Derivations of tensor fields

In the representation using the principal axes, the component matrix contains only diagonal elements. 

There is a special set of invariants, the principal invariants Jk, that can be formed from the eigenvalues  

Written for the tensor itself, they can be calculated as follows  

These principal invariants are important in the context of yielding of materials and are used in section 3.3.1. 

Physical quantities are frequently not defined by a single tensor, but by assigning a tensor to each point in space, thus defining a field. Examples are a temperature field (a scalar field), where a temperature is specified at each point in space, a velocity field in a flowing fluid (a vector field), where the flow direction and speed are stated at each point, or the stress field in a material (a tensor field of second order), assigning a value of the stress tensor to each point. 

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Compatibility conditions are representation formulas for the jumps of partial derivatives of tensor fields in general in terms of the jumps of the tangential, the normal, and the displacement derivatives of the tensor field at its singular surface. 

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