Editing Gradient (section)

==Further properties and applications==

===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.

===Conservative vector fields and the gradient theorem===
{{main|Gradient theorem}}

The gradient of a function is called a gradient field. A (continuous) gradient field is always a [[conservative vector field]]: its [[line integral]] along any path depends only on the endpoints of the path, and can be evaluated by the gradient theorem (the fundamental theorem of calculus for line integrals). Conversely, a (continuous) conservative vector field is always the gradient of a function.

===Gradient is direction of steepest ascent===
The gradient of a function <math>f \colon \R^n \to \R</math> at point {{math|''x''}} is also the direction of its steepest ascent, i.e. it maximizes its [[directional derivative]]:

Let <math> v \in \R^n</math> be an arbitrary unit vector. With the directional derivative defined as

<math display="block">\nabla_v f (x) = \lim_{h \rightarrow 0} \frac{f(x + vh) - f(x)}{h},</math>

we get, by substituting the function <math>f(x + vh)</math> with its [[Taylor series]],

<math display="block">\nabla_v f (x) = \lim_{h \rightarrow 0} \frac{(f(x) + \nabla f \cdot vh + R) - f(x)}{h},</math>

where <math>R</math> denotes higher order terms in <math>vh</math>.

Dividing by <math>h</math>, and taking the limit yields a term which is bounded from above by the [[Cauchy-Schwarz inequality]]<ref>{{cite book |author1=T. Arens | title=Mathematik |edition=5th |publisher=Springer Spektrum Berlin |year=2022 | doi=10.1007/978-3-662-64389-1 |isbn=978-3-662-64388-4 |url = https://doi.org/10.1007/978-3-662-64389-1}}</ref>

<math display="block">|\nabla_v f (x)| = |\nabla f \cdot v| \le |\nabla f| |v| = |\nabla f|.</math>

Choosing <math>v^* = \nabla f/|\nabla f|</math> maximizes the directional derivative, and equals the upper bound

<math display="block">|\nabla_{v^*} f (x)| = |(\nabla f)^2/|\nabla f|| = |\nabla f|.</math>