Why have physicists had the idea to define a potential function of a gradient vector field $\vec F$ to be a function $g$ such that $\vec F=-\nabla g$? What changes if we don't put the negative sign? I mean suppose a particle travel along a flow line of a gradient vector field what would be the changes on the motion of that particle if we don't put the negative sign?
2026-03-28 09:44:19.1774691059
On
On
A silly question about potential functions
134 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
3
There are 3 best solutions below
0
On
Energy conservation. You have that one form converts to another; if you have a minus on $V$ you have that their difference is constant; which would be still OK, but it is easier to think about it the other way round
0
On
In the context of electrostatics and gravitation it indicates that movement of a particle in the direction of the vector field acts to decrease it's potential energy. Somewhat fortuitously, it also makes life easier in deriving the field from the potential and vice versa in electrostatics by removing all the negative signs that crop up upon integration/differentiation.
The $g$ represents energy. And forces act to decrease energy.