What kind of tensor is the electromagnetic field tensor?

[center o bass]

The covariant form of the Lorentz force can be written as

m \ddot x^\mu =q F^{\mu \nu} \dot x_\nu

and such a relation should prove by the quotient rule that F is indeed a tensor.
But what kind of tensor is it? One can show that it transforms from an unprimed
to a primed system like

F'^{\mu \nu} = \Lambda^\mu_{\ \alpha} \Lambda^{\ \nu}_{\beta} F^{\alpha \beta} = \frac{\partial x'^\mu}{\partial x^\alpha} \frac{\partial x^\nu}{\partial x'^\beta} F^{\alpha \beta},

where \Lambda^{\ \nu}_{\beta} is the inverse Lorentz transformation matrix. But what kind of tensors transforms like this? Does it have a name? I know about covariant, contravariant and mixed tensors. The closest I get is a mixed tensor but it would transform like

T^\mu_\nu = \frac{\partial x'^\mu}{\partial x^\rho }\frac{\partial x^\sigma}{\partial x'^\nu} T^\rho_\sigma.

 

[tiny-tim]

hi center o bass!

a contravariant second-rank tensor (or second-rank contravariant tensor)

see eg http://mathworld.wolfram.com/Tensor.html

 

[center o bass]

Hi Tim!

But then it should transform like

F'^{\mu\nu} = \frac{\partial x'^\mu}{\partial x^\alpha} \frac{\partial x'^\nu}{\partial x^\beta} F^{\alpha \beta}

and not as

F'^{\mu \nu} = \frac{\partial x'^\mu}{\partial x^\alpha} \frac{\partial x^\nu}{\partial x'^\beta} F^{\alpha \beta}

which arises from the fact that the inverse lorentz transformation gives the equality

\frac{\partial x^\mu}{\partial x'^\nu} = \Lambda_\nu^{\ \ \mu}

and that I have shown that

F'^{\mu \nu} = \Lambda^\mu_{\ \ \alpha} \Lambda^{\ \ \nu}_{\beta} F^{\alpha \beta}

from transforming the lorentz force law in covariant form.

 

[center o bass]

Maybe there is a flaw in my proof? It goes as follows: Ignoring the constants m and q

\ddot x'^\mu = F'^{\mu \nu} \dot x'_\nu =\Lambda^\mu_{\ \alpha} (\dot x^\alpha) = \Lambda^\mu_{\ \alpha} (F^{\alpha \beta} \dot x_{\beta}) = \Lambda^\mu_{\ \alpha} F^{\alpha \beta} \Lambda_\beta^{\ \ \nu} \dot x'_\nu,
from which it should follow that

F&#039;^{\mu \nu} = \Lambda^\mu_{\ \alpha} \Lambda_\beta^{\ \ \nu} <br /> F^{\alpha \beta} = \frac{\partial x&#039;^\mu}{\partial x^\alpha} \frac{\partial x^\nu}{\partial x&#039;^\beta} F^{\alpha \beta}.

Here I have used the lorentz transformation and the inverse transformation
x&#039;^\mu = \Lambda^\mu_{\ \ \nu} x^\nu

x&#039;_\mu = \Lambda_\mu^{\ \ \nu} x_\nu

from which it follows that

\frac{\partial x^\mu}{\partial x&#039;^\nu} = \Lambda_\nu^{\ \ \mu}
and

\frac{\partial x&#039;^\mu}{\partial x^\nu} = \Lambda^\nu_{\ \ \mu}.