Del

Del can also be expressed in other coordinate systems, see for example del in cylindrical and spherical coordinates.

In particular, this notation is powerful because the gradient product rule looks very similar to the 1d-derivative case:

However, the rules for dot products do not turn out to be simple, as illustrated by:

The divergence is roughly a measure of a vector field's increase in the direction it points; but more accurately, it is a measure of that field's tendency to converge toward or diverge from a point.

The formula for the vector product is slightly less intuitive, because this product is not commutative:

The curl at a point is proportional to the on-axis torque that a tiny pinwheel would be subjected to if it were centred at that point.

The vector product operation can be visualized as a pseudo-determinant:

Unfortunately the rule for the vector product does not turn out to be simple:

The Laplace operator is a scalar operator that can be applied to either vector or scalar fields; for cartesian coordinate systems it is defined as:

and the definition for more general coordinate systems is given in vector Laplacian.

DCG chart: A simple chart depicting all rules pertaining to second derivatives. D, C, G, L and CC stand for divergence, curl, gradient, Laplacian and curl of curl, respectively. Arrows indicate existence of second derivatives. Blue circle in the middle represents curl of curl, whereas the other two red circles (dashed) mean that DD and GG do not exist.

And one of them can even be expressed with the tensor product, if the functions are well-behaved:

Most of the above vector properties (except for those that rely explicitly on del's differential properties—for example, the product rule) rely only on symbol rearrangement, and must necessarily hold if the del symbol is replaced by any other vector. This is part of the value to be gained in notationally representing this operator as a vector.

Though one can often replace del with a vector and obtain a vector identity, making those identities mnemonic, the reverse is not necessarily reliable, because del does not commute in general.

Central to these distinctions is the fact that del is not simply a vector; it is a vector operator. Whereas a vector is an object with both a magnitude and direction, del has neither a magnitude nor a direction until it operates on a function.

For that reason, identities involving del must be derived with care, using both vector identities and differentiation identities such as the product rule.