Tensor calculus is an organized expression, which contains sophisticated geometric insights.
- Combines geometric and analytical perspectives.
- Enables use of the co-ordinate system without the loss of geometric insight.
- Provides a framework for establishing equations valid in all coordinate systems.
- Provides a language that is concise and powerful.
- Based on a handful of operations.
Summation convention in Tensor algebra plays a crucial role. The convention is such that any lower-case alphabetic subscript that appears exactly twice in any term of an expression is understood to be summed over all the values that a subscript in that position can take.
The subscripted quantities may appear in the numerator and/or denominator of a term in an expression.
Subscripts that are summed over are called dummy subscripts, and others are free subscripts.
While introducing a dummy subscript into an expression, care needs to be taken not to use which is already present, either as a free / dummy subscript.
aijbjkckl cannot and must not, be replaced by
ailblkckl – NO
aimbmkckl – YES
aimbmncnl – YES