Difference between Stokes' Theorem and Divergence Theorem

Question

Recently learnt the two and I really can't tell the difference. I'm not sure if I'm missing something, but it really seems to me that they evaluate the same thing just using different methods.
With Stokes' Theorem, it seems to me that we evaluate the flux surface integral of a vector field with the double integral of the curl of the vector field dotted with the tangent vector component.
Then with the Divergence Theorem, it seems that we evaluate the same thing, except taking the triple integral of the divergence of the vector field...

I'm not sure if I'm misunderstanding it and I tried searching online but they only tell me methods to use them. Can someone explain in simple terms how to differentiate these?

Here is a question that might help clear my confusion.
Use two methods to calculate the flux integral $$\int_{S}(\nabla \times {\textbf{F}})\cdot dS$$
where ${\textbf{F}} = (y,z,x^2y^2)$ and $S$ is the surface given by $z=x^2 + y^2$ and $0\leq z \leq 4$.

I don't necessarily need a full method on this, but just (I guess) which theorems relate to this.
It looks like Stokes' Theorem, with the curl there... so this would be Stokes Theorem'. So first method is via the given equation, and then second would be evaluating the line integral with boundary of $S$? But what would be the boundary...? Would it be the circle when $z=0$ or the one at $z=4$?

Answer 1

Long story short, Stokes' Theorem evaluates the flux going through a single surface, while the Divergence Theorem evaluates the flux going in and out of a solid through its surface(s).

Think of Stokes' Theorem as "air passing through your window", and of the Divergence Theorem as "air going in and out of your room". Clearly, if you only had the window, both results should coincide, but what if you also count the door in your room?

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Just FYI: They do look similar because both theorems are actually special cases of a more generalized version of Stokes' Theorem.

Answer 2

In a sense, Stokes', Green's, and Divergence theorems are all special cases of the generalized Stokes theorem for differential forms $$\int_{\partial \Omega} \omega = \int_\Omega d\omega$$ but I don't think that's what you're asking about.

The usual (3-dimensional) Stokes' and Divergence theorems both involve a surface integral, but they are in rather different circumstances.

In the Divergence theorem, the surface $S$ is the boundary of a bounded region $R$ of space, and you're taking the flux through this surface of a vector field $\bf F$ defined in $R$ and on its boundary:

$$ \iint_S {\bf F} \cdot d{\bf S} = \iiint_R \text{div}\;{\bf F}\; dV $$

In Stokes' theorem, the surface is generally not the boundary of a region: instead it has a boundary which is a curve $C$; you're taking the flux, not of an arbitrary vector field, but of the curl of some other field:

$$ \iint_S \text{curl}\; {\bf G} \cdot d{\bf S} = \oint_C {\bf G} \cdot d{\bf r} $$

There is one situation where both apply: suppose your surface $S$ is the boundary of a bounded region $R$, and your vector field $\bf F$ happens to be the curl of some other field $\bf G$. Since the divergence of the curl is $0$, the Divergence theorem says the result is $0$. On the other hand, for Stokes the surface has no boundary (it's a closed surface), so Stokes integrates $\bf G$ around an empty curve and gets $0$ as well.