Editing Level set (section)

==Level sets versus the gradient==
[[Image:level grad.svg|right|thumb|Consider a function ''f'' whose graph looks like a hill. The blue curves are the level sets; the red curves follow the direction of the gradient. The cautious hiker follows the blue paths; the bold hiker follows the red paths. Note that blue and red paths always cross at right angles.]]

:'''[[Theorem]]:''' If the function {{mvar|f}}  is [[differentiable function|differentiable]], the [[gradient]] of  {{mvar|f}}  at a point is either zero, or perpendicular to the level set of  {{mvar|f}} at that point.

To understand what this means, imagine that two hikers are at the same location on a mountain. One of them is bold, and decides to go in the direction where the slope is steepest. The other one is more cautious and does not want to either climb or descend, choosing a path which stays at the same height. In our analogy, the above theorem says that the two hikers will depart in directions perpendicular to each other.

A consequence of this theorem (and its proof) is that if {{mvar|f}} is differentiable, a level set is a [[hypersurface]] and a [[manifold]] outside the [[critical point (mathematics)|critical points]] of {{mvar|f}}. At a critical point, a level set may be reduced to a point (for example at a [[local extremum]] of {{mvar|f}} ) or may have a 
[[singular point of an algebraic variety|singularity]] such as a [[intersection theory|self-intersection point]] or a [[cusp (singularity)|cusp]].