# Verifying Gauss divergence theorem

The question says that:

If $$\vec{F}=(2x^2-3z)\vec{i}-2xy\vec{j}-4x\vec{k}$$, we are supposed to calculate $$\iiint \operatorname{div} \vec{F}\, dV$$, where $$V$$ is the closed region bounded by the planes $$S_1: x=0, S_2: y=0, S_3: z=0$$ and $$S_4: 2x+2y+z=4$$. It suggests the use of Gauss Divergence theorem here, which states: $$\iint_S \vec{F}\, \vec{dS} = \iiint_V \operatorname{div} \vec{F}\, dV$$ The volume integral is easy to calculate, and it reduces to the volume of the tetrahedron: $$\iiint 2x dV=\frac{8}{3}$$

I am trying to verify whether the flux enclosed by the given region in the first octant is the same as above. $$\iint_S \vec{F}\, \vec{dS}=\iint_{S_1} \vec{F}\, \vec{dS}+\iint_{S_2} \vec{F}\, \vec{dS}+\iint_{S_3} \vec{F}\, \vec{dS}+\iint_{S_4} \vec{F}\, \vec{dS}=16+0+\frac{16}{3}-\frac{56}{3} = \frac{8}{3}$$ I am unsure about the flux calculation, especially through $$S_4$$. Kindly help me out in verifying, I am self-learning this topic and have been stuck for long in this question. Thanks in advance.

EDIT 1: $$\iint_{S_4} \vec{F}\, \vec{dS}= \iint_{2x+2y+z=4} \vec{F}.\vec{n_4} dS$$ where $$\vec{n_4}$$ is the unit normal vector on $$S_4$$ along $$\nabla{2x+2y+z}$$ which is: $$\frac{2}{3}\vec{i}+\frac{2}{3}\vec{j}+\frac{1}{3}\vec{k}$$.

So, $$\iint_{S_4} \vec{F}\, \vec{dS}=\frac{1}{3} \iint_{2x+2y+z=4} (4x^2-6z-4xy-4x) dS$$

Now, I took the orthogonal projection of the surface $$S_4$$ on the $$xy$$ plane to calculate it; so the surface element was substituted as: $$dS |{\vec{n_4}.\vec{k}}|=dx.dy$$ in the integral and on further solving, I got the expression:

$$\iint_{0 \le x \le 2, 0 \le y \le (2-x)} 4(x^2-xy+2x+3y-6) dx dy = \frac{-56}{3}$$

EDIT 2: I rechecked my calculation, as per the suggestion of the valuable comment and got my mistake in claculating $$S_1$$. Here $$x=0$$ and $$\vec{n_1}=\vec{-i}$$. Accordingly,

$$\iint_{S_1} \vec{F}\, \vec{dS} = \iint_{0 \le y \le 2,\ 0 \le z \le (4-2y)} 3z \ dz dy = 16$$

Kindly suggest if the calculation is okay, and if it can be done otherwise. Thanks!

• Can you add some more details as to how you evaluated each integral? Describing general steps would suffice. I'm getting a non-zero value for $S_1$. Commented Nov 8, 2023 at 22:36
• What are your unit normals on $S_1,S_2,S_3$? I do not understand your $dS \mod \vec n\cdot\vec k = dx\,dy$. Commented Nov 8, 2023 at 22:52
• @user170231 I got my mistake, please have a look. Thanks for suggesting!
– S.S
Commented Nov 8, 2023 at 23:20
• No, you yourself said that on $S_1$ we have $\vec n = -\vec i$. Commented Nov 8, 2023 at 23:27
• Looks good to me. Since you mentioned you're self-learning and wrote "if it can be done otherwise", are you also looking for other solutions? If so, you might find this post helpful. Commented Nov 9, 2023 at 17:12