Gauss divergence theorem: verification in first octant

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I can't tell where you went wrong: perhaps you forgot to integrate along all the surface that bounds your volume $V$?

Namely, the divergence theorem says

$$ \int_S \langle F, dS\rangle = \int_V \mathrm{div}Fdxdydz , \ $$

where $S = \partial V = S_1 + S_2 + S_3 + S_4$, where $S_1$ is your plane $2x + 3y + 6z = 12$ and $S_2, S_3, S_4$ the (portions of the) coordinates planes that help $S_1$ bound $V$. That is, the integral on the left hand side is actually composed of four integrals:

$$ \int_S \langle F, dS\rangle = \int_{S_1} \langle F, dS \rangle + \dots + \int_{S_4} \langle F, dS \rangle \ . $$

EDIT. Ok, so a few more computations. On the one hand,

$$ \int_{S_1}\langle F, dS \rangle = \int_{D_1} \langle F\circ\varphi, \varphi_x \times \varphi_y\rangle dxdy \ , $$

where $D_1 = \left\{ 0\leq x \leq 6, 0 \leq y \leq \frac{12 -2x}{3}\right\}$ and $\varphi (x,y) = (x,y, (12 -2x-3y )/6)$. Hence our integral is:

$$ \int_0^6\int_0^{(12 -2x)/3} \left( 3x\frac{12-2x-3y}{6}, y + \frac{12-2x-3y}{6}x, (xy+1)\frac{12-2x-3y}{6} \right) \begin{pmatrix} 1/3 \\ 1/2 \\ 1 \end{pmatrix}dydx \ . $$

That is,

$$ \int_0^6\int_0^{(12 -2x)/3} \left(\frac{y}{2} + \frac{12 -2x -3y}{4}x + \frac{12-2x-3y}{6} \right)dydx \ . $$

If $S_2$ is the face $y=0$ of $V$, then

$$ \int_{S_2}\langle F, dS \rangle = \int_{D_2}\left( 3xz, zx , z\right) \begin{pmatrix} 0 \\ -1 \\ 0 \end{pmatrix} dzdx = \int_0^6\int_0^{(12-x)/3}(-zx)dzdx \ . $$

And you should try to write down now $\int_{S_3}$ and $\int_{S_4}$ -if you haven't done yet.

On the other hand,

$$ \int_V \mathrm{div}F dxdydz = \int_0^6\int_0^{(12-2x)/3}\int_0^{(12-2x-3y)/6} (3z + 2 + xy)dzdydx \ . $$

So, if you understand those previous integrals and are able to write down those correspondint to the triangles $S_3$ and $S_4$, this would be enough. The rest of the computations are quite long and boring.

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Updated on February 02, 2020

Comments

  • manmohan
    manmohan almost 4 years

    Please help me. I have checked several times, and resulted wrong in verification of this particular problem. Please solve for me.

    Verify Gauss divergence theorem for $F = (3xz)i + (y + zx)j + (xyz + z)k,$ where $'V'$ is the volume bounded by the coordinate planes and the plane $2x + 3y + 6z = 12$ in the first octant.

    work done: I have taken the limits for integrating: for x: 0 to 6, for y: 0 to (12-2x)/3 and z: 0 to (12-2x-3y)/6 and then I got the value after integration is 37.6. when I verified the same by integral f.n with respect to s, I got 43.6. which is unmatched to my previous answer. Please let me know, where I am wrong?

    -manmohan

    • Chinny84
      Chinny84 almost 10 years
      Show some working out then it is easy to see where you may or may not gone wrong.
    • manmohan
      manmohan almost 10 years
      @chinny84! please verify. I am unable to type here my math equation, as I am new to this.
    • manmohan
      manmohan almost 10 years
      Chinny84! I have taken the limits for integrating: for x: 0 to 6, for y: 0 to (12-2x)/3 and z: 0 to (12-2x-3y)/6 and then I got the value after integration is 37.6. when I verified the same by integral f.n.ds, I got 43.6. which is unmatched to my previous answer. Please let me know, where I am wrong?
  • manmohan
    manmohan almost 10 years
    a.r.! I am extremely so sorry for asking you to solve the entire problem. Still I am not recognized, where I did mistake? Please solve and check. Please
  • Agustí Roig
    Agustí Roig almost 10 years
    Did you compute the four integrals I'm talking about?
  • manmohan
    manmohan almost 10 years
    A.r.! Thank you for your response. I did and may be I am wrong in calculation part. Moreover, I am unable to type here, whatever I did. Please solve, so that I will learn from you. I am trying this problem from last few days and I am very dispointed. Please solve. I think, I may get solution by you.
  • manmohan
    manmohan almost 10 years
    a.r! I understand the complete computation part. I need little help.when you find $S_1$, there is a column matrix (1/3, 1/2, 1). How you got this column matrix? please explain. Then everything is ok for me. I want to know the computation part of column matrix.
  • Agustí Roig
    Agustí Roig almost 10 years
    This column vector is $\varphi_x \times \varphi_y$.
  • manmohan
    manmohan almost 10 years
    a.r! please explain this column vector in more detailed... please
  • Agustí Roig
    Agustí Roig almost 10 years
    $\varphi_x, \varphi_y$ are the partial derivatives of $\varphi$ and $\varphi_x \times \varphi_y$ their cross product en.wikipedia.org/wiki/Vector_product .