Question about electromagnetism

In the approximately ideal magnetizable iron circle with two different air gaps two coils are arranged concentrically. Calculate the mutual inductance without regard to the scattering.


I have results but I don't understand some parts so if someone would like to help me, I would be thankful.
First they use Ampere's law:


I don't understand this at all, why this sum, why those formulas?
Than Gauss's law:

I understand this part.

And than they use:

From where comes this minus?

 

Hello,

Did you try this problem using reluctance instead? Often comes out much simpler.
I'd have to look that up again myself, but you could try that yourself first.

 

I haven't learned that in school