[tex]f(x,y,z)=x^2y\,\vec\imath+xy^2\,\vec\jmath+5xyz\,\vec k[/tex]
[tex]\implies\nabla\cdot f=2xy+2xy+5xy=9xy[/tex]
The plane has intercepts in the [tex]x,y[/tex] plane at (4, 0, 0) and (0, 1, 0), so the flux is given by (via the divergence theorem)
[tex]\displaystyle\iint_Sf\cdot\mathrm dS=9\int_{x=0}^{x=4}\int_{y=0}^{y=1}\int_{z=0}^{z=4-x-4y}xy\,\mathrm dz\,\mathrm dy\,\mathrm dx=-48[/tex]
In this exercise we have to know about the divergent theorem and so using the given function calculate the flow through the integrals, in this way we will have to:
The flux is given by [tex]-48[/tex]
Knowing that the function is given by :
[tex]f(x, y, z) = x^2y+xy^2+5xyz[/tex]
Making the divergent function we will find that:
[tex]\nabla f = 2xy+2xy+5xy= 9xy[/tex]
The plane has intercepts in the xy plane at (4, 0, 0) and (0, 1, 0), so the flux is given by (via the divergence theorem):
[tex]\int\limits \int\limits_S{f} \, ds = 9 \int\limits^4_0 \int\limits^1_0 \int\limits^{4-x-4y}_0 {xy} \, dzdydx= -48[/tex]
See more about divergent at brainly.com/question/4890593
