Is the definition of the stress vector the following?
The stress vector is the force per unit surface.
The stress tensor is the matrix $\{\sigma_{ij}(x,t)\}$ and its $(i,j)$-component is the $i$-component of the force per unit surface that is exerted at an element of the surface perpendiculart to the direction $j$. Is this definition of the stress tensor correct?
Which is the form of the stress tensor at a calm fluid?
Is the definition of the (static) pressure the following?
The (static) pressure is the diagonal entries of the tensor matrix.
Best Answer
Your definition of the stress tensor seems correct. If you slice through a stressed body, the stress tensor has a component vector acting across the cut surface. That's sometimes called the 'stress vector' but is better called the 'traction vector', the word 'stress' being reserved for the tensor.
Aside: in Voigt notation, stress is represented by a column matrix also sometimes called the 'stress vector'. This is purely a notational convenience to allow us to write the (4th order) elasticity tensor on a flat piece of paper; stress is properly a (2nd order) tensor.