3
$\begingroup$

Let $\omega$ be a $2$-form on $\mathbb{R}^3-\{(1,0,0),(-1,0,0)\}$, $$\omega=((x-1)^2+y^2+z^2)^{-3/2}((x-1)dy\wedge dz+ydz\wedge dx+zdx \wedge dy)+ ((x+1)^2+y^2+z^2)^{-3/2}((x+1)dy\wedge dz+ydz\wedge dx+zdx \wedge dy)$$ and $S=\{(x,y,z)\in \mathbb{R^3}: x^2+y^2+z^2=5 \}$.

In this condition, we calculate $\int_{S}\omega$, where the orientation of $S$ is the natural orientation induced by $D=\{(x,y,z)\in \mathbb{R^3}: x^2+y^2+z^2 \leq 5 \}$.

I can't calculate this, so if you solve this, please teach me the answer for this.

  • 2
    The $\omega$ you give is a 1-form. Then the integral makes no sense. Probably a typo? Also, what did you try already? The formulation of the exercise is screaming for Stoke's theorem.2012-08-08
  • 0
    Sorry,I correct it then.2012-08-08
  • 0
    Hint: Use polar coordinates.2012-08-09

2 Answers 2