# Definition:Differential Operator

## Theorem

Let $A$ be a mapping from a function space $\FF_1$ to another function space $\FF_2$.

Let $f \in \FF_2$ be a real function such that $f$ is the image of $u \in \FF_1$ that is:　$f = A \sqbrk u$

A differential operator is represented as a linear combination, finitely generated by $u$ and its derivatives containing higher degree such as

$\ds \map P {x, D} = \sum _{\size \alpha \mathop \le m} \map {a_\alpha} x D^\alpha$

where:

$\alpha = \set {\alpha_1, \alpha_2, \dotsc \alpha_n}$ is a set of non-negative integers forming a multi-index
$\size \alpha = \alpha_1 + \alpha_2 + \dotsb + \alpha_n$ is the length of $\alpha$
the $\map {a_\alpha} x$ are real functions on a open domain in a real cartesian space of $n$ dimensions
$D^\alpha = D_1^{\alpha_1} D_2^{\alpha_2} \dotsm D_n^{\alpha_n}$.

### Divergence Operator

Let $\mathbf V$ be a vector field acting over a region of space $R$.

The divergence of $\mathbf V$ at a point $P$ is the total flux away from $P$ per unit volume.

It is a scalar field.

Let $R$ be a region of space.

Let $F$ be a scalar field acting over $R$.

The gradient of $F$ at a point $A$ in $R$ is defined as:

$\grad F = \dfrac {\partial F} {\partial n} \mathbf {\hat n}$

where:

$\mathbf {\hat n}$ denotes the unit normal to the equal surface $S$ of $F$ at $A$
$n$ is the magnitude of the normal vector to $S$ at $A$.

### Curl Operator

Let $\mathbf V$ be a vector field acting over a region of space $R$.

Let a small vector area $\mathbf a$ of any shape be placed at an arbitrary point $P$ in $R$.

Let the contour integral $L$ be computed around the boundary edge of $A$.

Then there will be an angle of direction of $\mathbf a$ to the direction of $\mathbf V$ for which $L$ is a maximum.

The curl of $\mathbf V$ at $P$ is defined as the vector:

whose magnitude is the amount of this maximum $L$ per unit area
whose direction is the direction of $\mathbf a$ at this maximum.

## Also defined as

Some sources, particularly those dealing with specific physical phenomena such as electromagnetism and quantum mechanics, use the term differential operators to mean:

the divergence operator $\operatorname {div}$
the gradient operator $\grad$
the curl operator $\curl$.

## Also see

• Results about differential operators can be found here.