[AntonioLao]

analogous to Dirac's bra and ket vectors, the del operator has a dual called the ta operator. In 3D Cartesian system, the ta is denoted by



where i=1,2,3

the connection between the del and the ta operator is similar to the covariance and contravariance transformation of tensor analysis.

Traditionally, the outer product of the del operator with a vector is the curl operator. Similarly, the outer product of the ta operator with a vector is the toe operator.



when del operates on ta is the same as ta operates on del such that

[AntonioLao]

although the ta operator uses the same symbol as the difference operator, the ta is a special operator for rank 0 and rank 1 tensors. At first glance, it looks like the inverse of a differential operator. So for now, it could be called temporarily as an integral operator.

http://mathworld.wolfram.com/DifferentialOperator.html

analogous to the gradient vector operator, the tangent vector operator is the ta operator of a scalar, e.g., frequency.



the tangent of frequency is velocity (a vector).

http://mathworld.wolfram.com/Gradient.html

analogous to the divergence operator, the inner product of ta and a vector is called the convergence.



http://mathworld.wolfram.com/Divergence.html

[AntonioLao]

furthermore, the inner product of ta by ta is the surface or area operator.



and the triple products gives a volume operator



an alternate definition for the ta operator is that it is a length operator, i.e., it changes a scalar into a vector.

[AntonioLao]

correction to post#1