Everything in the course converges on one line: $\int_Md\omega=\int_{\partial M}\omega$. Differentiating inside and taking the boundary outside are adjoint operations. The fundamental theorem of calculus, Green's theorem, the divergence theorem and classical Stokes are the cases $k=1,2,3$ in various dimensions — not analogues of each other, but the same statement.
Conservation laws are boundary statements: what accumulates inside equals what crosses the surface. Writing that once, in a form independent of dimension and coordinates, is what makes it usable in general relativity, in fluid mechanics, and on any manifold at all.
Gauss's law, the continuity equation, circulation and lift, and every finite-volume numerical scheme are applications. The discrete analogue — exterior calculus on meshes — is used to build numerical methods that conserve exactly rather than approximately.
The classical version appeared on an 1854 Cambridge exam set by Stokes, who had it in a letter from Kelvin. Cartan's general form arrived around 1899. The pairing between $d$ and $\partial$ became de Rham cohomology, and the same duality reappears in Poincaré duality and in Hodge theory.
Let $M$ be a compact oriented $k$-manifold with boundary, and let $\omega$ be a $C^1$ $(k-1)$-form on $M$. Give $\partial M$ the induced orientation. Then
If $\partial M=\emptyset$, the right side is zero, so every exact form integrates to zero over a closed manifold. Compactness can be traded for compact support; without one of the two the theorem is false, as $\int_0^\infty$ of a derivative shows.
At a boundary point, choose an outward-pointing vector $\nu$. A basis $(v_1,\dots,v_{k-1})$ of the boundary tangent space is positively oriented when $(\nu,v_1,\dots,v_{k-1})$ is positively oriented in $M$. "Outward normal first" is the whole rule, and every sign convention in classical vector calculus is a consequence of it.
In dimension one this says the right endpoint counts $+$ and the left endpoint $-$, which is where the $F(b)-F(a)$ of the fundamental theorem comes from.
Under the machinery, the theorem is the one-variable fundamental theorem applied once, in the right coordinate, with everything else arranged so that only that one application is needed.
$M=[a,b]$, a $1$-manifold with boundary $\{a,b\}$; $\omega=f$ a $0$-form. Then $d\omega=f'\,dx$ and the induced orientation gives $b$ the sign $+$ and $a$ the sign $-$:
$M\subseteq\mathbb{R}^2$ a region with boundary curve, $\omega=P\,dx+Q\,dy$. Then $d\omega=(Q_x-P_y)\,dx\wedge dy$ and
with the boundary traversed counterclockwise — which is what the outward-normal-first rule produces in the plane.
$M\subseteq\mathbb{R}^3$ a solid region, $\omega$ the $2$-form corresponding to a field $F$. Then $d\omega=(\nabla\cdot F)\,dV$ and
with $n$ the outward normal — again straight from the induced orientation.
$M$ an oriented surface in $\mathbb{R}^3$ with boundary curve, $\omega$ the $1$-form of a field $F$. Then $d\omega$ corresponds to $\nabla\times F$ and
the curve oriented by the right-hand rule relative to $n$.
Each case above is the same identity with $k=1,2,3$ and the appropriate degree of $\omega$. The apparent differences — a difference of values, a line integral, a flux integral — come entirely from the vector calculus dictionary of Unit 12 and the coincidence $\binom32=3$ in $\mathbb{R}^3$.
Taking $\omega=\tfrac12(-y\,dx+x\,dy)$ gives $d\omega=dx\wedge dy$, so the area of a plane region equals $\tfrac12\oint(-y\,dx+x\,dy)$ over its boundary. That integral is what a planimeter mechanically evaluates, and it is the shoelace formula in the polygonal case.
The angle form $\omega$ on the punctured plane is closed with $\oint_{S^1}\omega=2\pi$. If it were exact, say $\omega=df$, then Stokes on the circle — a closed manifold — would give $\oint\omega=0$. So it is not exact, and the obstruction is the puncture.
$\int_{S^2}d\eta=0$ for every $1$-form $\eta$, since $S^2$ has no boundary. Consequently the volume form of $S^2$, whose integral is $4\pi$, is not exact — a statement about the topology of the sphere obtained by pure calculus.
Define $H^k_{dR}(M)$ as closed $k$-forms modulo exact ones. Stokes says integration over a closed $k$-manifold is well defined on this quotient, and de Rham's theorem says the quotient computes the topology of $M$. That is the doorway out of this course, into MAT327H1 and beyond.
• Getting the induced orientation wrong. Outward normal first, every
time; the classical right-hand rules are consequences, not extra
conventions.
• Dropping compactness or compact support. Both sides can then be
infinite or unequal.
• Applying it across a singularity. The punctured disc is not a manifold
with boundary containing the puncture, which is exactly why the angle form
survives.
• Mismatching degrees. On a $k$-manifold, $\omega$ must be a
$(k-1)$-form.
• Concluding a form is exact because its integral vanishes over one
cycle. It must vanish over every cycle, and even then the conclusion needs de
Rham.
Original problems written for this guide in the style of Spivak and Munkres exercises. Write a full solution before opening the answer.
Evaluate $\oint_C(y^2\,dx+x^2\,dy)$ over the unit circle counterclockwise, two ways.
Compute the flux of $F=(x,y,z)$ out of the unit sphere.
Use the boundary formula on $x=a\cos t$, $y=b\sin t$.
Why is $\iint_{S}(\nabla\times F)\cdot n\,dS=0$ for a closed surface $S$?
Two surfaces share a boundary curve. Compare the flux of a curl through each.
Let $C$ be a loop in the punctured plane winding twice around the origin. Compute $\oint_C\omega$ for the angle form.
Derive $\int_M d\alpha\wedge\beta =\int_{\partial M}\alpha\wedge\beta-(-1)^{k}\int_M\alpha\wedge d\beta$ for a $k$-form $\alpha$.
Show the volume of a solid $M\subseteq\mathbb{R}^3$ equals $\tfrac13\iint_{\partial M}(x,y,z)\cdot n\,dS$.
Course coverage. This unit is part of the complete MAT257Y1 topic map. It should cover the following items before you move on:
Spivak's final chapter states and proves the theorem in the form given here, then derives the classical cases; Munkres does the same at greater length with more attention to the orientation bookkeeping. Confirm which classical cases the syllabus expects you to derive.
Manifolds and orientation give the left and right sides meaning. Partitions of unity reduce the proof to a chart. The fundamental theorem of calculus finishes it. $d^2=0$ paired with $\partial\partial=\emptyset$ is what the whole framework was built to express.
Use these after the main practice set. They are designed to force definition-level reasoning, theorem selection, and mixed application rather than routine calculation only.
Write Stokes' theorem in one line and say what each side means.
State the induced boundary orientation.
What does the theorem give on a closed manifold?
Which two facts are dual to each other under the theorem?