Talk:PlanetPhysics/Curl

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To the \htmladdnormallink{cross product}{http://planetphysics.us/encyclopedia/VectorProduct.html} of the \htmladdnormallink{gradient operator}{http://planetphysics.us/encyclopedia/Gradient.html} $\nabla \times$ Maxwell gave the name {\bf curl}.

$$\nabla \times {\bf V} = curl \,\, {\bf V}$$

The curl of a \htmladdnormallink{vector function}{http://planetphysics.us/encyclopedia/VectorFunctions.html} ${\bf V}$ is itself a vector function of \htmladdnormallink{position}{http://planetphysics.us/encyclopedia/Position.html} in space. As the name indicates, it is closely connected with the angular \htmladdnormallink{velocity}{http://planetphysics.us/encyclopedia/Velocity.html} or \htmladdnormallink{spin}{http://planetphysics.us/encyclopedia/QuarkAntiquarkPair.html} of the \htmladdnormallink{flux}{http://planetphysics.us/encyclopedia/AbsoluteMagnitude.html} at each point. But the interpretation of the curl is neither so easily obtained nor so simple as that of the \htmladdnormallink{divergence}{http://planetphysics.us/encyclopedia/DivergenceOfAVectorField.html}.

Consider as before that ${\bf V}$ represents the flux of a fluid. Take at a definite instant an infinitesimal sphere about any point $(x, y, z)$. At the next instant what has become of the sphere? In the first place it may have moved off as a whole in a certain direction by an amount $d{\bf r}$. In other words it may have a translational velocity of $d{\bf r}/dt$. In other words it may have undergone such a \htmladdnormallink{deformation}{http://planetphysics.us/encyclopedia/CohomologicalProperties.html} that it is no longer a sphere. It may have been subjected to a \emph{strain} by virtue of which it becomes slightly ellipsoidal in shape. Finally it may have been rotated as a whole about some axis through an angle $dw$. That is to say, it may have an angular velocity the \htmladdnormallink{magnitude}{http://planetphysics.us/encyclopedia/AbsoluteMagnitude.html} of which is $dw/dt$. An infinitesimal sphere therefore may have any one of these distinct \htmladdnormallink{types}{http://planetphysics.us/encyclopedia/Bijective.html} of \htmladdnormallink{motion}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} or all of them combined. First, a translation with definite velocity. Second, a strain with three definite rates of elongation along the axes of an ellipsoid. Third, an angular velocity about a difinite axis. It is this third type of motion which is given by the curl. In fact, the \emph{curl} of the flux $V$ is a \htmladdnormallink{vector}{http://planetphysics.us/encyclopedia/Vectors.html} which has at each point of space the direction of the instantaneous axis of rotation at that point and a magnitude equal to twice the instantaneous angular velocity about that axis.

The analytic discussion of the motion of a fluid presents more difficulties than it is necessary to introduce in treating the curl. The motion of a \htmladdnormallink{rigid body}{http://planetphysics.us/encyclopedia/CenterOfGravity.html} is sufficiently complex to give an adequate idea of the \htmladdnormallink{operation}{http://planetphysics.us/encyclopedia/Cod.html}. It was seen that the velocity of the \htmladdnormallink{particles}{http://planetphysics.us/encyclopedia/Particle.html} of a rigid body at any instant is given by the \htmladdnormallink{formula}{http://planetphysics.us/encyclopedia/Formula.html} $${\bf v} = {\bf v}_0 + {\bf a} \times {\bf r}$$

$$ curl \,\, {\bf v} = \nabla \times {\bf v} = \nabla \times {\bf v}_0 + \nabla \times \left ( {\bf a} \times {\bf r} \right) $$

Let

$$ {\bf a} = a_1 {\bf \hat{i}} + a_2 {\bf \hat{j}} + a_3 {\bf \hat{k}} $$

$$ {\bf r} = r_1 {\bf \hat{i}} + r_2 {\bf \hat{j}} + r_3 {\bf \hat{k}} = x {\bf \hat{i}} + y {\bf \hat{j}} + z {\bf \hat{k}} $$

expand $\nabla \times \left ( {\bf a} \times {\bf r} \right)$ formally as if it were the \htmladdnormallink{Vector Triple Product}{http://planetphysics.us/encyclopedia/BACKCAB.html} of $\nabla$, ${\bf a}$, and ${\bf r}$. Then

$$ \nabla \times {\bf v} = \nabla \times {\bf v}_0 + \left ( \nabla \cdot {\bf r} \right ) {\bf a} - \left ( \nabla \cdot {\bf a} \right ) {\bf r} $$

${\bf v}_0$ is a constant vector. Hence the term $\nabla \times {\bf v_0}$ vanishes.

$$ \nabla \cdot {\bf r} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} = 3  $$

Since ${\bf a}$ is a constant vector, it may be placed upon the other side of the differential \htmladdnormallink{operator}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html}, $\nabla \cdot {\bf a} = {\bf a} \cdot \nabla$

$$ {\bf a} \cdot \nabla {\bf r} = \left ( a_1 \frac{\partial}{\partial x} + a_2 \frac{\partial}{\partial y} + a_3 \frac{\partial}{\partial z} \right ) {\bf     r} = a_1 {\bf \hat{i}} + a_2 {\bf \hat{j}} + a_3 {\bf \hat{k}} = {\bf a}  $$

Hence

$$ \nabla \times {\bf v} = 3 {\bf a} - {\bf a} = 2 {\bf a} $$

Therefore in the case of the motion of a rigid body the curl of the linear velocity at any point is equal to twice the angular velocity in magnitude and in direction.

$$ \nabla \times {\bf v} = curl \,\, {\bf v} = 2 {\bf a} $$ $$ {\bf a} = \frac{1}{2} \nabla \times {\bf v} = \frac{1}{2} curl \,\, {\bf v}$$ $$ {\bf v} = {\bf v_0} + \frac{1}{2} \left ( \nabla \times {\bf v} \right ) \times {\bf r} = {\bf v_0} + \frac{1}{2} \left ( curl \,\, {\bf v} \right ) \times {\bf r} $$

The expansion of $\nabla \times \left ( {\bf a} \times {\bf r} \right )$ formally may be avoided by multiplying ${\bf a} \times {\bf r}$ out and then applying the operator $\nabla \times$ to the result.

\subsection{References}

[1] Wilson, E. "Vector Analysis." Yale University Press, New Haven, 1913.

This entry is a derivative of the Public \htmladdnormallink{domain}{http://planetphysics.us/encyclopedia/Bijective.html} \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} [1].

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