Vector operator

A vector operator is a differential operator used in vector calculus. Vector operators include the gradient, divergence, and curl:

• Gradient is a vector operator that operates on a scalar field, producing a vector field.
• Divergence is a vector operator that operates on a vector field, producing a scalar field.
• Curl is a vector operator that operates on a vector field, producing a vector field.

Defined in terms of del:

${\begin{aligned}\operatorname {grad} &\equiv \nabla \\\operatorname {div} &\equiv \nabla \cdot \\\operatorname {curl} &\equiv \nabla \times \end{aligned}}$

The Laplacian operates on a scalar field, producing a scalar field:

$\nabla ^{2}\equiv \operatorname {div} \ \operatorname {grad} \equiv \nabla \cdot \nabla$

Vector operators must always come right before the scalar field or vector field on which they operate, in order to produce a result. E.g.

$\nabla f$

yields the gradient of f, but

$f\nabla$

is just another vector operator, which is not operating on anything.

A vector operator can operate on another vector operator, to produce a compound vector operator, as seen above in the case of the Laplacian.

Set $O_{x,\alpha }^{n}(h)$ of Fractional Operators

The Fractional Calculus of Sets (FCS), first mentioned in the article titled "Sets of Fractional Operators and Numerical Estimation of the Order of Convergence of a Family of Fractional Fixed-Point Methods", is a methodology derived from fractional calculus. The main concept behind FCS is the characterization of elements of fractional calculus using sets due to the large number of fractional operators available. This methodology originated from the development of the fractional Newton-Raphson method and subsequent related works.

Illustration of some lines generated by the fractional Newton–Raphson method for the same initial condition $x_{0}$ but with different orders $\alpha$ of the fractional operator implemented. Source: Applied Mathematics and Computation

Fractional calculus, a branch of mathematics dealing with derivatives of non-integer order, emerged almost simultaneously with traditional calculus. This emergence was partly due to Leibniz's notation for integer-order derivatives: ${\frac {d^{n}}{dx^{n}}}$. Thanks to this notation, L’Hopital was able to ask Leibniz in a letter about the interpretation of taking $n={\frac {1}{2}}$ in a derivative. At that time, Leibniz could not provide a physical or geometrical interpretation for this question, so he simply replied to L’Hopital in a letter that "... it is an apparent paradox from which, one day, useful consequences will be drawn."

The term "fractional calculus" originates from a historical question, as this branch of mathematical analysis studies derivatives and integrals of a certain order $\alpha \in \mathbb {R}$. Currently, fractional calculus lacks a unified definition of what constitutes a fractional derivative. Consequently, when it is not necessary to explicitly specify the form of a fractional derivative, it is typically denoted as follows:

${\frac {d^{\alpha }}{dx^{\alpha }}}.$

Fractional operators have various representations, but one of their fundamental properties is that they recover the results of traditional calculus as $\alpha \to n$. Considering a scalar function $h:\mathbb {R} ^{m}\to \mathbb {R}$ and the canonical basis of $\mathbb {R} ^{m}$ denoted by $\{{\hat {e}}_{k}\}_{k\geq 1}$, the following fractional operator of order $\alpha$ is defined using Einstein notation:

$o_{x}^{\alpha }h(x):={\hat {e}}_{k}o_{k}^{\alpha }h(x).$

Denoting $\partial _{k}^{n}$ as the partial derivative of order $n$ with respect to the $k$-th component of the vector $x$, the following set of fractional operators is defined :

$O_{x,\alpha }^{n}(h):=\left\{o_{x}^{\alpha }:\exists o_{k}^{\alpha }h(x){\text{ and }}\lim _{\alpha \to n}o_{k}^{\alpha }h(x)=\partial _{k}^{n}h(x)\ \forall k\geq 1\right\},$

whose complement is:

$O_{x,\alpha }^{n,c}(h):=\left\{o_{x}^{\alpha }:\exists o_{k}^{\alpha }h(x)\ \forall k\geq 1{\text{ and }}\lim _{\alpha \to n}o_{k}^{\alpha }h(x)\neq \partial _{k}^{n}h(x){\text{ for at least one }}k\geq 1\right\}.$

As a consequence, the following set is defined:

$O_{x,\alpha }^{n,u}(h):=O_{x,\alpha }^{n}(h)\cup O_{x,\alpha }^{n,c}(h).$

Extension to Vector Functions

For a function $h:\Omega \subset \mathbb {R} ^{m}\to \mathbb {R} ^{m}$, the set is defined as:

${}_{m}O_{x,\alpha }^{n,u}(h):=\left\{o_{x}^{\alpha }:o_{x}^{\alpha }\in O_{x,\alpha }^{n,u}([h]_{k})\ \forall k\leq m\right\},$

where $[h]_{k}:\Omega \subset \mathbb {R} ^{m}\to \mathbb {R}$ denotes the $k$-th component of the function $h$.

See also

• del
• d'Alembert operator

Further reading

• H. M. Schey (1996) Div, Grad, Curl, and All That: An Informal Text on Vector Calculus, ISBN 0-393-96997-5.