I am reading magnetostatics from Introduction to Electrodynamics Textbook by David J. Griffiths
So here ampere's circuital law in differential form was derived from biot-sarvart law and an assumption was made that j tends to 0 as we go to infinity.and the integral form was derived from the differential form.
But in next page ampere's circuital law in integral form was used to calculate magnetic field of a long infinite wire(whose current density won't be 0 at infinity)
My doubt is,is the condition for ampere's circular law is weaker (no requirement of 0 current density at infinity) or the author forgot to justify the usage of ampere's law in this case.
I know that ampere's law is a law and its validity is verified by experiments but my question is on the basis of sequence of book and hence I want a mathematical reasoning.