I'm teaching freshman electricity and magnetism this semester, and as usual in this type of course, I will need to teach my students a lot of vector calculus before they see it in a math course. The usual way this works in most curricula in my country (US) is that students only learn the integral form of Maxwell's equations. This semester, I'm attempting to do more of a balanced presentation, so that students will have some clue as to the local form of these equations, and of the correspondence between the differential and integral forms. This is meant to be mostly at a pretty simple level, without a lot of emphasis on coordinate-based calculations. They will see Gauss's law next week (the second week of the semester), but will not see Stokes's theorem until the very last week of the semester.
In this context, I'm looking for simple, easy examples of this kind of interplay of global and local perspectives in math. The simplest example I've come up with, to get across the idea that there can be local statements and global ones, is a linguistic one:
There are three errors in the sentence, only one of which can be detected if you're looking through a keyhole and can only see one letter.
A second example would be that if one of the kitchen staff at a fancy restaurant is stealing the gold-plated silverware, you can detect it either by seeing them put it in their pocket, or by seeing that the count of pieces at the end of the day is short. This is the discrete analog of the fundamental theorem of calculus.
As a third, visual example very specifically applied to Gauss's law, I have this picture:
Here a vector field in two dimensions is represented by field lines. If this is a vacuum, then Gauss's law is being violated. Such a violation at the global level (line escaping the large square) can always also be detected at the local level as well (line escaping the small square).
I feel like there ought to be a nice example from some branch of mathematics that would interpolate between examples 2 and 3 in terms of obviousness or surprisingness. Basically I would like an example that is sort of interesting in its own right, and easy to verify once you see it, but that isn't like the silverware example in being kind of trivial and uninteresting.
It seems like Euler's polyhedral formula is exactly right in spirit, and maintains some simplicity because it's entirely discrete. However, I feel like I would probably need too big a chunk of a class session to do justice to this example. Is there some simpler example from geometry or topology that demonstrates the ideas without getting into calculus?