# User Contributed Dictionary

see del

### Proper noun

Del- A diminutive of the male given name Derek.

Del

- Delaware

### Noun

See del- defn English

# Extensive Definition

∇

Del operator, represented by the nabla
symbol.

Del is a mathematical tool serving primarily as a
convention
for mathematical
notation; it makes many equations easier to
comprehend, write, and remember. Depending on the way del is
applied, it can describe the gradient (slope), divergence (degree to which
something converges or diverges) or curl (rotational motion at points
in a fluid). More intuitive descriptions of each of the many
operations del performs can be found below.

Mathematically, del can be viewed as the derivative in
multi-dimensional space. When used in one dimension, it takes the
form of the standard derivative of calculus. As an operator, it acts on vector
fields and scalar
fields with analogues of traditional multiplication. As with
all operators, these analogues should not be confused with
traditional multiplication; in particular, del does not commute.

## Definition

In the three-dimensional Cartesian coordinate system R3 with coordinates (x, y, z), del is defined as- \nabla = \mathbf + \mathbf + \mathbf

where is the standard
basis in R3.

Though this page chiefly treats del in three
dimensions, this definition can be generalized to the n-dimensional
Euclidean
space Rn. In the
Cartesian coordinate system with coordinates (x1, x2, …, xn),
del is:

- \nabla = \sum_^n \vec e_i

where \ is the standard basis in this
space.

More compactly, using the
Einstein summation notation, del is written as

- \nabla = \vec e_i \partial_i

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

## Notational uses of del

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.### Gradient

The vector derivative of a scalar field f is called the gradient, and it can be represented as:- \mbox\,f = \mathbf + \mathbf + \mathbf = \nabla f

It always points in the direction of greatest
increase of f, 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 h(x,y), the 2d projection of the
gradient at a given location will be a vector in the xy-plane (sort
of like 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:

- \nabla(f g) = f \nabla g + g \nabla f

The rules for products do not always turn out so
simple, as illustrated by:

- \nabla (\vec u \cdot \vec v) = (\vec u \cdot \nabla) \vec v + (\vec v \cdot \nabla) \vec u + \vec u \times (\nabla \times \vec v) + \vec v \times (\nabla \times \vec u)

### Divergence

The divergence of a vector field v(x,y,z) = vx i + vy j + vz k is a scalar function that can be represented as:- \mbox\,\vec v = + + = \nabla \cdot \vec v

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

The power of the del notation is shown by the
following product rule:

- \nabla \cdot (f \vec v) = f \nabla \cdot \vec v + \vec v \cdot \nabla f

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

- \nabla \cdot (\vec u \times \vec v) = \vec v \cdot \nabla \times \vec u - \vec u \cdot \nabla \times \vec v

### Curl

The curl of a vector field v(x, y, z) = v_x\mathbf + v_y\mathbf + v_z\mathbf is a vector function that can be represented as:- \mbox\;\vec v = \left( - \right) \mathbf + \left( - \right) \mathbf + \left( - \right) \mathbf = \nabla \times \vec v

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

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

- \nabla \times \vec v = \left|\begin \mathbf & \mathbf & \mathbf \\ \\ & & \\ \\ v_x & v_y & v_z \end\right|

Again the power of the notation is shown by the
product rule:

- \nabla \times (f \vec v) = (\nabla f) \times \vec v + f \nabla \times \vec v

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

- \nabla \times (\vec u \times \vec v) = \vec u \, \nabla \cdot \vec v - \vec v \, \nabla \cdot \vec u + (\vec v \cdot \nabla) \vec u - (\vec u \cdot \nabla) \vec v

### Directional derivative

The directional derivative of a scalar field f(x,y,z) in the direction a(x,y,z) = ax i + ay j + az k is defined as:- \vec\cdot\mbox\,f = a_x + a_y + a_z = (\vec a \cdot \nabla) f

This gives the change of a field f in the
direction of a. 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.

### Laplacian

The Laplace operator is a scalar operator that can be applied to either vector or scalar fields; it is defined as:- \Delta = + + = \nabla \cdot \nabla = \nabla^2

The Laplacian is ubiquitous throughout modern
mathematical
physics, appearing in Poisson's
equation, the heat
equation, the wave
equation, and the Schrödinger
equation — to name a few.

### Tensor derivative

Del can also be applied to a vector field with the result being a tensor. The tensor derivative of a vector field \vec is a 9-term second-rank tensor, but can be denoted simply as \nabla \otimes \vec , where \otimes represents the dyadic product. This quantity is equivalent to the Jacobian matrix of the vector field with respect to space.For a small displacement \delta \vec, the change
in the vector field is given by:

- \delta \vec = (\nabla \otimes \vec) \sdot \delta \vec

## Second derivatives

When del operates on a scalar or vector, generally a scalar or vector is returned. Because of the diversity of vector products, one application of del already gives rise to three major derivatives — the divergence, gradient, and curl. 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 Laplacian gives two more:- \mbox\,(\mbox\,f ) = \nabla \cdot (\nabla f)
- \mbox\,(\mbox\,f ) = \nabla \times (\nabla f)
- \Delta f = \nabla^2 f
- \mbox\,(\mbox\, \vec v ) = \nabla (\nabla \cdot \vec v)
- \mbox\,(\mbox\,\vec v ) = \nabla \cdot (\nabla \times \vec v)
- \mbox\,(\mbox\,\vec v ) = \nabla \times (\nabla \times \vec v)
- \Delta \vec v = \nabla^2 \vec v

These are of interest principally because they
are not always unique or independent of each other. As long as the
functions are well-behaved,
two of them are always zero:

- \mbox\,(\mbox\,f ) = \nabla \times (\nabla f) = 0
- \mbox\,(\mbox\,\vec v ) = \nabla \cdot \nabla \times \vec = 0

Two of them are always equal:

- \mbox\,(\mbox\,f ) = \nabla \cdot (\nabla f) = \nabla^2 f = \Delta f

The 3 remaining vector derivatives are related by
the equation:

- \nabla \times \nabla \times \vec = \nabla (\nabla \cdot \vec) - \nabla^2 \vec

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

- \nabla ( \nabla \cdot \vec ) = \nabla \cdot (\nabla \otimes \vec)

## 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 del is replaced by any
other vector. This is part of the tremendous value gained in
representing this operator as a vector in its own right.

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

A counterexample that relies on del's failure to
commute:

- \vec u \cdot \vec v = \vec v \cdot \vec u
- \nabla \cdot \vec v \ne \vec v \cdot \nabla

A counterexample that relies on del's
differential properties:

- (\nabla x) \times (\nabla y) = \mathbf
- (\vec u x )\times (\vec u y) = \mathbf

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 precise numerical magnitude and
direction, del doesn't have a precise value for either until it is
allowed to operate on something.

For that reason, identities involving del must be
derived from scratch, not derived from pre-existing vector
identities.

## See also

## References

- Div, Grad, Curl, and All That, H. M. Schey, ISBN 0-393-96997-5
- Jeff Miller, Earliest Uses of Symbols of Calculus (Aug. 30, 2004).
- Cleve Moler, ed., "History of Nabla", NA Digest 98 (Jan. 26, 1998).

## External links

- A survey of the improper use of ∇ in vector analysis (1994) Tai, Chen

Del in Catalan: Operador nabla

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