How to Motivate the Gauss Divergence Theorem in a simplest possible setup in multi D?

Question

I usually motivate the Fundamental Theorem of Calculus by breaking down the difference $F(b) - F(a)$ into a sum of small differences, then using the Mean Value Theorem to show how it connects to the integral $\int_a^b f'(x) , dx$.

Now, I'm looking to motivate the Gauss Divergence Theorem in a similarly intuitive way. My goal is to provide a simple setup that gradually leads them to understand the theorem's significance. Specifically, I’d like to explain how the sum of local divergences within a volume relates to the total flux across the boundary.

What would be a good approach or analogy to help undergraduates grasp the mathematical motivation behind the Gauss Divergence Theorem, starting from basic concepts (for example using fundamental theorem of caluclus in 1D to understand divergence theorem)? Any insights or ideas on simplifying this concept would be greatly appreciated!

P.S.: I am more interested in understanding the mathematical motivation in a simplest possible way behind its development rather than teaching pedagogy and hence posted it here and not in MathEducators.StackExhange

Would analyzing a the net flux through a differential rectangular prism be considered too rudimentary for your students? It is pretty easy to start with side lengths $dx$, $dy$, $dz$ and show that $P~dydz + Q~dxdz + R~dxdy = (P_x+Q_y+R_z)dxdydz$. You can use a physical field or object to motivate the importance of flux depending on their backgrounds — whpowell96, Aug 23 '24 at 04:19
It may be easier for some of the students to get the idea of flux, if you lead up to it with a 2-dimensional toy example. Imagine a big city with a lot of roads going East-West and North-South. Traffic at the point $(x,y)$ has E/W component $f_1(x,y)$, and a N/S component $f_2(x,y)$. Assume the city limit is a parametrized curve. Express the net traffic across the city limit A) as a line integral over the limit curve and B) as a 2-d integral of the divergence. A tiny parallelogram is like a single square block etc. And divergence measures the contribution of tha block. — Jyrki Lahtonen, Aug 23 '24 at 04:39
I very much approve of the idea of starting with 1-dimensional blob (with a 0-dimensional boundary handled by FTC). My suggestion basically is that instead of jumping straight to a 3-dimensional blob (with a 2-dimensional boundary) you can do the intermediate step of a 2-dimensional blob and 1-dimensional bounday. — Jyrki Lahtonen, Aug 23 '24 at 04:42
I'm uncertain whether this question is better handled here or at MathEducators.StackExhange. Most of the answers may be "anecdotal" (like my comment). But, who knows, somebody somewhere may have carried out a comparative study or some such :-) — Jyrki Lahtonen, Aug 23 '24 at 04:54
@JyrkiLahtonen Thanks for your comments. I am more interested in understanding the mathematical motivation in a simplest possible way behind its development rather than teaching pedagogy. — Celestina, Aug 23 '24 at 05:17
@whpowell96 Thanks for your comment. I am more interested in understanding the mathematical motivation in a simplest possible way behind its development rather than teaching pedagogy. — Celestina, Aug 23 '24 at 05:17

How to Motivate the Gauss Divergence Theorem in a simplest possible setup in multi D?

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