# Definition:Differential Operator

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## 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
- $D^\alpha = D_1^{\alpha_1} D_2^{\alpha_2} \dotsm D_n^{\alpha_n}$.

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### 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.

### Gradient Operator

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:

## 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**.