Definition:Curl Operator

From ProofWiki
Jump to navigation Jump to search

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 line 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.


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.


Rotational-and-irrotational-motion.png


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 line 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 line integral is greatest.

We have from Ampère's 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 curl 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