  This paper expands on previous work to derive and motivate the Lagrangian
formulation of field theories. In the process, we take three deliberate steps.
First, we give the definition of the action and derive Euler-Lagrange equations
for field theories. Second, we prove the Euler-Lagrange equations are
independent under arbitrary coordinate transformations and motivate that this
independence is desirable for field theories in physics. We then use the
Lagrangian for Electrodynamics as an example field Lagrangian and prove that
the related Euler-Lagrange equations lead to Maxwell's equations.
