By, now you know that if you want to change only the magnitude of a vector without changing its direction, you will go for the multiplication of the vector with a scalar quantity.
In case you want to create a new vector with a different magnitude as well as direction (than the initial vector) then you have to multiply the initial vector with another type of mathematical entity called a tensor.
The tensor is a more generalized form of scalar and vector. Or, the scalar, vector are the special cases of tensor.
- If a tensor has only magnitude and no direction (i.e., rank 0 tensor), then it is called scalar.
- If a tensor has magnitude and one direction (i.e., rank 1 tensor), then it is called vector.
- If a tensor has magnitude and two directions (i.e., rank 2 tensor), then it is called dyad.
- And so on...
Please note that there are differences between the term “direction" and the term “dimension." All the types of tensor (scalar, vector, and dyad) can be defined in a three dimensional space or co-ordinate system.
For describing a rank-1 tensor, one subscript should be sufficient. Refer the Fig.1 and the matrix representation of the vector a above for better clarity. You can think of a force vector for practical example.
For describing a rank-2 tensor or dyad, I will use the example of mechanical stress tensor below:
Please observe that each of the stress components of the stress tensor matrix has two subscripts, the first subscript is for the direction of area normal (the surface normal of the x2 –x3 surface is 1 and so on) and the second subscript is for direction of the stress component.
So, the stress tensor (a dyad or rank-2 tensor) has two directions namely direction of area normal and the direction of stress component.
Image credit: Wikipedia