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Please excuse me if this question is a duplicate. I tried my best but I didn't find an existing question for this.

In physics class, I was first introduced to gravity in terms of a force, specifically, the inverse square law. At the same time, there is also the concept of gravitational potential energy, and the gravitational force vector always pointed from high potential to low potential.

Mathematically, you could describe that as $F_g = -\nabla g$, where $g$ is the gravitational potential field. On the other hand, you could also say in terms of work $g = -\int F_g \cdot dx$ (assuming a point at $\infty$ has 0 potential).

The way I typically thought about it is that something like flux as the "root source" of the fields, and the flux comes from a property such as mass for gravity or charge for electromagnetism. Then, the potentials are a result of the force. That makes sense to me since the force field is like the flux area density, which would be proportional to the inverse square of the distance from the source of the flux. For example, the electric force weakens by a factor of 4 when your distance doubles, since the flux is being spread over a Gaussian surface that's 4 times larger.

My question is, is the flux responsible for setting up the forces, from which the potentials come as a result, or does it set up the potentials, and the forces simply act along the gradient? Or, are these interpretations the same?

In problems I usually thought about it in terms of the differential way but I would love to get some insight on this. Thanks.

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Thinking in terms of "root source" is somehow like going in circles: you might always choose the starting point of your theory however you want and argue that the remaining quantities easily follow from the definitions (or as properties, or else). What makes more sense, in physics, is trying to understand what are the observable quantities (i. e. quantities that one can empirically measure with experiments) versus the mathematical framework that you introduce to describe the theory and make predictions.

In your example the observable quantities are the forces and the charges (by charge we mean gravitational/electrical unit of particles) because most experiments are designed to directly access them; in principle not even fields are observables as it is experimentally difficult to decouple the effect of the force from the unit of charge giving raise to such a force. One might argue that in some cases fields potentials may be observable (see the Aharonov-Bohm effect, for instance), but this is a refined topic and open for discussion.

On the other hand, once you are provided with a full mathematical description of your theory, you see that (if some particular conditions hold) conservative fields are uniquely specified in each point in space once you assign the divergence and the curl: essentially one can prove that the solutions to those differential equations are enough to reconstruct the fields at every point. Then it is up to you what quantity you want to use for your equations: usually it comes down to choosing the one such that the equations take the simplest form in terms of symmetries and solutions.

My question is, is the flux responsible for setting up the forces, from which the potentials come as a result, or does it set up the potentials, and the forces simply act along the gradient? Or, are they the same?

Forces exist because they are generated by charges, thus neither the flux, nor the potential, nor the field, nor anything else are responsible for setting them up. Moreover, they are not the same, as one is a scalar function and the other is a vector field that, only is some particular cases, can be derived from each other.

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