# Connection field lines/potential/vector field

by Funzies
Tags: connection, field
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,568 Let f(x,y,z) be the value of the potential field at each point (x,y,z). Then the vector $grad f= \nabla f$ points in the direction of the "field lines", the lines of fastest increase of the function f. Further, the rate of increase of f in the direction of unit vector $\vec{v}$ is given by $\nabla f\cdot \vec{v}$. That is, the direction in which the derivative is 0, the "equipotential lines" (strictly speaking, in three dimensions, they would be equipotential surfaces) is exactly the direction in which that dot product is 0, the direction in which the vector $\vec{v}$ is perpendicular to $\nabla f$ and so perpendicular to the field lines.