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Good day all!

Edit: I'm currently doing a personal study on differentiation on $\Bbb{R}^n$ but I have this challenging problem. Although, some answers have been provided on how to show that show that $f$ is differentiable on $\Bbb{R}^n$ but I would further like to compute $f'$ on $\Bbb{R}^n$.

There is this function $$f:\Bbb{R}^n\to \Bbb{R}$$ $$x\mapsto f(x)=\frac{1}{2}\langle x,u(x)\rangle+\langle x,b\rangle$$ where $u:\Bbb{R}^n\to\Bbb{R}^n$ is linear and symmetric $:$ $(\forall\;x,y\in \Bbb{R}^n,\langle x,u(y)\rangle=\langle u(x), y \rangle)$ and $b\in \Bbb{R}^n.$

Honestly, I am just coming across this kind of function. I want to know what name it's called. How do I show that $f$ is differentiable on $\Bbb{R}^n$ and how do I compute $f'$?

Thanks for your help!

Michael Hardy
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Omojola Micheal
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  • $u$ is symmetric? or do you mean the inner product is symmetric? – dmtri Aug 13 '18 at 17:24
  • While your notation is perfectly fine, linear operations are often written like matrix multiplication, so instead of $u(x)$ you will often see something like $Ux$. In general, a good first attempt at computing $f'$ is to compute $f(x+h)-f(x)$ and look for the terms that are linear in $h$. There are some notational subtleties that are irrelevant for $\mathbb{R}^n$ but become important with other spaces. – copper.hat Aug 13 '18 at 17:42

3 Answers3

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Using the symmetry and the linearity of $u$ you can easily check that $$ f(x+h) = f(x) + \langle u(x) , h\rangle + \langle h, b \rangle + O(|h|^2), $$ i.e. $$ f(x+h) = f(x) + L_x(h) + o(|h|), $$ with $L_x(h) := \langle u(x) ,h\rangle + \langle h, b \rangle$. Hence, by definition, the linear map $L_x$ is the differential of $f$ at $x$.

Rigel
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sorry but I dont have enough score to write this as a comment

I think the best way is to passe by the defintion

Let $P$ be a point in $\mathbb{R}^n$ and $w$ a vector in $\mathbb{R}^n$

you can define a curve $\alpha :(-\epsilon , \epsilon)\rightarrow \mathbb{R}^n $ such that $\alpha(0)=p$ and $\alpha^{'}(0)=w$

now we can define a curve $\beta = F o \alpha\; $ and the answer is $dF_p(w)=\beta^{'}(o)$ this is the idea you have just to write the calculations Good luck

T A R I K
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Consider the rule for differentiating inner products described here, and use the usual derivative properties (addition, chain rule, differentiating constant and linear functions).

Sambo
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