# Math question about the path formed by following a gradient **URL:** \ **Category:** Factual Questions\ **Created:** [August 31, 2015, 10:47pm UTC](https://boards.straightdope.com/t/math-question-about-the-path-formed-by-following-a-gradient/729552 "2015-08-31T22:47:40Z")\ **Posts on this page:** 6\ **Page:** 2
**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\ **Post date:** [September 1, 2015, 1:37pm UTC](https://boards.straightdope.com/t/math-question-about-the-path-formed-by-following-a-gradient/729552/21 "2015-09-01T13:37:40Z")
> [@Chronos](#): > > **watchwolf49** , you might want to educate yourself on the terms being used before you chime in. The word “gradient”, used as a noun, has a very precise definition in mathematics, and everyone else in this thread is using it in this sense. I’m not sure what you mean by the word, but it’s clearly not the same thing everyone else in this thread means. 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. ---
**Author:** ![MikeS](https://avatars.discourse-cdn.com/v4/letter/m/919ad9/32.png) [@MikeS](https://boards.straightdope.com/u/MikeS)\ **Post date:** [September 1, 2015, 2:09pm UTC](https://boards.straightdope.com/t/math-question-about-the-path-formed-by-following-a-gradient/729552/22 "2015-09-01T14:09:13Z")
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](https://en.wikipedia.org/wiki/Method_of_steepest_descent), for example. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [September 1, 2015, 2:39pm UTC](https://boards.straightdope.com/t/math-question-about-the-path-formed-by-following-a-gradient/729552/23 "2015-09-01T14:39:00Z")
> [@](#): > > Notice how LSLGuy actually provided this definition in mathematics. He states his opinion on my educational level BY giving this definition. 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. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 1, 2015, 3:25pm UTC](https://boards.straightdope.com/t/math-question-about-the-path-formed-by-following-a-gradient/729552/24 "2015-09-01T15:25:38Z")
> [@MikeS](#): > > 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](https://en.wikipedia.org/wiki/Method_of_steepest_descent), for example. I think the more relevant “method of steepest descent” article is the one on [“gradient descent”](https://en.wikipedia.org/wiki/Gradient_descent) (which, as noted in that link and in **Hellestal** ’s link, is closely related to “gradient flow”). ---
**Author:** ![Learjeff](https://avatars.discourse-cdn.com/v4/letter/l/94ad74/32.png) [@Learjeff](https://boards.straightdope.com/u/Learjeff)\ **Post date:** [September 1, 2015, 3:48pm UTC](https://boards.straightdope.com/t/math-question-about-the-path-formed-by-following-a-gradient/729552/25 "2015-09-01T15:48:02Z")
> [@Indistinguishable](#): > > Though, the OP had us assume smoothness, so the gradient will be everywhere defined; the only potential issue is points where the gradient hits zero. At such points, the slope in every direction will be accordingly zero, and thus there will be no particular direction of maximum slope. 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.) > [@watchwolf49](#): > > I’m going to stick my neck in this noose here and comment what I think, and it’s not pretty. The word “gradient” is more commonly used as an adjective, it describes a specific quality of whatever noun it is modifying. 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). ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [September 1, 2015, 3:55pm UTC](https://boards.straightdope.com/t/math-question-about-the-path-formed-by-following-a-gradient/729552/26 "2015-09-01T15:55:44Z")
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] [Previous page](https://boards.straightdope.com/t/math-question-about-the-path-formed-by-following-a-gradient/729552.md?page=1)