Answer :
The divergence theorem is a mathematical announcement of the bodily reality that, withinside the absence of the advent or destruction of matter, the density inside a area of area can extrade simplest with the aid of using having it circulate or farfar from the area thru its boundary.
Use the Divergence Theorem to discover the outward flux of
- mathbf FF
- throughout the boundary of the area D.
- mathbf = x ^ mathbf - 2 x y mathbf + three x z mathbf F=x 2 i−2xyj+3xzk
- D: The area reduce from the primary octant with the aid of using the sphere
- x ^ + y ^ + z ^ = 4.x 2 +y 2 +z 2 =4.
- VerifiedStep eleven of 2The divergence Theorem of this vector discipline is
- begin nabla cdot boldsymbol &=fracleft(x^right)+frac(-2 x y)+frac(three x z) &=2 x-2 x+three x=three x end∇⋅F = ∂x∂ (x 2 )+ ∂y∂ (−2xy)+ ∂z∂ (3xz)=2x−2x+3x=3x
- Let SS be the floor that encloses the area DD. Then, from the Divergence Theorem the flux throughout this boundary is given with the aid of using
- operatorname=intint_ boldsymbol cdot boldsymbol d sigma=intintint_ nabla cdot boldsymbol d VFlux=∫∫ S F⋅ndσ=∫∫∫ D ∇⋅FdV
- In round coordinates we've that
- begin nabla cdot boldsymbol &=three x=three r cos theta sin phi d V &=r^ sin phi d r d phi d theta D &=zero leq theta leq pi / 2, zero leq phi leq pi / 2, zero leq r leq 2} end∇⋅FdVD =3x=3rcosθsinϕ=r 2 sinϕdrdϕdθ={(r,θ,ϕ)∣zero≤θ≤π/2, zero≤ϕ≤π/2, zero≤r≤2}
- Therefore the outward flux throughout the boundary of DD is
- begin intint_ boldsymbol cdot boldsymbol d sigma &=int_{zero}^ int_{zero}^ int_{zero}^ three r^{three} cos theta sin ^ phi d r d phi d theta &=three int_{zero}^ cos theta d theta int_{zero}^ sin ^ phi d phi int_{zero}^ r^{three} d r &=three[sin theta]_{zero}^ int_{zero}^left[fracright] d phileft[frac}right]_{zero}^ &=three(1)left[frac-frac sin (2 theta)right]_{zero}^(4) &=12left(fracright)=three pi end∫∫ S F⋅ndσ =∫ 0π/2 ∫ 0π/2 ∫ 02 3r three cosθsin 2 ϕdrdϕdθ=three∫ 0π/2 cosθdθ∫ 0π/2 sin 2 ϕdϕ∫ 02 r three dr=three[sinθ] 0π/2 ∫ 0π/2 [ 21−cos(2ϕ) ]dϕ[ 4r 4 ] 02 =three(1)[ 2θ − 41 sin(2θ)] 0π/2 (4)=12( 2π/2 )=3π.
Read more about divergence;
https://brainly.com/question/17177764
#SPJ4
