## Tensors

25 We now seek to generalize the concept of a vector by introducing the tensor (T), which essentially exists to operate on vectors v to produce other vectors (or on tensors to produce other tensors!). We designate this operation by T-v or simply Tv.

26 We hereby adopt the dyadic notation for tensors as linear vector operators u = T-v or ui = TijVj (1.39-a)

27 | In general the vectors may be represented by either covariant or contravariant components Vj or vj. Thus we can have different types of linear transformations

Ui = ±1 Vj; ui = Tj vj involving the covariant components Tj, the contravariant components Tij and the mixed components Tj or Ti3.

28 Whereas a tensor is essentially an operator on vectors (or other tensors), it is also a physical quantity, independent of any particular coordinate system yet specified most conveniently by referring to an appropriate system of coordinates.

29 Tensors frequently arise as physical entities whose components are the coefficients of a linear relationship between vectors.

30 A tensor is classified by the rank or order. A Tensor of order zero is specified in any coordinate system by one coordinate and is a scalar. A tensor of order one has three coordinate components in space, hence it is a vector. In general 3-D space the number of components of a tensor is 3" where n is the order of the tensor.

31 A force and a stress are tensors of order 1 and 2 respectively.

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