Tensors
摘要
A tensor of order n \((\in \{0,1,2,3,...\})\) in a three-dimensional space is a mathematical object comprised of \(3^n\) numbers, the numerical value of which may depend on the coordinate system we choose to express that tensor. These \(3^n\) numbers are called the scalar components of the tensor in the chosen coordinate system. Even though these scalar components, in general, are dependent on the choice of the coordinate system used to express it, the tensor as a mathematical entity and the physical quantity that it represents must be independent of the choice of the working coordinate system. In other words, the tensor itself is invariant to the choice of the coordinate system being used to express it. This property of the invariance of the tensor is ensured by a set of relationships that exists between the set of \(3^n\) components of the tensor in one coordinate system and another set of the \(3^n\) components of the same tensor in the other coordinate system. Such relationships are called the transformation rules of tensors. A tensor of order zero is a special case wherein the transformation rule is trivial. Such a tensor is described by one ( \(3^0\) ) number, which is independent of the choice of the working coordinate system.