Consider the force field $\vec{F}(x,y,z) = (\sin(y+z),\cos(x+z),\sin(x+y))$
- Compute the diverge of $F$
- Be $S$ the sphere which center in origin (0,0,0) and radio 1. $S = \{ (x,y,z)\in R^3 , x^2+y^2+z^2 =1 \}$. Determine $\int\int_S \vec{F} \cdot \hat{n} dA$.
My attempt:
$\bigtriangledown • \vec{F} = 0$ (partial derivative of each member of vector are zero. e.g $\partial / \partial x \sin(y+z) = \cos (y+z) \cdot 0 = 0$
How Can I compute internal product of F and $n = 1 \hat{a}_R$? Is The force field on spherical coordinate on right form?
HINT
As you noted
$$div(\vec F)= \frac{\partial\sin(y+z)}{\partial x}+\frac{\partial\cos(x+z)}{\partial y}+\frac{\partial\sin(x+y)}{\partial z}=0$$
and now recall that
$$\int\int_S \vec{F} \cdot \hat{n} dA=\iiint_V div{F} dV $$