# Use the Divergence Theorem to compute the flux of F across

Consider the vector field $$F=<0,0,x+z>$$,and the surface S which is the part of the plane $$x+y+2z=4$$ inside the first octant. Assume the unit normal vector n to S has positive third component.Use the Divergence Theorem to compute the flux of F across S by finding instead the flux of F across a different surface and an appropriate triple integral.

Here is my work: $$div(\mathbf F) = \frac{\partial 0}{\partial x} + \frac{\partial 0}{\partial y}+\frac{\partial}{\partial z}(x+z) = 1$$ By divergence theorem, $$\int\int\int_Ddiv\mathbf FdV=\int\int_S(F\cdot n)d\sigma$$ $$\int\int\int_D1dV = the\;volume\;under\;"x+y+2z=4"$$ so I got $$\frac {16}3$$, but the correct answer is 16. Does anyone know where I did it wrong?

That is because applying Divergence Theorem will give you flux through the entire closed region which is a tetrahedron with $$4$$ surfaces enclosed by $$x + y + 2z = 4$$ in the first octant and the coordinate planes, including the XY plane between $$(0, 0, 0), (4, 0, 0), (0, 4, 0)$$ and similarly part of $$YZ$$ and $$XZ$$ planes.

But I think your question asks you to find flux only through the surface $$x + y + 2z = 4$$. So you need to subtract the flux through the other $$3$$ surfaces.

Given our vector field is $$(0, 0, x + z)$$, the flux through $$XZ$$ plane and $$YZ$$ planes will be zero as the unit normal vectors to these planes will have no $$z$$ component and dot product $$\vec{F} \cdot \hat{n}$$ will be zero. But that is not the case for $$XY$$ plane.

The outward pointing unit normal vector through the surface which is $$XY$$ plane is $$(0, 0, -1)$$.

Flux through this surface will be $$\displaystyle \int_0^4 \int_0^{4-y} (0, 0, x + 0) \cdot (0, 0, -1) \, dx \, dy = -\frac{32}{3}$$.

So the flux through surface ($$S, x + y + 2z = 4$$) will be the flux through the entire closed region minus the flux through other $$3$$ surfaces.

$$= \frac{16}{3} - (-\frac{32}{3}) = 16$$.

• @Boba did this help? Do you have any questions? Dec 15, 2020 at 18:12