Summationskonvention

The Summationskonvention, often translated as the summation convention, is a notational shortcut used primarily in physics and engineering to simplify expressions involving sums. Introduced by Tullio Levi-Civita, it eliminates the need to explicitly write summation symbols. The convention states that whenever an index variable appears twice in a single term, once as a subscript and once as a superscript, it is implied that a summation is to be performed over all possible values of that index.

For example, in vector notation, the dot product of two vectors a and b in n-dimensional space can be written as $a \cdot b = \sum_{i=1}^{n} a_i b_i$. Using the Summationskonvention, this expression is simplified to $a_i b_i$. Here, the repeated index 'i' implies a summation over all its possible values, typically from 1 to n in this context.

This convention is particularly useful in tensor calculus and differential geometry, where expressions can become very lengthy and complex. By reducing the number of symbols, it enhances readability and reduces the chance of typographical errors. It's important to note that the convention only applies to repeated indices within a single term. If an index appears only once, it is a free index and its value is not summed over. If an index appears more than twice in a single term, it is considered an error in notation. The range of summation for an index is usually understood from the context of the problem or explicitly stated elsewhere.