This article is about the mathematical operator represented by the nabla symbol. For the symbol itself, see nabla symbol. For the operation associated with the symbol ∂, also sometimes referred to as "del", see Partial derivative. For other uses, see Del (disambiguation).
one-dimensional domain, it denotes the standard derivative of the function as defined in calculus. When applied to a field (a function defined on a multi-dimensional domain), it may denote any one of three operations depending on the way it is applied: the gradient or (locally) steepest slope of a scalar field (or sometimes of a vector field, as in the Navier–Stokes equations); the divergence of a vector field; or the curl
(rotation) of a vector field.
Del is a very convenient
commute
with other operators or products. These three uses, detailed below, are summarized as:
Where the expression in parentheses is a row vector. In
three-dimensional
Cartesian coordinate system with coordinates 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.
More specifically, for any scalar field and any vector field , if one defines
then using the above definition of , one may write
Del is used as a shorthand form to simplify many long mathematical expressions. It is most commonly used to simplify expressions for the
Laplacian
.
Gradient
The vector derivative of a scalar field 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. The magnitude of the gradient is the value of this steepest slope.
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 power of the del notation is shown by the following product rule:
The formula for the
vector product
is slightly less intuitive, because this product is not commutative:
Curl
The curl of a vector field 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:
This gives the rate of change of a field in the direction of , scaled by the magnitude of . In operator notation, the element in parentheses can be considered a single coherent unit;
convective derivative
—the "moving" derivative of the fluid.
Note that is an operator that takes scalar to a scalar. It can be extended to operate on a vector, by separately operating on each of its components.
Laplacian
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
) is the following, where is the outer product tensor:
Second derivatives
When del operates on a scalar or vector, either a scalar or vector is returned. Because of the diversity of vector products (scalar, dot, cross) one application of del already gives rise to three major derivatives: the gradient (scalar product), divergence (dot product), and curl (cross product). 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
vector Laplacian
gives two more:
These are of interest principally because they are not always unique or independent of each other. As long as the functions are well-behaved ( 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:
Precautions
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.
A counterexample that demonstrates the divergence () and the advection operator () 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. 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.