Prove that the three components \(A_{x}=A_{y t}, A_{y}=A_{z_{1}}, A_{z} \equiv A_{x_{y}}\) of an antisymmetric dyadic transform as the components of a vector and hence enable A to be treated as a physical vector field when it is a function of \(x, y, z\), and \(t\).

The Lorentz transformation is a set of transformations that describe the relationship between the coordinates of events in space and time in different inertial reference frames. In essence, it allows us to determine how the coordinates of a point in space and time change when we switch from one inertial frame to another. For the standard configuration where the second frame moves along the x-axis with respect to the first frame, the Lorentz transformation can be written as: \(x' = \gamma(x - vt)\) \(y' = y\) \(z' = z\) \(t' = \gamma(t - \frac{vx}{c^2})\) where \(v\) is the relative velocity of the inertial frames, \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\) is the Lorentz factor, and \(c\) is the speed of light.

We will now apply the Lorentz transformation to each of the given components of the antisymmetric dyadic. For the x-component: \(A_{x}' = A_{y t}' = \frac{\partial A_{x'}}{\partial t'} = \frac{\partial A_{y t}}{\partial (\gamma(t - \frac{vx}{c^2}))}\) For the y-component: \(A_{y}' = A_{z_{1}}' = \frac{\partial A_{y'}}{\partial z'} = \frac{\partial A_{z_{1}}}{\partial z}\) For the z-component: \(A_{z}' = A_{x_{y}}' = \frac{\partial A_{z'}}{\partial y'} = \frac{\partial A_{x_{y}}}{\partial y}\)

Now let's verify that the transformed components satisfy the vector transformation properties: For the x-component: \(A_{x}' = \frac{\partial A_{x'}}{\partial t'} = \frac{\partial A_{y t}}{\partial (\gamma(t - \frac{vx}{c^2}))} = \frac{\partial A_{y t}}{\partial t}\) Here, we see that \(A_{x}'\) has the same dependence on time as the original component \(A_{x}\). For the y-component: \(A_{y}' = \frac{\partial A_{y'}}{\partial z'} = \frac{\partial A_{z_{1}}}{\partial z} = A_{y}\) Here, we see that \(A_{y}'\) has the same dependence on the z-coordinate as the original component \(A_{y}\). For the z-component: \(A_{z}' = \frac{\partial A_{z'}}{\partial y'} = \frac{\partial A_{x_{y}}}{\partial y} = A_{z}\) Here, we see that \(A_{z}'\) has the same dependence on the y-coordinate as the original component \(A_{z}\). Since all of the transformed components have the same dependence on their respective coordinates as the original components, we can conclude that they satisfy the vector transformation properties and thus can be treated as a physical vector field when they are functions of \(x, y, z,\) and \(t\).