This is a problem that arose in my electromagnetics class, but it is purely maths.
Problem
We have the current density vector defined in two different areas (in cylindrical coordinates) $$\mathbf{J}=\mathbf{a_r}\cdot(\sigma_1\cdot \rho \cdot \frac{a}{r\varepsilon_1}) \: \: \: \: \text{for} \: a<r<c$$ $$\mathbf{J}=\mathbf{a_r}\cdot(\sigma_2\cdot \rho \cdot \frac{a}{r\varepsilon_2}) \: \: \: \: \text{for} \: c<r<b $$ Where $\sigma_1,\sigma_2,\rho,a,\varepsilon_1,\varepsilon_2$ are just constants, and $\mathbf{a_r}$ is a unit vector.
There is the following relationship between the magnetic field $\mathbf{H}$, and the current density $\mathbf{J}$: $$\nabla \times \mathbf{H}=\mathbf{J} $$ It is also known that $$\nabla \cdot \mathbf{J}=0 $$
Find $\mathbf{H}$ for both cases of $\mathbf{J}$.
My attempt
The curl of a vector field $A$ in cylindrical coordinates can be found using this "formula". $$\nabla\times A=\mathbf{a_r}\cdot\bigg(\frac{\partial A_z}{r\partial\phi}-\frac{\partial A_{\phi}}{\partial z}\bigg)+\mathbf{a_{\phi}}\cdot \bigg(\frac{\partial A_r}{\partial z}-\frac{\partial A_z}{\partial r} \bigg) +\mathbf{a_z} \cdot \frac{1}{r}\bigg[\frac{\partial (rA_{\phi})}{\partial r}-\frac{\partial A_r}{\partial \phi}\bigg] $$
In our case, $\mathbf{J}$ only has an $\mathbf{a_r}$ component, so we end up with these three equations. $$\frac{\partial H_z}{r\partial\phi}-\frac{\partial H_{\phi}}{\partial z} =\sigma_1\rho_1 \cdot\frac{a}{r\varepsilon_1}$$ $$\frac{\partial H_r}{\partial z}-\frac{\partial H_z}{\partial r}=0 $$ $$\frac{\partial (rH_{\phi})}{\partial r}-\frac{\partial H_r}{\partial \phi}=0 $$
And here I have some difficulties, because I don't know how to solve these equations. $A$ is a vector field, and not just a regular set of variables.
I don't even know if this is the intended way to solve the problem, or if I'm supposed to use some other methods.
