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The nabla operator is a symbol that is used in the vector analysis to context-dependent to designate one of the three differential operators gradient, divergence or rotation. He is denoted by the nabla symbol ( or even to emphasize the formal similarity to conventional vector magnitudes ). Its name comes from the name of a harp-like stringed instrument Hebrew, which roughly had the shape of this sign.

Definition

Formally, the nabla operator is a vector whose components are the partial derivative operators:

It can be used both as the column vector (e.g., degrees) as well as the row vector ( for example, div) occur. In three-dimensional Cartesian coordinate system one also writes:

In this case, and the unit vectors of the coordinate system.

Calculations are performed with the nabla operator as a vector, where the "product" is interpreted by a right of standing function as a partial derivative.

Representation of other differential operators

In the n- dimensional space

Be an open subset of a differentiable function and a differentiable vector field.

The (formal ) product of the function gives the gradient

The (formal ) scalar product with the vector field whose divergence gives

The (formal ) scalar product of with itself yields the Laplace operator, because it is

In three-dimensional space

Now let be an open subset, a differentiable function and a differentiable vector field. In three-dimensional space with the Cartesian coordinates, the above formulas represent as follows:

The nabla operator applied to the scalar field gives the gradient of the scalar field

The result is a vector field. Here are the unit vectors of.

The nabla operator applied to the vector field gives the divergence of the vector field as a formal scalar product with the vector field to

Thus, a scalar field.

A special feature of the three-dimensional space is the rotation of a vector field. It is calculated by ( right-side ) function via the formal cross product as

So once again a vector field.

Notation with subscript

Does the Nablaoperator only to certain components of a function with a multidimensional argument, this will be indicated by a subscript. For a function with, for example, is

In contrast to

This term is common when the nabla symbol simple differential (i.e., the one-line Jacobian matrix ) or a part thereof is referred to.

Occasionally, alternative spellings of the nabla symbol on the spelling.

Calculation rules

Calculation rules for the nabla operator can be formally derived from the calculation rules for scalar and cross product, together with the derivation rules. One has to apply the product rule when the nabla operator is left of a product.

Are and scalar fields (functions) and and vector fields, then:

Further calculation rules, as well as the representation in different coordinate see Gradient, divergence and curl.