

\[\textbf{A} \cdot \textbf{B} = A_{1}B_{1}+A_{2}B_{2}+A_{3}B_{3} = \sum_{i=1}^{3} \, A_{i}B_{i}\] This is a significant improvement over having to write three terms. Note that the index \(i\) is repeated and is being summed over. The Einstein's convention further simplifies this expression by eliminating the need to write the summation sign \(\sum\). It does so by implicitly assuming that whenever an index is repeated, it is being summed over. Using this notation the above dot product can be written as:

$$ \textbf{A} \cdot \textbf{B} = \sum_{i=1}^{3} \, A_{i}B_{i} = A_{i}B_{i} $$

If the dot product is to be evaluated between vectors in a 10-dimensional space, say \(\textbf{p} = (p_{1},p_{2},...,p_{10})\) and \(\textbf{q} = (q_{1},q_{2},...,q_{10})\), the dot product, following Einstein convention, is written as: $$ \textbf{p} \cdot \textbf{q} = \sum_{i=1}^{10} \, p_{i}q_{i} = p_{i}q_{i} $$

SO, next time you come across an expression such as \(e = f_{i}g_{i}\), you know what it really means is \(e = \sum_{i=1}^{n} f_{i}g_{i}\), where \(i\) can takes values in the range {\(1,...,n\)}. If, however, the expression is \(e = f_{i}g_{i}h_{j}\), it means \(e = \sum_{i=1}^{n} f_{i}g_{i}h_{j} = (f_{1}g_{1}+f_{2}g_{2}+...+f_{n}g_{n})h_{j}\); \(h_{j}\) does not get summed over since the index \(j\) is not repeated. Here, such an index \(j\) is also known as a free index and it remains unaffected by the summation process. Whereas an index that is repeated, hence summed over, is known as a dummy index. The index \(i\) here is a dummy index.

Einstein introduced this notational convention in physics in 1916 to facilitate the process of summation over an indexed elements of a set. This may appear to accomplishing a mere brevity in notation but the implications of this convention are remarkably useful in tensor calculus where tensors products between multi-dimensional covariant and contravariant vectors and tensors tend to get tedious. However, we don't need to delve into the tensor calculus to realize the usefulness of this simple rule of notation. Instead, we will apply this convention to solve regular problems based on gradient, divergence and curl of regular vectors. But there is one more thing we need to get out of the way before we can move to the problem solving part. In the next section, we will learn about another very useful object in mathematics known as the Levi-Civita symbol. We need this to work with cross-products using index notation method.

NEXT: Levi-Civita symbol, cross-products and determinants