I have only a high school physics background, so when I study the later parts of multivariable calculus, e.g., Greens, Gauss, and Stokes' theorems, there are some topics that I only know the mathematics of - but not the physical significance of the theorems.
One question in particular is this: why does a conservative vector field integrate to zero, on a closed curve? This is supposed to mean that the vector field did "no work".
What's a good, real world example of this result? (I don't need a math proof.)
Thanks,
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$\begingroup$Gravity! Imagine if you have some object at a height $h$. From some physical intuition, you can imagine that, as long as you get back to the same position (and in fact, you could end up at any point with the height $h$!), there will be no energy gain from the (Newtonian) gravity.
$\endgroup$ 7 $\begingroup$The most common, simple examples of conservative vector fields are the gravitational and electronic fields around a point particle. These are conservative fields, so the concept of "potential energy" of gravitation or static electricity makes sense. This also allows orbits, as @JyrkiLahtonen just pointed out in a commment. (Remember that for a while, electrons were though to orbit the nucleus).
The point particles are the simplest, but of course the fields can be generalized to uniform spherical objects, spherical objects where the density depends only on the distance from the center, and so on.
@KennethGoodenough brought up a simplification, where the field is uniform. I suppose that would be the simplest of all.
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