The kernel of an operator (or matrix) is a vectorspace. The operator transforms the elements of the kernel to the null vector. For the Jordan form, the following dimensions should be known
dim kernel (A-λI)k,
where k=1...r and r is the algebraic multiplicity of λ.
For k=1,
dim kernel...
It is good to start with determining the D vector from the Gauss theorem. D is constant inside the device
D=Q/A,
where Q is the charge on the metal plate of the capacitor. Then, the electric field is
E(x)=D/( ε_0 ε_r(x) )
The polarization vector is
P(x)=ε_0 χ(x) E(x),
where...
Just add together the results of sheets as they would be connected as a series
1/C=Ʃ_i Δx_i/ (ε_0 ε_r(x_i) A)
Taking the limit Δx_i -> 0
1/C=\int_0^L dx/(ε_0 ε_r(x) A)
where Δx_i is the thickness of one sheet, L is the distance between the plates and A is the area. Finally, you...