# Definition:Differential Operator

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## Theorem

A **differential operator** is a symbol indicating that differentiation needs to be performed.

It can be written as $D$ or $\dfrac \d {\d x}$, and so on.

### 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**.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**differential operator** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**differential operator**