2

I need to find flux of vector field $F=(x,y,z)$ over area $T=\{|x|+|y|+|z|\le 4|(x,y,z)\in \mathbb R^3\}$ oriented in the direction of outside normal, using:a)divergence theorem; b)directly. First, using divergence theorem i get integral $\iiint 3dV$. Then, for the bounds, $z$ should always be in $[0,4]$. In $OXY$ plane I get square which contains two symmetric triangles. Then I thought to find bounds for the triangle in first and fourth quadrant like $0\le x\le 4, x-4\le y\le 4-x$(and then multiply integral with $2$). And what is left is to integrate. Is this correct? For the second part I should calculate integral $F\cdot n_0$ where $n_0$ is normal vector, but to find it, first I need to calculate gradient which I guess won't be that easy because of absolute values.

Trevor
  • 533
  • Draw the region. See that calculating the volume is a lot easier than that integral. – Arthur Jan 29 '21 at 12:49
  • OK, but is my working correct? I know that probably there is easier way, but is this correct? – Trevor Jan 29 '21 at 12:51
  • I thought actually to calculate integral $\int\limits_{0}^{4}dx\int\limits_{x-4}^{4-x}dy\int\limits_{0}^{4}dz$. This is not correct? – Trevor Jan 29 '21 at 13:35
  • 1
    No if you keep $z$ independent, it will give you the volume of the cylinder with triangular base. $z$ is not independent. – Math Lover Jan 29 '21 at 13:40
  • Oh, I see you actually done this in $OYZ$ plane, and you were only looking at first quadrant... My bad. And without divergence theorem we are calculating volume of pyramid? – Trevor Jan 29 '21 at 13:44
  • 1
    Yes that is correct. – Math Lover Jan 29 '21 at 13:45
  • You found the divergence $3$ so you have to multiply the volume by $3$. By the way again reading your comment, thought of clarifying - I am actually taking the $x$ limits from plane $- x + y+z = 4$ to $ x+y+z = 4$ (which is not just first octant but both first and second octant). – Math Lover Jan 29 '21 at 13:58
  • Also are you clear how you would do this directly, without applying divergence theorem? – Math Lover Jan 29 '21 at 14:00
  • 1
    Yes, I think I understand everything. Thanks :) – Trevor Jan 29 '21 at 15:39

1 Answers1

2

Yes you have correctly applied the divergence theorem. Now it is about finding the volume of the region.

See the base of the pyramid in the below diagram, which is in $XY$ plane. Now there are two square pyramids, one above $XY$ plane and one below with vertex at $(0,0,4)$ and $(0,0,-4)$.

Side of the base $ b = 4\sqrt2$ and height is $h = 4$. You can simply use the formula for pyramid volume $V = \frac{1}{3}b^2h$. Multiply by $2$ as you have two pyramids.

If you are going integral route, take the base above $x-$axis and that part of the pyramid for positive $z$. Find volume of it and then multiply by $4$.

$V = 4 \displaystyle \int_0^4 \int_{0}^{4-z} \int_{y+z-4}^{4-y-z} \ dx \ dy \ dz$

enter image description here

Math Lover
  • 51,819