Editing Gradient (section)

===Level sets===
{{see also|Level set#Level sets versus the gradient}}
A level surface, or [[isosurface]], is the set of all points where some function has a given value.

If {{math|''f''}} is differentiable, then the dot product {{math|(∇''f''&thinsp;)<sub>''x''</sub> ⋅ ''v''}} of the gradient at a point {{math|''x''}} with a vector {{math|''v''}} gives the directional derivative of {{math|''f''}} at {{math|''x''}} in the direction {{math|''v''}}. It follows that in this case the gradient of {{math|''f''}} is [[orthogonal]] to the [[level set]]s of {{math|''f''}}. For example, a level surface in three-dimensional space is defined by an equation of the form {{math|1=''F''(''x'', ''y'', ''z'') = ''c''}}. The gradient of {{math|''F''}} is then normal to the surface.

More generally, any [[embedded submanifold|embedded]] [[hypersurface]] in a [[Riemannian manifold]] can be cut out by an equation of the form {{math|1=''F''(''P'') = 0}} such that {{math|''dF''}} is nowhere zero. The gradient of {{math|''F''}} is then normal to the hypersurface.

Similarly, an [[affine algebraic variety|affine algebraic hypersurface]] may be defined by an equation {{math|1=''F''(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>) = 0}}, where {{math|''F''}} is a polynomial. The gradient of {{math|''F''}} is zero at a singular point of the hypersurface (this is the definition of a singular point). At a non-singular point, it is a nonzero normal vector.