I am trying to calculate the Noether current, more specifically, the energy density of the Yang-Mills-Higgs Lagrangian. Please refer to the equations in the Harvey lectures on Magnetic Monopoles, Duality, and Supersymmetry. I am trying to get the equation 1.25 from the Lagrangian 1.20. My Lagrangian is as follows: $$\mathcal{L}=-\frac{1}{4}Tr(F_{\mu \nu}F^{\mu \nu})+\frac{1}{2}Tr(D_\mu \Phi D^\mu \Phi)-V(\Phi)$$
As the Lagrangian is invariant under gauge transformation, the Noether current is
$$\Theta = \frac{\partial \mathcal{L}}{\partial(\partial_\mu A_\nu)}\partial_\nu A_\mu + \frac{\partial \mathcal{L}}{\partial(\partial_\mu \Phi)}\partial_\nu \Phi $$
The first derivative of the current can be calculated as follows:
$$\frac{\partial(F_{\alpha\beta}F^{\alpha\beta})}{\partial(\partial_\mu A_\nu)}=2F^{\alpha \beta}\frac{\partial F_{\alpha \beta}}{\partial{(\partial_\mu A_\nu)}}$$ Expanding out $F_{\alpha\beta}$, and using $\frac{\partial(\partial_\alpha A_\beta)}{\partial(\partial_\mu A_\nu)}=\delta^\mu_\alpha\delta^\nu_\beta$, we get $$\frac{\partial(F_{\alpha\beta}F^{\alpha\beta})}{\partial(\partial_\mu A_\nu)}=2F^{\alpha\beta}(\delta^\mu_\alpha\delta^\nu_\beta-\delta^\mu_\beta\delta^\nu_\alpha)=4F^{\mu\nu}$$
Exactly in the same way, the calculation for the second derivative can be done. So I get the following: $$\Theta ^{\mu \nu}=-F^{\mu\nu} \partial_\nu A_\mu + D_\mu \Phi \partial_\nu \Phi$$
From this how do I get the answer written in Harvey's lectures. $$\Theta^{\mu \nu}=F^{\mu \rho}F^{\rho \nu}+D^\mu \Phi D^\nu \Phi$$
I have tried hard but I am not getting the 'extra' terms in the answer to cancel and give me my anser, or I am making some fundamental error. Thanks in advance.
EDIT: As Ron has pointed out, we are expected to calculate the symmetric energy tensor. Please could anyone tell me how to get the symmetric stress-energy tensor directly and from the canonical energy tensor, what is the conceptual difference between the two? If possible suggest a reference.