An element of a flowing liquid or gas will endure forces from the surrounding fluid, including viscous stress forces that cause it to gradually deform over time. These forces can be mathematically first order approximated by a viscous stress tensor, usually denoted by .
The deformation of a fluid element, relative to some previous state, can be first order approximated by a strain tensor that changes with time. The time derivative of that tensor is the strain rate tensor, that expresses how the element's deformation is changing with time; and is also the gradient of the velocity vector field  at that point, often denoted .
The tensors  and  can be expressed by 3×3 matrices, relative to any chosen coordinate system. The fluid is said to be Newtonian if these matrices are related by the equation
where  is a fixed 3×3×3×3 fourth order tensor that does not depend on the velocity or stress state of the fluid.