Introduction to tensor notation

In the derivation of balance equations the tensor notation is used because it allows the equations to be written in a clearer and simpler fashion. We have restricted ourselves in this to cartesian coordinates. In the following, the essential features of cartesian tensor notation are only illustrated to an extent required for the derivation of the balance equations extensive publications are available for further reading. We will start with an example. The velocity w of a point of mass is known as a vector which can be set in a cartesian coordinate system using its components wxl wy, wz  

If the unit vector in a cartesian coordinate system is indicated by ex, ey, ez, it holds that 

In tensor notation, the indices x, y, z are replaced by the indices 1, 2, 3 and we write instead 

The velocity vector w is characterised completely by its components w, i = 1, 2, 3. In tensor notation, the velocity vector is indicated by the abbreviation with = 1,2, 3. Correspondingly, the position vector x(x, y, z) is determined by its components x1 = x, x2 = y, x3 = z, and in tensor notation by the abbreviated xf with = 1,2, 3. According to this, a vector is indicated by a single index. 

It is possible to differentiate between tensors of different levels. Zero level tensors are scalars. They do not change by transferring to another coordinate system. Examples of scalars are temperature , pressure p and density g. No index is necessary for their characterisation. 

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