Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Is there some kind of relation between weak derivative and partial derivative. I have been reading weak derivative as the weakening the partial derivative. But I found it rather difficult to conceptualise the weak derivative as a derivative . Any help would be great . Thanks

share|cite|improve this question
Integration by parts is the link between weak derivatives and actual derivatives. The wikipedia article has a pretty nice exposition on it. – Braindead Jul 15 '12 at 19:28

You don't say anything about context here, so let's try it with distributions. I'll stick with one variable for simplicity. So let $T$ be a distribution, and $\varphi$ a test function (i.e., smooth with compact support). Write $\varphi_h(x)=\varphi(x-h)$, and $\langle T_h,\varphi\rangle=\langle T,\varphi_{-h}\rangle$. Then by definition $$ \langle T',\varphi\rangle =-\langle T,\varphi'\rangle =\Bigl\langle T,\lim_{h\to0}\frac{\varphi_h-\varphi}{h}\Bigr\rangle =\lim_{h\to0}\Bigl\langle\frac{T_{-h}-T}{h},\varphi\Bigr\rangle, $$ so that $T'$ is the weak limit of $$\frac{T_{-h}-T}{h}$$ as $h\to0$. As part of the calculation, you need to know that $$\varphi'=-\lim_{h\to0}\frac{\varphi_h-\varphi}{h}$$ in the topology of the space of test functions.

The signs are a bit awkward here, but I try to stick with the common sign convention for function translation. If $T$ were a function, of course $$\frac{T_{-h}-T}{h}(x)=\frac{T(x+h)-T(x)}{h}$$ as expected, and then you get to take the weak (distributional)limit of that.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.