Calculating a distributional derivative - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T04:27:12Z http://mathoverflow.net/feeds/question/101008 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/101008/calculating-a-distributional-derivative Calculating a distributional derivative dcs24 2012-06-30T16:40:46Z 2012-07-01T09:06:47Z <p>Suppose that we have a sequence of functions $u_j$ that are in $L^{\infty}(0,1)$. Then the sequence of maps $N_j(s) := \|u_j(s)\|^2$ are also in $L^{\infty}(0,1)$. Hence they give rise to distributions and therefore has a distributional derivative. What is the explicit formula for $DN_j$? Is it related to the classical formula $2\langle u_j , Du_j\rangle$? </p> http://mathoverflow.net/questions/101008/calculating-a-distributional-derivative/101050#101050 Answer by Liviu Nicolaescu for Calculating a distributional derivative Liviu Nicolaescu 2012-07-01T09:06:47Z 2012-07-01T09:06:47Z <p>First, I do not understand why do you need a sequence of functions when the question involves an individual function. Suppose that $u$ is real valued. Then the product of the distributions $u$ and $u'$ may not even be defined. (This is the case when $u$ is the Heaviside function.) However, if the distributional derivative of $u$ is Lebesgue integrable, then </p> <p>$$\frac{d}{dt}(\; u^2\;) = 2u u'.$$</p>