David J. Griffiths in his "Introduction to Electrodynamics" (chapter2), says that the Electric field inside a cavity (in a conductor) which contains a point charge is nonzero. this is because we can choose a gaussian surface inside the cavity (just a little smaller than the cavity), then by applying Gauss's law:
$$\int \vec E \, .d\vec a = \frac{q}\epsilon_0 \rightarrow \vec E\neq0$$
and when the charge is outside of the conductor (the case of Faraday Cage), he argues that if we choose a closed path which is partly in the "meat" of the conductor and partly in the cavity, then we can prove that the Field is zero in the latter reigion ($C_1$ is the path in the conductor and $C_2$ is the path in the cavity):
$$\vec \nabla \times \vec E=0 \Rightarrow \int_{C}\vec E. \vec dl=\int_{C_1} \vec E_1.\vec dl +\int_{C_2}E_2.\vec dl=0$$
since $E_1$ (the field inside the conductor) is zero, $E_2$ should be zero as well.
can someone plaese explain his second argument? I don't really understand it, what exactly is preventing us to apply this argument to the first case? conceptually and mathematically (in the context of electrostatics)
Thanks in advance.