Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$ be differentiable and for any point in the open unit disc about the origin, we have $\nabla f(x)=0$. Show $f$ is the constant function in this region.
A necessary condition for constant functions via the $\nabla$ operator
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multivariable-calculus
derivatives
1 Answers
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Let $a$ and $b$ two arbitrary points in the open unit disc $D$ about the origin and $S=\{a+t(b-a): t \in [0,1]\}$. Then $ S \subseteq D$.
The Mean value theorem gives some $c \in S$ such that
$$f(b)-f(a)=(b-a)*\nabla f(c)=0.$$
( $*$ denotes the inner product on $\mathbb R^2$). It follws that $f(b)=f(a)$