How do I use Maxwell's equations when there is no symmetry in the situation Consider the case in which there is initially $0$ electric or magnetic field anywhere in space, and then in a circular region of radius $R$ a constantly increasing magnetic field is introduced (at a rate of say $p$ Teslas per second). 

(ok the picture isn't much use, but I had already made it in paint so I included it)
How do I calculate the electric field at every point in space? I've only been taught the integral forms of Maxwell's equations, and have always been told to only use them when there is a symmetry in the situation. Sadly, I don't think I can apply those laws in this case. However, I am aware there is a differential version of Maxwell's equations, which use operators such as gradient and divergence (which I'm not really all that familiar with). Would using these alternate forms help with solving such problems without any symmetry? If not, then how would you solve such situations, would it only be possible through computation?
The reason I wanted to solve this situation was that I could then take the limit as the circle size approached infinity, which was the original problem that I had in my head. If anyone has any thoughts on what would happen, much appreciated
 A: The differential versions of Maxwell's equations would tell you something about the fields at a point.
For example, inside the circle, where $\partial \vec{B}/\partial t = p \hat{k}$, then Faraday's law
$$ \nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t},$$
tells you what the "curl" of the E-field must be.
Similarly, Ampere's law, with no current present, tells you that
$$\nabla \times \vec{B} = \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}$$
But if you have not studied vector calculus then little is to be gained by discussing these.
You do not need these for the problem you pose. There is cylindrical symmetry around the z-axis (into the page). The integral version of Faraday's law tells you that
$$ \oint \vec{E} \cdot d\vec{l} = - \int \frac{\partial \vec{B}}{\partial t} \cdot d\vec{S}$$
In this case you specify the rate of change of magnetic field. You define a closed circular loop centred on the z-axis. You can argue that the E-field must circulate around the region, parallel with this loop. A symmetric radial field is ruled out since no field lines can end or originate without any free charge and a time-dependent z-component of the E-field is ruled out since that would have an associated B-field that circulated in the plane of your diagram$^1$. You then have a simple left hand side.
The solution has a different form for the E-field inside and outside the circular area you have drawn.
$^1$ NB: This argument also relies on the projected arrangement you have drawn extending infinitely along the z-axis. 
A: As I understand it, your system does have a great deal of symmetry. I'm assuming the problem is cylindrical, i.e. the picture you have drawing is valid at all values of $z$, if we make the $z$ co-ordinate out of the page. So, I'm assuming your problem definition is $\vec{H}(t) = k\,t\, \hat{z};\forall\,t>0;\, r<r_0$ where $r_0$ is the radius of the magnetized region and the initial conditions are $H(r,z,0)=E(r,z,0)=0$.
By symmetry, you argue that the fields depend on $r$ and $t$ only. So it remains to find out the $r,\,t$ dependence and the directions of the fields. Use the integral forms: these are as good to work with as the differential forms for a problem like this. You can work out the governing equations easily from these symmetries, but you're going to need either series of Bessel functions (which you may not yet be familiar with) or numerical analysis to solve them.
Consider the radial components $E_r(r,t)$ and $B_r(r,t)$. Apply Gauss's laws to an arbitrary radius cylinder centered on the circular region. There is no electric nor magnetic charge in side this surface, and the fluxes through it per unit length are $2\,\pi\,r\,E_r(r,t)$ and $2\,\pi\,r\,H_r(r,t)$ so these must both be nought. The radial components of all fields vanish.
Now consider a circular loop concentric with the field system. Work out the flux of $\vec{B}$ through it and the line integral of $\vec{E}$ around it and apply Faraday's law:
$$-2\,\pi\,\mu_0\,\frac{\mathrm{d}}{\mathrm{d}\,t} \int_0^r\,u\,H_z(u,\,t)\,\mathrm{d}\,u=2\,\pi\,r\,E_\phi(r,\,t)\tag{1}$$
where $E_\phi$ is the component of $\vec{E}$ tangential to the loop. You may have heard this called the azimuthal component. Now do the same for the flux of $\vec{E}$ through the loop and the line integral of $\vec{H}$ around it and apply Ampère's law:
$$2\,\pi\,\epsilon_0\,\frac{\mathrm{d}}{\mathrm{d}\,t} \int_0^r\,u\,\,E_z(u,\,t)\,\mathrm{d}\,u=2\,\pi\,r\,H_\phi(r,\,t)\tag{2}$$
Lastly, we must apply Faraday's and Ampere's laws around a square loop lying in a radial plane stretching from the center of the problem $r=0$ out to the radius $r$. If you do this you get:
$$\mu_0\,\frac{\mathrm{d}}{\mathrm{d}\,t} \int_0^r\,H_\phi(u,\,t)\,\mathrm{d}\,u=E_z(r,\,t)-E_z(0,\,t)\tag{3}$$
$$-\epsilon_0\,\frac{\mathrm{d}}{\mathrm{d}\,t} \int_0^r\,E_\phi(u,\,t)\,\mathrm{d}\,u=H_z(r,\,t)-H_z(0,\,t)\tag{4}$$
Now the usual method is to differentiate all these equations with respect to $r$; this will yield exactly the same results that you would have gotten with the differential form Maxwell equations:
$$-\mu_0\,\,r\,\frac{\partial}{\partial\,t}\,H_z(r,\,t)=r\,\frac{\partial}{\partial\,r}\,E_\phi(r,\,t) + E_\phi(r,\,t)\tag{1a}$$
$$\epsilon_0\,\,r\,\frac{\partial}{\partial\,t}\,E_z(r,\,t)=r\,\frac{\partial}{\partial\,r}\,H_\phi(r,\,t) + H_\phi(r,\,t)\tag{2a}$$
$$\mu_0\,\frac{\partial}{\partial\,t} H_\phi(r,\,t)=\frac{\partial}{\partial\,r}\,E_z(r,\,t)\tag{3a}$$
$$-\epsilon_0\,\frac{\partial}{\partial\,t} E_\phi(r,\,t)=\frac{\partial}{\partial\,r}\,H_z(r,\,t)\tag{4a}$$
and you now differentiate  (1a) and (2a) with respect to $t$ and thensubstitude (3a) and (4a) into the time-derviatives of (1a) and (2a) to get two, second order differential equations in $E_z$ and $H_z$. These are the cylindrical wave equations for $E_z$ and $H_z$. Notice how $(H_\phi,\,E_z)$ and $(E_\phi,\,H_z)$ are independent pairs - neither of $H_\phi,\,E_z$ has any bearing on $E_\phi,\,H_z$ nor contrariwise. The wave equation for $H_z$ is at last:
$$\mu_0\,\epsilon_0\,r\,\frac{\partial^2}{\partial\,t^2}\,H_z(r,\,t)=r\,\frac{\partial^2}{\partial\,r^2}\,H_z(r,\,t) + \frac{\partial}{\partial\,r}\,H_z(r,\,t)\tag{5}$$
and our task is now to match up solutions of this one with your initial conditions.
This is actually highly nontrivial to solve properly. I'll sketch how it is done. One usually builds solutions up as superpositions of solutions that vary sinusoidally with time, and uses Fourier analysis to do so. Outside the central "excitation" region i.e. where $r>r_0$, when the solution varies sinusoidally with with, i.e. $H_z(r,\,t) = h_z(r)\,e^{i\,\omega\,t}$, the solution to (5) is an outwardly propagating wave:
$$H_z(r,\,t) = H^{(1)}_0\left(\frac{\omega\,r}{c}\right)\,e^{i\,\omega\,t}\tag{6}$$
where $H^{(1)}_0$ is the Hankel function; you probably haven't met this one yet, but for large distances it looks like a wave of the form $e^{i\,k\,r}/\sqrt{r}$, so your system is radiating waves travelling at the speed of light $c$. The disturbance takes time $r/c$ to reach a radius $r$. The general solution is a superposition of this over all frequencies:
$$H_z(r,\,t) =\int_{-\infty}^\infty A(\omega)\, H^{(1)}_0\left(\frac{\omega\,r}{c}\right)\,e^{i\,\omega\,t}\,\mathrm{d}\,\omega\tag{7}$$
and the task reduces to one of finding the superposition function $ A(\omega)$ to match your problem. 
