Editing Tensor (section)

===Computer vision and optics===
The concept of a tensor of order two is often conflated with that of a matrix.  Tensors of higher order do however capture ideas important in science and engineering, as has been shown successively in numerous areas as they develop.  This happens, for instance, in the field of [[computer vision]], with the [[trifocal tensor]] generalizing the [[fundamental matrix (computer vision)|fundamental matrix]].

The field of [[nonlinear optics]] studies the changes to material [[Polarization density#Relation between P and E in various materials|polarization density]] under extreme electric fields.  The polarization waves generated are related to the generating [[electric field]]s through the nonlinear susceptibility tensor.  If the polarization '''P''' is not linearly proportional to the electric field '''E''', the medium is termed ''nonlinear''.  To a good approximation (for sufficiently weak fields, assuming no permanent dipole moments are present), '''P''' is given by a [[Taylor series]] in '''E''' whose coefficients are the nonlinear susceptibilities:

:<math> \frac{P_i}{\varepsilon_0} = \sum_j \chi^{(1)}_{ij} E_j  +  \sum_{jk} \chi_{ijk}^{(2)} E_j E_k + \sum_{jk\ell} \chi_{ijk\ell}^{(3)} E_j E_k E_\ell + \cdots. \!</math>

Here <math>\chi^{(1)}</math> is the linear susceptibility, <math>\chi^{(2)}</math> gives the [[Pockels effect]] and  [[second harmonic generation]], and <math>\chi^{(3)}</math> gives the [[Kerr effect]].  This expansion shows the way higher-order tensors arise naturally in the subject matter.