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I have a vector valued function $U(x,y)=\Big(u_{1}(x,y),u_{2}(x,y)\Big)$. I want to find $\|\nabla U\|_{L_{2}(0,1)}$, but i could not figure how can do it. Do you have any idea?

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    @Brhn What is the context of the problem? Is there any more information you can provide?2012-08-04

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It is unusual to denote the derivative of a vector valued function $(u_1, u_2)$ by gradient, but if one does there is not much choice -- it will involve the gradient of both $u_1, u_2$ in some way.

One possible and, I guess, reasonable, defintion would be $\nabla U:= (\nabla u_1, \nabla u_2)$, which is matrix of course. One possible defintion of norm would then be $|\nabla u_1|+ |\nabla u_2|$, but for the definion of an $L_2$ - norm $(|\nabla u_1|^2+ |\nabla u_2|^2)^{1/2}$ looks more natural to me, with $|\nabla u_i| = \left[\sum_k \left( \frac{\partial u_i}{\partial x^k}\right)^2\right]^{1/2}$ as norm of choice for each $i$.

The $L_2$ - norm of this would then be defined as $ \int_\Omega (|\nabla u_1|^2+ |\nabla u_2|^2)^{1/2} dx^1d x^2 $

Note that it does not make sense to define an $L_2((0,1))$-norm for a function of two variables, the domain of definition should allow for two variables.

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    You are correct, Silly me. Sorry for the bother.2012-08-04