I have seen two forms of linear response:
One is in the calculation of susceptibilities using Green functions.
The other is in the evaluation of response currents, say, London current of a superconductor. The latter is very neat. Provided an effective action $S$, simply we have $$j^\mu(x)=\frac{\delta S}{\delta A_\mu(x)},$$ where $A$ is the vector potential.
My questions:
Are the two formalisms equivalent?
Can I calculate the response current to some fields other than $A$ via the second formalism?