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