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Modelling is a very specific extraordinary section in Mathematics.

Although many systems can only be described by partial differential equations, and may be tensors equations, they can be modeled, approximately and even in some cases exactly, by very simpler equations, O.D.E.s and may be Vector Field Multi-Dimensional Functions, just assume the variables you are not in need for, pick up the property you are in need, and derive the equation.

For example, conservative field force are described by an O.D.E. (or P.D.E. in the sense of general force form), but can be analyzed very simply by the form of Potential.

If there is some thing wrong, would some one advise?

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Not sure what the question is here... Yes, sometimes under specific circumstances simplifications to the model can be made without introducing much error at all. The art and knowledge of modelling is to know when such simplifications are appropriate, how much they affect the quality of the final answer, and balancing the cost versus speed of arriving at an answer.

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• 2 weeks later...

Part of the trick with modelling is to know what sort of answer you're looking for. Exact answers like "at time T, the value x will be f(T)" require an entirely different approach to holistic answers like "the system oscillates under these conditions, and doesn't under these", neither approach is better or worse since you can't expect outright accuracy from any model. But yeah I have no idea what we're being asked either.

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But yeah I have no idea what we're being asked either.

About the only thing sensible to me in the OP is the connection between a conservative vector field and the existence of a a potential function, which is true in a simply-connected domain.

I think it was some sort of free association without much real content, and very little appreciation for the underlying mathematics.

Mathematical modeling can be as much art as science, depending on the specific problem. Much of the value comes from understanding the nature of the idealizations and approximations that are made, knwing the limitations of them, and in interpreting the results of the model. As always, both with the model itself and with the interpretive capabilities of the modeler, GIGO.

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I was just sharing.

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