PlanetPhysics/Examples of Lamellar Field

In the examples that follow, show that the given vector field is lamellar everywhere in and determine its scalar potential .\\

Example 1. \, Given

For the rotor (curl) of the field we obtain </math>\nabla\!\times\!\vec{U} = \left|\begin{matrix} \vec{i} & \vec{j} & \vec{k}\\ \frac{\partial}{\partial{x}} & \frac{\partial}{\partial{y}} & \frac{\partial}{\partial{z}}\\ y & x\!+\!\sin{z} & y\cos{z} \end{matrix}\right| \\= \left(\frac{\partial(y\cos{z})}{\partial{y}}-\frac{\partial(x\!+\!\sin{z})}{\partial{z}}\right)\vec{i} +\left(\frac{\partial{y}}{\partial{z}}-\frac{\partial(y\cos{z})}{\partial{x}}\right)\vec{j} +\left(\frac{\partial(x\!+\!\sin{z})}{\partial{x}}-\frac{\partial{y}}{\partial{y}}\right)\vec{k}Failed to parse (syntax error): {\displaystyle ,\\ which is identically <math>\vec{0}} for all , , .\, Thus, by the definition given in the parent entry, is lamellar.\\ Since \,,\, the scalar potential \,\, must satisfy the conditions Thus we can write where may depend on or . Differentiating this result with respect to and comparing to the second condition, we get Accordingly, where may depend on .\, So Differentiating this result with respect to and comparing to the third condition yields This means that is an arbitrary constant. Thus the form expresses the required potential function.\\

Example 2. \, This is a particular case in :

Now,\; </math>\nabla\!\times\!\vec{U} = \left|\begin{matrix} \vec{i} & \vec{j} & \vec{k}\\ \frac{\partial}{\partial{x}} & \frac{\partial}{\partial{y}} & \frac{\partial}{\partial{z}}\\ \omega y & \omega x & 0 \end{matrix}\right| = \left(\frac{\partial(\omega x)}{\partial{x}}-\frac{\partial(\omega y)}{\partial{y}}\right)\vec{k}=\vec{0}\vec{U} with\, .\, We deduce successively: Thus we get the result which corresponds to a particular case in .\\

Example 3. \, Given

The rotor is now\, </math>\nabla\!\times\!\vec{U} = \left|\begin{matrix} \vec{i} & \vec{j} & \vec{k}\\ \frac{\partial}{\partial{x}} & \frac{\partial}{\partial{y}} & \frac{\partial}{\partial{z}}\\ ax & by & -(a+b)z \end{matrix}\right|= \vec{0}.\nabla u=\vec{U} Differentiating (1) and (2) with respect to and using (3) give We substitute and again into (1) and (2) and deduce as follows: putting , into (1), (2) then gives us whence, by comparing,\, ,\, so that by (3), the expression and itself have been found, that is,

Unlike Example 1, the last two examples are also solenoidal, i.e.\, ,\, which physically may be interpreted as the continuity equation of an incompressible fluid flow.\\

Example 4. \, An additional example of a lamellar field would be with a differentiable function \,;\, if is a constant, then is also solenoidal.