Del

Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇.

When applied to a function defined on a one-dimensional domain, it denotes the standard derivative of the function as defined in calculus.

Del is a very convenient mathematical notation for those three operations (gradient, divergence, and curl) that makes many equations easier to write and remember.

As a vector operator, it can act on scalar and vector fields in three different ways, giving rise to three different differential operations: first, it can act on scalar fields by a formal scalar multiplication—to give a vector field called the gradient; second, it can act on vector fields by a formal dot product—to give a scalar field called the divergence; and lastly, it can act on vector fields by a formal cross product—to give a vector field called the curl.

These three uses, detailed below, are summarized as: In the Cartesian coordinate system

and standard basis or unit vectors of axes

, del is written as As a vector operator, del naturally acts on scalar fields via scalar multiplication, and naturally acts on vector fields via dot products and cross products.

Del is used as a shorthand form to simplify many long mathematical expressions.

It is most commonly used to simplify expressions for the gradient, divergence, curl, directional derivative, and Laplacian.

is called the gradient, and it can be represented as: It always points in the direction of greatest increase of

, and it has a magnitude equal to the maximum rate of increase at the point—just like a standard derivative.

In particular, if a hill is defined as a height function over a plane

, the gradient at a given location will be a vector in the xy-plane (visualizable as an arrow on a map) pointing along the steepest direction.

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 of a vector field

is a scalar field that can be represented as: 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.

is a vector function that can be represented as: The curl at a point is proportional to the on-axis torque that a tiny pinwheel would be subjected to if it were centered at that point.

The vector product operation can be visualized as a pseudo-determinant: Again the power of the notation is shown by the product rule: The rule for the vector product does not turn out to be simple: The directional derivative of a scalar field

is defined as: Which is equal to the following when the gradient exists This gives the rate of change of a field

In operator notation, the element in parentheses can be considered a single coherent unit; fluid dynamics uses this convention extensively, terming it the convective derivative—the "moving" derivative of the fluid.

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.

refers to a Laplacian or a Hessian matrix depends on the context.

Del can also be applied to a vector field with the result being a tensor.

(in three dimensions) is a 9-term second-rank tensor – that is, a 3×3 matrix – but can be denoted simply as

This quantity is equivalent to the transpose of the Jacobian matrix of the vector field with respect to space.

The divergence of the vector field can then be expressed as the trace of this matrix.

Applying these three sorts of derivatives again to each other gives five possible second derivatives, for a scalar field f or a vector field v; the use of the scalar Laplacian and vector Laplacian gives two more: These are of interest principally because they are not always unique or independent of each other.

in most cases), two of them are always zero: Two of them are always equal: The 3 remaining vector derivatives are related by the equation: 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.

) are not commutative: A counterexample that relies on del's differential properties: Central to these distinctions is the fact that del is not simply a vector; it is a vector operator.