Pretty snarky there, Chronos, not sure how many people are interested in your opinion of my educational level. Notice how LSLGuy actually provided this definition in mathematics. He states his opinion on my educational level BY giving this definition. I’m sure I’m not the only reader who has learned from this.
I’ve heard the path you’re describing called (rather unimaginatively) the “path of steepest descent/ascent.” There’s a mathematical technique for approximating certain integrals called the method of steepest descent, for example.
Yes, and if the talk in this thread happens to drift to divergence, or curl, as it well might, then you’d be similarly ignorant of the technical meanings of those terms, and would still have to do further study.
I think the more relevant “method of steepest descent” article is the one on “gradient descent” (which, as noted in that link and in Hellestal’s link, is closely related to “gradient flow”).
I believe the technical term for that, geographically, is “lake.” 
But even then, there’s at least one direction with the lowest positive gradient. Of course, that doesn’t guarantee there’s no infinite loop (the lowest gradient can lead to another lake, whose lowest gradient leads back to the first one.)
Can you give an example? I’ve never heard it used that way (except above, in “gradient flow,” which is pretty technical.) The most common usage is as a ratio of rise to distance, as used in driving (especially trucking).
For what it’s worth, “gradient” is a noun even in “gradient flow” (just as it is in “gradient descent”, or “vector” is in “vector flow” or “vector field”, or “pizza” is in “pizza pie”, or such things). A noun-noun phrase doesn’t mean the first noun is actually an adjective; it just happens to be a fact about nouns that they can often be combined in this way.
[This is a pedantic linguistic point, not a mathematical one]