Proofs and Connections

The proofs of the Divergence and Stokes Theorems are not new ideas — they are the Fundamental Theorem of Calculus applied carefully in higher dimensions. The key insight is that all integral theorems of vector calculus share one structural pattern: integrate a derivative over a region, get the values on the boundary. Once you see this, the four named theorems (FTC, Green, Stokes, Divergence) reveal themselves as four projections of one statement, which chapter 6 will write as $\int_M d\omega = \int_{\partial M}\omega$.

Proof strategy for the Divergence Theorem

Suppose $V$ is a box $[a_1, b_1]\times[a_2, b_2]\times[a_3, b_3]$ and $\mathbf{F} = (F_1, F_2, F_3)$. We want to show

Hold $y$ and $z$ fixed and apply the FTC in $x$:

The right side is the difference of boundary values on the two $x$-faces. Integrating over $(y, z)\in[a_2,b_2]\times[a_3,b_3]$ gives the $x$-face flux contributions. Repeating for $F_2, F_3$ recovers all six faces. For more general regions, decompose into small boxes and pass to the limit (boundary terms on shared interior faces cancel in pairs).

Proof strategy for Stokes' Theorem

Parametrise the surface $S$ by $\mathbf{r}: D\to S$ for a planar domain $D$. The line integral on $\partial S$ pulls back to a line integral on $\partial D$, and the surface integral on $S$ pulls back to a double integral on $D$. After the change of variables, the equality reduces to Green's Theorem on $D$. Green's Theorem itself is proved by decomposing $D$ into simple regions and applying the FTC in one variable at a time — exactly the strategy for divergence, but in 2D.

The universal pattern

Look at the four classical theorems side by side:

• FTC (1D): $\int_a^b f'(x),dx = f(b) - f(a)$. Integral of a derivative over $[a,b]$ = boundary values.
• Green (2D): $\iint_D (Q_x - P_y),dA = \oint_{\partial D}(P,dx + Q,dy)$. Integral of "derivative" over $D$ = boundary line integral.
• Stokes (2D surface in 3D): $\iint_S (\nabla\times\mathbf{F})\cdot d\mathbf{S} = \oint_{\partial S}\mathbf{F}\cdot d\mathbf{r}$.
• Divergence (3D): $\iiint_V \nabla\cdot\mathbf{F},dV = \iint_{\partial V}\mathbf{F}\cdot d\mathbf{S}$.

Each says the same thing with one extra dimension. In chapter 6 they become one theorem written as $\int_M d\omega = \int_{\partial M}\omega$, where $\omega$ is a differential form, $d$ is the exterior derivative, and $M$ is a manifold with boundary.

Why $d^2 = 0$

The identities $\nabla\times(\nabla f) = \mathbf{0}$ and $\nabla\cdot(\nabla\times\mathbf{F}) = 0$ are not coincidences. In the language of forms (chapter 6), both become a single fact: applying the exterior derivative twice gives zero. The geometric statement is even more striking: the boundary of a boundary is empty, $\partial(\partial M) = \emptyset$. A disc has a circle as its boundary, and a circle has no boundary at all. Algebra and geometry agree.

(These proofs are abstract enough that a static widget would not add clarity. Working through the box-decomposition argument by hand on a unit cube is the recommended drill.)

Pause and think: Stokes and Divergence both have the form “integral of a derivative over a region = something on the boundary.” What plays the role of the derivative in each case? What is the dimensional pattern?

Try it

• For $\mathbf{F} = (P, Q, 0)$ and $S=D$ a region in the $xy$-plane, show that Stokes' Theorem reduces to Green's Theorem. Verify by writing out $\nabla\times\mathbf{F}$ and $d\mathbf{S}$.
• Use the box-decomposition strategy to convince yourself: divergence of $\mathbf{F}=(x,0,0)$ on the unit cube gives $\int 1,dV = 1$, which equals $F_1(1,y,z) - F_1(0,y,z) = 1$ integrated over the $yz$-face area $1$. Both sides agree.
• Explain in one sentence why $\nabla\cdot(\nabla\times\mathbf{F}) = 0$ is the divergence-theorem analogue of "the boundary of a closed surface is empty."
• True or false: every classical integral theorem of vector calculus is implied by one statement on differential forms. Identify that statement.

A trap to watch for

The proofs as written rely on smooth boundaries, but real applications often have corners (the boundary of a cube, for example). The theorems hold for piecewise smooth boundaries with the natural inherited orientation, but extending the proof requires care at edges and vertices where two faces meet. The integrand $\mathbf{F}$ also has to be $C^1$ everywhere on $V$ — singularities (like $1/r^2$ at the origin) break the theorem unless you carve out a small ball around the singularity and treat its boundary separately. This is exactly the trick that derives Gauss's law for a point charge.

What you now know

You can sketch the proofs of the Divergence and Stokes Theorems via FTC plus decomposition, recognise the four classical integral theorems as projections of a single statement, and anticipate the generalised form on manifolds. Chapter 6 builds the algebraic machinery (wedge products, exterior derivative, differential forms) needed to state and use that one theorem.

References

• Garrity, T. (2002). All the Mathematics You Missed. Cambridge University Press, ch. 5.
• Spivak, M. (1965). Calculus on Manifolds. W. A. Benjamin, ch. 5 (the classical proofs).
• Marsden, J. E., Tromba, A. J. (2011). Vector Calculus (6th ed.). W. H. Freeman.
• Bachman, D. (2012). A Geometric Approach to Differential Forms (2nd ed.). Birkhäuser.
• Schey, H. M. (2004). Div, Grad, Curl, and All That (4th ed.). W. W. Norton.