Another way to look a tensors is as follows: Consider an object having a specific location and simple numerical value. That is a scalar. Now imagine that a second like object is placed in its near vicinity but with the opposite value. The difference between them multilied by the displacement is a dipole moment, and has not only a magnitude but also a direction, and in 3 dimensional space would be expressed by 3 components. Assuming linear mathematics, this would be a vector. Vectors can of course be combined by the usual procedures to give other vectors, that likewise each have (in this case) 3 components and are equivalent to a magnitude times a displacement. Now imagine you have a *vector* and then displaced from it another like vector except of opposite sign. This configuration is now associated with the underlying original value, the magnitude and direction of the first displacement, and the magnitude and direction of the second displacement. This object is vector-like but is more complex than a vector, and sums of such objects cannot necessarily be expressed as a simple replica of the exact style of object that one such object would be based only on the two-displacement model. (Although the similarity of this to a quadrupole moment is obvious, it is not correct to state that the structure is a quadrupole because one needs to distinguish between the case of starting out with a vector in the x direction followed by displacement in the y direction for the opposite vector, as distinguishted between starting out with the vector in the y direction and then displacing for the opposite vector in the x direction. These two procedures are not in general equivalent, and to have a true quadrupole, as in an electromagnetic sense, it is necessary that both processes be present simultaneously, something that is not true of all processes). In general, in 3 dimensional space, an object of this type require 9 components for its expression and admits of additional complexities, specifically, that tensors (of the second rank, which is what we are talking about here) can be expressed as combinations of (using ordinary geometrical language such as one would use in things such as mechanical engineering) symmetric, shear, and antisymmetric forms, or some combination of these. An alternative expression is in terms of dyads, none of which fit these descriptions but all of which are aligned along the coordinate axes of interest (here assumed rectangular) and can be combined to form the aforesaid forms. An example of a symmetric tensor is the surface of a lens, the gradient of whose surface constitutes an (effectively linear) vector field, so that the value of that gradient (which is a vector) as a function of the distance from the center (which is also a vector) is simply a numerical constant times the distance vector. This is a special kind of symmetric tensor, known as a unitary tensor, and is represented by a matrix having 0 as all off-diagonal values and a constant for the diagonal values. The multiplication whereby this tensor would actuate upon the position vector to give the gradient vector is simply to do the matrix multiplication of the diagonal-only matrix by the vector, which, of course, in this case simply multiplies each component of the position vector by the corresponding diagonal element in the matrix, resulting in simply a constant multiplication everywhere. If the lens is astigmatic, having different curvatures on perpendicular axes, the vector field varies from place to place in both magnitude and direction, so that in general the gradient is neither simply proportional to the distance from the center nor is it necessarily parallel to that displacement. It would be expressed by a matrix whose non-diagonal elements are still 0 but whose diagonal elements are unequal. That is of course if the coordinate system used is aligned with the principle axes of the lens. If they are not, then we will end up with something having nonzero off-diagonal components but they will be symmetrically balanced across the diagonal (hence the term symmetric tensor). In general, symmetric tensors can be associated with ellipsoidal or hyperboloidal configurations , whether in 2, 3, or more dimensions, and as such will be characterized by principle axes, and when these axes are aligned with the coordinate system being used, will result in a matrix expression having only diagonal values, the off-diagonals being zero. If the values are all of one sign, then the configuration will be ellipsoidal; otherwise hyperboloidal. (If some of the diagonal elements are zero, you get a configuration with cylindrical characteristics). Then we have the antisymmetric tensor. That is characterized by, when aligned with the coordinate axes, of a matrix expression that has zero for all diagonal positions, and in the off-diagonal positions, values on opposite sides of the diagonal are of opposite sign. An example of this is the velocity distribution of a spinning object. The dyadic tensor, mentioned earlier, is one that is aligned strictly on one pair of coordinate axes, so that it is like a vector along the x direction, and another like but opposite vector displace from it along the y direction, but antiparallel to the original. For some purposes these are the easiest and simplest to work with, and whenever you have a matrix expression for the tensor, each value in the matrix actually corresponds to one such dyad. A physical example of a pure dyadic tensor is the stress present in a beam when one face of it is being pulled in a direction parallel to its length while the other face is being pulled in the opposite direction.