Tensors in a two-dimensional Euclidean plane Consider a two-dimensional Euclidean plane with coordinates $(x^1,x^2)$
If we define a set of new coordinate $z^1$ and $z^2$
$$z^1=x^1+ix^2$$
$$z^2=x^1-ix^2$$
A question is if a symmetric tensor $T^{\mu\nu}$ satisfies the equation 
${T^\mu}_\mu=0$ then what are the consequences
for its co-variant components?
I don't really get the question, can anyone help?
 A: It's page 20 of the book.
Anyways, let's first work out things in $(x^1,x^2)$ coordinates. We are starting with
$$
T^{\mu\nu} = \begin{pmatrix}T^{11}&T^{12}\\T^{21}&T^{22}\end{pmatrix}
$$
Since the plane is Euclidean, we also know that
$$
g_{\mu\nu} = \begin{pmatrix}1&0\\0&1\end{pmatrix}
\quad,\quad
g^{\mu\nu} = \begin{pmatrix}1&0\\0&1\end{pmatrix}
$$
In terms of contravariant coordinates, the traceless tensor obeys:
$$
T^{\mu}_{\;\;\mu} = T^{\mu\nu} g_{\mu\nu} = T^{11} + T^{22} = 0
$$
Now, let's calculate the covariant components.
$$
T_{\mu\nu}
\stackrel{\text{def.}}{=}\begin{pmatrix}T_{11}&T_{12}\\T_{21}&T_{22}\end{pmatrix}
\stackrel{\text{def.}}{=}g_{\mu\rho}g_{\nu\sigma}T^{\rho\sigma}
=
\begin{pmatrix}T^{11}&T^{12}\\T^{21}&T^{22}\end{pmatrix}
$$
Keep in mind that the latter equality holds due to the particular form of the metric. Had we chosen it to be Lorentzian, we would end up with ${\{T_{11}=T^{11},T_{12}=-T^{12},T_{21}=-T^{21},T_{22}=T^{22}\}}$. Anyways, from the latter equation we conclude that ${\{T_{11}=T^{11},T_{22}=T^{22}\}}$. Therefore for covariant coordinates one has
$$
T^{\mu}_{\;\;\mu} = T_{11} + T_{22} = 0
$$
We now want to switch to $(z^1,z^2)$ coordinates. The partial derivatives will be calculated according to
$${x^1=\dfrac{1}{2}(z^1+z^2)
\quad,\quad
x_2=\dfrac{1}{2i}(z^1-z^2)}$$
The covariant components of the tensor in the new basis are:
$$
T_{\alpha\beta} = \dfrac{\partial x^\mu}{\partial z^\alpha}
\dfrac{\partial x^\nu}{\partial z^\beta} T_{\mu\nu} = 
\dfrac{1}{4}\begin{pmatrix}T_{11}-i(T_{12}+T_{21})-T_{22}&T_{11}+T_{22}\\T_{11}+T_{22}&T_{11}+i(T_{12}+T_{21})-T_{22}\end{pmatrix}
$$
Wherefrom it follows that the condition $T^{\mu}_{\;\;\mu} = T_{11} + T_{22} = 0$ implies vanishing of non-diagonal components in the $(z^1,z^2)$ basis.
