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:

(1) Are the two formalisms equivalent?

(2) Can I calculate the response current to some fields other than $A$ via the second formalism?

I have seen two forms of linear response:

1. One is in the calculation of susceptibilities using Green functions.

2. 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:

1. Are the two formalisms equivalent?

2. Can I calculate the response current to some fields other than $A$ via the second formalism?