In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig of old ideas of Lord Kelvin. The differences here lie in certain weights attached to the selected entries of the tensor. Nomenclature may vary according to what is traditional in the field of application. For example, a 2×2 symmetric tensor X has only three distinct elements, the two on the diagonal and the other being off-diagonal. Thus it can be expressed as the vector . As another example: The stress tensor (in matrix notation) is given as In Voigt notation it is simplified to a 6-dimensional vector: The strain tensor, similar in nature to the stress tensor—both are symmetric second-order tensors --, is given in matrix form as Its representation in Voigt notation is where , , and are engineering shear strains. The benefit of using different representations for stress and strain is that the scalar invariance is preserved. Likewise, a three-dimensional symmetric fourth-order tensor can be reduced to a 6×6 matrix. Mnemonic rule A simple mnemonic rule for memorizing Voigt notation is as follows: Write down the second order tensor in matrix form (in the example, the stress tensor) Strike out the diagonal Continue on the third column Go back to the first element along the first row. Voigt indexes are numbered consecutively from the starting point to the end (in the example, the numbers in blue). Mandel notation For a symmetric tensor of second rank only six components are distinct, the three on the diagonal and the others being off-diagonal. Thus it can be expressed, in Mandel notation, as the vector The main advantage of Mandel notation is to allow the use of the same conventional operations used with vectors, for example: A symmetric tensor of rank four satisfying and has 81 components in three-dimensional space, but only 36 components are distinct. Thus, in Mandel notation, it can be expressed as Applications The notation is named after physicist Woldemar Voigt & John Nye (scientist). It is useful, for example, in calculations involving constitutive models to simulate materials, such as the generalized Hooke's law, as well as finite element analysis, and Diffusion MRI. Hooke's law has a symmetric fourth-order