In this paper, we review two related aspects of field theory: the modeling of the fields
by means of exterior algebra and calculus, and the derivation of the field dynamics, i.e., the Euler–
Lagrange equations, by means of the stationary action principle. In contrast to the usual tensorial
derivation of these equations for field theories, that gives separate equations for the field components,
two related coordinate-free forms of the Euler–Lagrange equations are derived. These alternative
forms ...

In this paper, we review two related aspects of field theory: the modeling of the fields
by means of exterior algebra and calculus, and the derivation of the field dynamics, i.e., the Euler–
Lagrange equations, by means of the stationary action principle. In contrast to the usual tensorial
derivation of these equations for field theories, that gives separate equations for the field components,
two related coordinate-free forms of the Euler–Lagrange equations are derived. These alternative
forms of the equations, reminiscent of the formulae of vector calculus, are expressed in terms
of vector derivatives of the Lagrangian density. The first form is valid for a generic Lagrangian
density that only depends on the first-order derivatives of the field. The second form, expressed
in exterior algebra notation, is specific to the case when the Lagrangian density is a function of the
exterior and interior derivatives of the multivector field. As an application, a Lagrangian density for
generalized electromagnetic multivector fields of arbitrary grade is postulated and shown to have,
by taking the vector derivative of the Lagrangian density, the generalized Maxwell equations as
Euler–Lagrange equations.

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