# vanishing of gradient in domain

Theorem. If the function $f$ is defined in a domain (http://planetmath.org/Domain2) $D$ of ${\mathbb{R}}^{n}$ and all the partial derivatives^{} of a $f$ vanish identically in $D$, i.e.

$$\nabla f\equiv \overrightarrow{0}\mathit{\hspace{1em}}\text{in}D,$$ |

then the function has a constant value in the whole domain.

*Proof.* For the sake of simpler notations, think that $n=3$; thus we have

${f}_{x}^{\prime}(x,y,z)={f}_{y}^{\prime}(x,y,z)={f}_{z}^{\prime}(x,y,z)=\mathrm{\hspace{0.33em}0}\mathit{\hspace{1em}}\text{for all}(x,y,z)\in D.$ | (1) |

Make the antithesis that there are the points ${P}_{0}=({x}_{0},{y}_{0},{z}_{0})$ and ${P}_{1}=({x}_{1},{y}_{1},{z}_{1})$ of $D$ such that
$f({x}_{0},{y}_{0},{z}_{0})\ne f({x}_{1},{y}_{1},{z}_{1})$.
Since $D$ is connected, one can form the broken line ${P}_{0}{Q}_{1}{Q}_{2}\mathrm{\dots}{Q}_{k}{P}_{1}$ contained in $D$. When one now goes along this broken line from ${P}_{0}$ to ${P}_{1}$, one mets the first corner where the value of $f$ does not equal
$f({x}_{0},{y}_{0},{z}_{0})$. Thus $D$ contains a line segment^{}, the end points^{} of which give unequal values to $f$. When necessary, we change the notations such that this line segment is ${P}_{0}{P}_{1}$. Now, ${f}_{x}^{\prime},{f}_{y}^{\prime},{f}_{z}^{\prime}$ are continuous^{} in $D$ because they vanish. The mean-value theorem for several variables guarantees an interior point $(a,b,c)$ of the segment such that

$$0\ne f({x}_{1},{y}_{1},{z}_{1})-f({x}_{0},{y}_{0},{z}_{0})={f}_{x}^{\prime}(a,b,c)({x}_{1}-{x}_{0})+{f}_{y}^{\prime}(a,b,c)({y}_{1}-{y}_{0})+{f}_{z}^{\prime}(a,b,c)({z}_{1}-{z}_{0}).$$ |

But by (1), the last sum must vanish. This contradictory result shows that the antithesis is wrong, which settles the proof.

Title | vanishing of gradient in domain |
---|---|

Canonical name | VanishingOfGradientInDomain |

Date of creation | 2013-03-22 19:11:58 |

Last modified on | 2013-03-22 19:11:58 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 8 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 26B12 |

Synonym | partial derivatives vanish |

Related topic | FundamentalTheoremOfIntegralCalculus |

Related topic | ExtremumPointsOfFunctionOfSeveralVariables |