Definition:Curl Operator
Definition
Physical Interpretation
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:
Geometrical Representation
Let $R$ be a region of space embedded in Cartesian $3$ space $\R^3$.
Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$.
Let $\mathbf V$ be a vector field acting over $R$.
The curl of $\mathbf V$ at a point $A$ in $R$ is defined as:
- $\curl \mathbf V = \paren {\dfrac {\partial V_z} {\partial y} - \dfrac {\partial V_y} {\partial z} } \mathbf i + \paren {\dfrac {\partial V_x} {\partial z} - \dfrac {\partial V_z} {\partial x} } \mathbf j + \paren {\dfrac {\partial V_y} {\partial x} - \dfrac {\partial V_x} {\partial y} } \mathbf k$
where:
- $V_x$, $V_y$ and $V_z$ denote the magnitudes of the components at $A$ of $\mathbf V$ in the directions of the coordinate axes $x$, $y$ and $z$ respectively.
Cartesian $3$-Space
Let $\map {\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions..
Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$.
Let $\mathbf f := \tuple {\map {f_x} {\mathbf x}, \map {f_y} {\mathbf x}, \map {f_z} {\mathbf x} }: \R^3 \to \R^3$ be a vector-valued function on $\R^3$.
The curl of $\mathbf f$ is defined as:
| \(\ds \curl \mathbf f\) | \(:=\) | \(\ds \nabla \times \mathbf f\) | where $\nabla$ denotes the del operator | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\mathbf i \dfrac \partial {\partial x} + \mathbf j \dfrac \partial {\partial y} + \mathbf k \dfrac \partial {\partial z} } \times \paren {f_x \mathbf i + f_y \mathbf j + f_z \mathbf k}\) | Definition of Del Operator | |||||||||||
| \(\ds \) | \(=\) | \(\ds \begin {vmatrix} \mathbf i & \mathbf j & \mathbf k \\ \dfrac \partial {\partial x} & \dfrac \partial {\partial y} & \dfrac \partial {\partial z} \\ f_x & f_y & f_z \end{vmatrix}\) | Definition of Vector Cross Product | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\dfrac {\partial f_z} {\partial y} - \dfrac {\partial f_y} {\partial z} } \mathbf i + \paren {\dfrac {\partial f_x} {\partial z} - \dfrac {\partial f_z} {\partial x} } \mathbf j + \paren {\dfrac {\partial f_y} {\partial x} - \dfrac {\partial f_x} {\partial y} } \mathbf k\) |
Thus the curl is a vector in $\R^3$.
Also known as
The curl of a vector quantity is also known in some older works as its rotation, denoted $\operatorname {rot}$.
However, curl is now practically universal, being unambiguous and compact.
Examples
Rotation of Rigid Body
Consider a rigid body $B$ in rotary motion with angular velocity $\omega$ about an axis $OA$, where $O$ is some fixed point inside $B$.
Let $P$ be an arbitrary point inside $B$.
Let $B$ also be subject to a linear velocity $\mathbf v_0$ in an arbitrary direction.
Let the instantaneous velocity of $P$ be $\mathbf V$
Then:
- $\bsomega = \dfrac 1 2 \curl \mathbf V$
where $\bsomega$ is the angular velocity (axial) vector along the axis $OA$ in the sense according to the right-hand rule.
Motion of Fluid
Consider an infinitesimal volume of fluid $F$.
It may have $3$ kinds of motion:
- $(1): \quad$ Moving with a linear velocity as a whole
- $(2): \quad$ Changing its shape
- $(3): \quad$ In rotary motion.
At any instant, $F$ may be regarded as a rigid body.
Hence from Curl of Rotation of Rigid Body, the curl of the velocity of $F$ is twice its angular velocity where its axis of rotation at that instant is the same as that of the curl.
Consider the diagram above.
On the left, the element $E_1$ has itself rotated in moving to ${E_1}'$.
If every element of the body of fluid has rotated the same amount, $\curl \mathbf V$ would be twice the angular velocity about $O$.
On the right, on the other hand, the element $E_2$ has not actually rotated in moving to ${E_2}'$.
Hence there is no $\curl \mathbf V$ and its angular velocity is zero.
Magnetic Field of Conductor
Consider a conductor of electricity $C$.
Let $C$ be carrying a steady current $I$.
Let $P$ be an arbitrary point in the magnetic field $\mathbf H$ induced by $I$.
Let a small plane surface be placed at $P$, turned into a position so that the contour integral of the magnetic force taken around its boundary is the greatest possible.
This value per unit area is the curl of $\mathbf H$.
This is the magneto-motive force per unit area at $P$.
If $P$ is within $C$ at the point where current density is $\mathbf J$, this will be the total current passing normally through the closed curve when the contour integral is greatest.
We have from the Ampère-Maxwell Law that:
- $\curl \mathbf H = \mathbf J$
For a point in the magnetic field outside the conductor there is no current density and so:
- $\curl \mathbf H = 0$
Also see
- Results about the curl operator can be found here.
Historical Note
During the course of development of vector analysis, various notations for the curl operator were introduced, as follows:
| Symbol | Used by |
|---|---|
| $\nabla \times$ or $\curl$ | Josiah Willard Gibbs and Edwin Bidwell Wilson |
| $\curl$ | Oliver Heaviside Max Abraham |
| $\operatorname {rot}$ | Vladimir Sergeyevitch Ignatowski Hendrik Antoon Lorentz Cesare Burali-Forti and Roberto Marcolongo |