# What exactly is the meaning of weak formulations and what is its purpose?

What is the purpose behind weak formulation of PDEs? I have read in the book by Zienkiewicz and Taylor that a weak formulation is more "permissive" that the original problem in the sense that it allows for discontinuities in coefficients of PDEs where as the original PDE in differential form must have a "truly" smooth solution to be solved analytically. So my question is, firstly what is a weak formulation in the physical sense? and secondly, is it only used to handle PDEs with coefficient discontinuities in the sense that an analytical integration gives a "too smooth" result which isn't relevant in problems of nature?