Given is the tensor $T$
in Cartesian coordinates $T=\operatorname{diag}\{T_{xx},T_{yy},T_{zz}\}$
in cylindrical coordinates $T=\operatorname{diag}\{T_{rr},T_{\theta\theta},T_{zz}\}$
How does one express $$\frac{\partial T_{zz}}{\partial \theta}$$ through $T_{xx},T_{yy},T_{zz}$ in Cartesian coordinates (no longer differentiating with respect to $r,\theta,z$ but with respect to $x,y,z$)?
If $T$ is a function, then one should use the chain rule, but how is this to be done when $T$ is a tensor?
To rephrase the question:
Let's say, in cylindrical coordinates $\frac{\partial T_{\theta \theta}}{\partial \theta}=0$. My question is - how do we express this same relation in Cartesian coordinates? It has to be a relation involving $T_{xx}, T_{yy},T_{zz}$ and the partial derivatives $\frac{\partial T_{ii}}{\partial x_j}$, where i,j are one of {x,y,z}
You can view your tensor as a triplet of three functions $T_{xx}, T_{yy}, T_{zz}$. Each of these functions can be expressed in either Cartesian or cylindrical coordinates. Thus, $T_{zz} = T_{zz}(x, y, z) = T_{zz}(r, \theta, z)$. You want to express $\partial T_{zz}/\partial \theta$ in Cartesian coordinates. We therefore begin with \begin{equation} dT_{zz} = \frac{\partial T_{zz}}{\partial x}dx + \frac{\partial T_{zz}}{\partial y}dy + \frac{\partial T_{zz}}{\partial z}dz \end{equation} and hence \begin{equation} \frac{\partial T_{zz}}{\partial \theta} = \frac{\partial T_{zz}}{\partial x}\frac{\partial x}{\partial\theta} + \frac{\partial T_{zz}}{\partial y}\frac{\partial y}{\partial\theta} + \frac{\partial T_{zz}}{\partial z}\frac{\partial z}{\partial\theta} \end{equation} You can repeat if for other components of the tensor.