Calculate the flux across the surface $S$ of $F=(x+3y^5, y+10xz, z-xy),$ $ S$ the hemisphere bounded by $x^2+y^2+z^2=1, z\ge0$.

I have done $n=(x,y,z)$

thus $\iint F\cdot n dS= \iint(x+3y^5, y+10xz, z-xy)\cdot(x,y,z)dS = \iint(1+3xy^5+9xyz)dS$

since $z=(1-x^2+y^2)^{1/2}$ it follows $\int_{-1}^1 \int_{-\sqrt {1-x^2}}^{\sqrt {1-x^2}}(1+3xy^5+9xy(1-x^2+y^2)^{1/2})dS$

then $\int_{-1}^1 \int_{-\sqrt {1-x^2}}^{\sqrt {1-x^2}}1dydx=\pi,$

$\int_{-1}^1 \int_{-\sqrt {1-x^2}}^{\sqrt {1-x^2}}3xy^5dydx=0$ and $\int_{-1}^1 \int_{-\sqrt {1-x^2}}^{\sqrt {1-x^2}}9xy(1-x^2+y^2)^{1/2}dydx=0,$

so $\iint F\cdot n dS=\pi$

I want to know if this result is correct by using the Divergence Theorem, Does anybody can help me?

  • $\begingroup$ you should add how you computed, so we can see where you went wrong. The answer should be $2\pi$ $\endgroup$ May 20 '18 at 1:39
  • $\begingroup$ I have add what I computed $\endgroup$ May 20 '18 at 2:08

$$div F=(F_1)'_x+(F_2)'_y+(F_3)'_z=3$$

$$\int\int\int_V3dxdydz=3.\frac {2}{3}\pi 1^3=2\pi $$


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