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Del

Encyclopedia : D : DE : DEL : Del

For other uses, see Del (disambiguation)}}}.
In vector calculus, del is a vector differential operator represented by the nabla symbol, ∇.

In the three-dimensional Cartesian coordinate system R3 with coordinates (x, y, z), del can be defined as

[\nabla = \begin, , \end]
or alternatively,

[\nabla = \mathbf + \mathbf + \mathbf]
where [(\mathbf, \mathbf, \mathbf)] is the standard basis in R3.

The del operator 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, and using the Einstein summation notation del is written as

[ \nabla = \hat e_i \partial_i. ]
Del can also be expressed in other coordinate systems, see for example nabla in cylindrical and spherical coordinates.

Contents

Uses of del

Intuitively, del can be described as the general form of the derivative in multiple dimensions. When used in one dimension, it takes the form of the standard derivative in calculus.

The vector derivative of a scalar field f is called the gradient, and it equals

[ \nabla f=\left(, , \right)]
(also denoted [\mbox\,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.

Another common use of del in vector calculus is in the directional derivative

[ (\vec\cdot\nabla)f = a_ + a_ + a_]
(also denoted [\vec\cdot\mbox\,f]) which gives the change of a field [f] in the direction of [\vec]. This is used extensively in the dynamics of fluids where it describes the convective derivative, or 'moving' derivative of the fluid.

Del can also be applied to a vector field, with the result being a tensor. The tensor derivative of a vector field [v] is [ \nabla \otimes v] where [\otimes] represents the dyadic product.

One may also take the dot product and cross product of del with a vector field, [\vec=(v_1, v_2, v_3)] when one obtains the divergence and curl, given respectively by the formulas

[\nabla \cdot \vec = + + ]
(also denoted [\mbox\;\vec v]) and

[\nabla \times \vec=\left(} - }, } - },} - }\right) ]
(also denoted [\mbox\;\vec v]).

Divergence is, briefly, a measure of "spreadiness" — it tells one how much of a field, if it represented a flow, is accumulated in a point. Curl is a measure of "spinniness" — it tells one how a field, if it were a force field, would spin a pinwheel.

Another useful operator involving del is the Laplacian, [\Delta=\nabla^2.]

Del and second derivatives

For a scalar field f, the first derivative is, again, [\nabla f], which is a vector. There is really only one field to form.

For vectors, there are three primary modes of multiplication — cross products, dot products, and dyadic products. Hence, there are three possible second derivatives for a scalar field. While there are three possible first derivatives for a vector field, one of these is a scalar, one a vector, and one a tensor. Since the cross product is not well defined on tensors, we get 1 + 3 + 2 = 6 second derivatives for a vector field:

For a scalar field [f]
[\nabla \cdot \nabla f] [\nabla \times \nabla f] [\nabla \otimes \nabla f]
For a vector field [v]
[\nabla \cdot \nabla \times \vec] [\nabla \times \nabla \times \vec] [\nabla \otimes \nabla \times \vec]
[\nabla ( \nabla \cdot \vec )] [\nabla \cdot \nabla \otimes \vec] [\nabla \otimes \nabla \otimes \vec]

As long as the functions are well-behaved,

[\nabla \times \nabla f = 0]
[\nabla \cdot \nabla \times \vec = 0.]
Also, so long as the functions are well-behaved, two of these second derivatives are always equal:
[\nabla \cdot \nabla \otimes \vec = \nabla ( \nabla \cdot \vec ).]
So, there are only really 6 nontrivial unique second derivatives for well-behaved functions:

[\nabla \cdot \nabla f] [\nabla \otimes \nabla f] [\nabla (\nabla \cdot \vec)]
[\nabla \times \nabla \times \vec] [\nabla \otimes \nabla \times \vec] [\nabla \otimes \nabla \otimes \vec]

The Laplacian [\nabla^2 f] is easily the most important of these second derivatives; however, for well-behaved functions the matrix [\nabla \otimes \nabla f] is a symmetric matrix, and, consequently, it is usually also a Hermitian matrix. Hermitian matrices have real eigenvalues and orthogonal eigenvectors.

Finally, four more identities hold:

(1) [\nabla \cdot \nabla f = \nabla^2 f]
(2) [\nabla \times \nabla \times \vec = \nabla(\nabla \cdot \vec) - \nabla^2 \vec]
(3) [\nabla(\vec\cdot\vec)=\vec\cdot\nabla\vec+\vec\cdot\nabla\vec+\vec\times(\nabla\times\vec)+\vec\times(\nabla\times\vec)]
If [\vec = \vec] then (3) becomes:
[\frac\nabla(\vec\cdot\vec)=\frac\nabla(v^2)=\vec\cdot\nabla\vec+\vec\times(\nabla\times\vec)]
(4)[\nabla\times(f\vec)+\nabla f \times\vec]

See also

References

External links

 

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