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To calculate (using Gauss) fluxus of the vectorial field $\bar{V}=z*\bar{i}+x*\bar{j}+y*\bar{k}$ through surface determined by $x^2+y^2+z^2=1,\ x=0,\ y=0,\ z=0$ with $x>0,\ y>0,\ z>0$

Please help to solve the problem. I've tried to solve it. I've read suggested articles on wikipedia and my course book, but I didn't understand it. And theory without practical solution is not useful Please refrain from general indications such as read that and try that. I received this problem as a possible exam problem, I don't know how to solve it or what to try, if I knew I wouldn't be posting it. Not all schools are perfect. My teachers don't have time to explain them to me. Neither do my colleagues. Ireally hope I can find help here. If you can please provide a complete solution with a step by step explanation as for a dummy. Thanks so much in advance!

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  • $\begingroup$ First of all your notation is a bit off for the vector field. I guess it's easier to compute the volume integral and then that will tell you the flow through the surface..that is divergence theorem in a nutshell. $\endgroup$
    – Chinny84
    Feb 6, 2014 at 1:30
  • $\begingroup$ Is it the volume integral you are having issues with? Since that looks like it is zero since the divergence of $\mathbf{V}$ is zero and hence the integral is zero for both volume and surface integral? $\endgroup$
    – Chinny84
    Feb 6, 2014 at 1:31
  • $\begingroup$ @Chinny84 Thanks for the comment, but I don't quite understand what you mean, could you please write it in math, or edit the post? $\endgroup$
    – Zhivago
    Feb 6, 2014 at 1:32
  • $\begingroup$ @Chinny84 Yes it's the volume integral. Please help. Thanks $\endgroup$
    – Zhivago
    Feb 6, 2014 at 1:34
  • $\begingroup$ @Chinny84 I don't understand like that. Please write an answer with LaTex if you can $\endgroup$
    – Zhivago
    Feb 6, 2014 at 1:37

1 Answer 1

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To compute the volume integral we compute $$ \int \int \int \nabla\cdot\mathbf{V}\mathrm{d}x\mathrm{d}y\mathrm{d}z = \oint_{S}\mathbf{V}\cdot\mathrm{d}\mathbf{S} $$ but if we compute the divergence of the $\mathbf{V}$ as stated above $$ \nabla\cdot\mathbf{V} = \left(\mathbf{i}\frac{\partial }{\partial x} + \mathbf{j}\frac{\partial }{\partial y} + \mathbf{k}\frac{\partial }{\partial z}\right)\cdot\left(z\mathbf{i} +x\mathbf{j} + y\mathbf{k}\right)\\ =\frac{\partial }{\partial x}(z) + \frac{\partial }{\partial y}(x) \frac{\partial }{\partial z}(y) = 0 $$

So this implies from gausses theorem that: $$ \oint_{S}\mathbf{V}\cdot\mathrm{d}\mathbf{S} = 0 $$

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  • $\begingroup$ Thanks very much. Very kind of you. Could you also please tell me why did you use nabla here for? Is that a formula I think I've seen it somewhere before. $\endgroup$
    – Zhivago
    Feb 6, 2014 at 1:47
  • $\begingroup$ Because thats the symbol for the taking the derivative in three co-ordinates. I will edited my post to be more explicit. $\endgroup$
    – Chinny84
    Feb 6, 2014 at 1:49

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