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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

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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

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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.

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