Why is vector calculus so much more important in classical electrodynamics than in classical mechanics?

Question

In this question "vector calculus" refers to the integration and differentiation of vector fields.

Why is vector calculus so much more important in classical electrodynamics than in classical mechanics?

I'm not looking for answers such as "there are these formulas which are prominent in electrodynamics and those formulas use vector calculus", because my question essentially is: what is fundamental about electrodynamics compared to mechanics that would require such formulas to appear in it in the first place?

In other words, what fundamental physical fact that separates the forces of electrodynamics from those of mechanics makes it likely that in the formulas of the electrodynamical model vector calculus will make an appearance?

Answer 1

Vector calculus is necessary when describing the dynamics of fields, which are described mathematically as functions of several variables (usually spatial coordinates and time). The electric and magnetic fields are typically the first example of dynamical fields that you encounter during your physics education. In classical particle mechanics, the fundamental dynamical variables are the particle's position and momentum, which are functions of only one variable (i.e. time) and so vector calculus is not necessary.

Vector calculus does appear in classical particle mechanics in the sense that particles often interact with background fields, but because those fields typically don't evolve with time it is often not necessary to do any intensive vector calculus with them, as you would if you had to study their evolution equations.

With that being said, the premise of the question is not necessarily true from a more advanced perspective. Continuum mechanics (e.g. fluid dynamics or elastodynamics) falls under the general heading of classical mechanics, but is also fundamentally a theory of fields and requires quite a lot of vector and tensor calculus. Classical statistical mechanics and thermodynamics also make use of vector calculus in a generalized sense, where the "fields" in question are no longer necessarily functions of spatial coordinates.

Answer 2

No, when it comes to fluids, say, or if you are interested in the internal motion of extended objects, or if you are interested in the general behaviour of a family of systems i.e. all the different energies, not just the single one specific to one instantiation thereof, vector calculus also becomes important to classical mechanics.

It is just that we are often just trying to teach the basics in classical mechanics, and exploiting conservation of momentum and energy and angular momentum and assuming rigid bodies, that the complicated equations simplify.

It is also the case that in electrodynamics we care about the fields in the vacuum of space, where the fields are not constant. In the gravitational case we often just do it with the gravitational potential and rarely need the full blown gravitational field. The electrostatic potential came from people having already studied the gravitational potential and field cases, and they realised those same ideas applied to electrostatics too.

Answer 3

Classical electrodynamics is a field theory. It describes the electric and magnetic fields as vector fields. Thus, vector calculus is so important in classical electrodynamics.

Vector calculus is needed to describe any theory involving fields. Representing some quantity as a field may help to develop intuition. Thus, vector calculus can also be used in classical mechanics if using a "field interpretation" is more intuitive than the alternatives for that particular problem. Example- fluid mechanics; Cartan's theory of gravity.