An action

$$S[\varphi]=\int d^{4} x L(\varphi, \varphi, a)$$

is given, where $\varphi(x)$ is a scalar field. Explain heuristically how to compute the functional derivative $\delta S / \delta \varphi$.

Consider the action for electromagnetism,

$$S\left[A_{a}\right]=-\int d^{4} x\left\{\frac{1}{4 \mu_{0}} F^{a b} F_{a b}+J^{a} A_{a}\right\} .$$

Here $J^{a}$ is the 4-current density, $A_{a}$ is the 4-potential and $F_{a b}=A_{b, a}-A_{a, b}$ is the Maxwell field tensor. Obtain Maxwell's equations in 4-vector form.

Another action that is sometimes suggested is

$$\widehat{S}\left[A_{a}\right]=-\int d^{4} x\left\{\frac{1}{2 \mu_{0}} A^{a, b} A_{a, b}+J^{a} A_{a}\right\} .$$

Under which additional assumption can Maxwell's equations be obtained using this action?

Using this additional assumption establish the relationship between the actions $S$ and $\widehat{S}$.