E&M: Prove the Divergence Theorem

[N8G]

Homework Statement

Griffiths Introduction to Electrodynamics 4th Edition
Example 1.10

Check the divergence theorem using the function:
v = y^2 (i) + (2xy + z^2) (j) + (2yz) (k)
and a unit cube at the origin.

Homework Equations

(closed)∫v⋅da = ∫∇⋅vdV
The flux of vector v at the boundary of the closed surface (surface integrals) is equal to the volume integral of the divergence of the vector field.

The Attempt at a Solution

I have completed the volume integral of ∇⋅v and it equals 2. However the issue I am running into is that I do not understand how to properly construct all 6 of the surface integrals necessary to determine the left hand side of the equation. Obviously it must equal 2 but I need help setting up the integrals.

This is an example problem in the Griffiths E&M text however it just shows the integrals and their solutions rather than how each integral is conceived for each surface of the cube.

If anyone can give me a walkthrough on how to set up those integrals, I would appreciate it immensely.

Thank you.

 

[kuruman]

You have to
1. Pick a cube face, say the one parallel to the yz-plane at x = a (a = 1 m).
2. Express the outward normal in Cartesian coordinates. Answer: $\hat n = \hat x$.
3. Calculate $\vec v \cdot \vec n$. Answer: $(y^2 \hat x +(2ay + z^2) \hat y +2yz \hat z)\cdot \hat x$
4. Multiply out the dot product.
5. Do the integral $\int{\vec v \cdot \vec n~dA},$ where for this particular choice of face $dA=dy~dz$.

Note that in step 3 I replaced $x$ with its constant value at that face. Similar replacements must be done at the other 5 faces where $x$, $y$ or $z$ are constant.

 

[Charles Link]

I'll give you an example of one surface integral: the surface that has $z=1$ with the outward normal pointing in the $+\hat{k}$ direction, where $x$ and $y$ each get integrated from $0$ to $1$. You just take $\int\limits_{0}^{1} \int\limits_{0}^{1} \vec{v} \cdot \hat{k} \, dx dy$. It takes a little work doing all 6 faces, and you need to make sure you pick the outward pointing normal in the right direction, but it's pretty straightforward.