I suggest not to have this tag. The mean value property is probably the most characteristic property of harmonic function. Indeed a $C^2$ function is harmonic iff it satisfies the mean value property. Thus if a question is related to mean value property, almost sure the OP will tag "harmonic fu...
In the general situation of $f:S\to \mathbb R^m$ where $S\subset \mathbb R^n$. There is a form of the mean value theorem: $a\cdot (f(y)-f(x))=a\cdot (f'(z)(y-x))$ which requires a vector $a$ and dot products. In Tom Apostol's Mathematical Analysis (Second Edition), page No. 355, I found that aft...
Can someone show me the proof of this form of the mean value theorem for vector valued functions? Let $f:R^n \rightarrow R^n$ be a function of class $C^1$ and $a,b\in R^n$, than there exists some $d\in R^n : a<d<b$ such that $(b-a)\cdot (f(b)-f(a))=(b-a)\cdot (f'(d)(b-a)) $ where $\cdot$ is a ...
Given that $f$ is differentiable on $\mathbb{R}$, I know that "between the zeroes of $f$, there is a zero of $f'$. But given that $f'$ has $k$ roots, then is it true that $f$ has at most $k+1$ roots.
In progress The tag adjoint needs renaming The tag adjoint is "[f]or questions about adjoints, in the category-theoretic or inner-product-space sense, as well as about adjugate matrices", which certainly falls under "two or more completely unrelated things". If it's needed, a separate tag a...
The divisors is for divisors in algebraic geometry. There have been discussions about this in the Tag management threads earlier: Tag management 2015 and Tag management 2016 Among the options is expanding the tag name to divisors-algebraic-geometry.
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