Analysis on Manifolds V: Stokes' Theorem

The conclusion of a five-part lecture series on differential forms and the generalised Stokes theorem.

0. The culmination

Five parts converge here.

Part I built analysis on from first principles: the topology of open balls and compact sets, the derivative as a linear map , the chain rule as composition of linear maps, and the inverse and implicit function theorems that let one pass between coordinate descriptions. The derivative was identified as the best linear approximation to a smooth map, a local, linear-algebraic shadow of global, nonlinear geometry.

Part II built the algebra of a single vector space: the dual , multilinear alternating maps, the wedge product, the basis of , determinants as top forms, the interior product , and the Hodge star that turns the algebra into a metric-enriched structure.

Part III glued the algebra onto smooth manifolds: tangent spaces , cotangent bundles , differential -forms as smooth sections of , the exterior derivative satisfying , and the pullback that makes differential forms covariant. The classical vector calculus operations grad, curl, and div emerged as the exterior derivative at dimensions 0, 1, and 2 in .

Part IV integrated. Orientation, manifolds with boundary, the induced orientation on , partitions of unity, the integral of a -form over an oriented -manifold, the Riemannian volume form, period integrals and their relationship to topology. The de Rham cohomology emerged as the obstruction to exactness, computed for spheres, tori, projective spaces, and the Möbius band via the Mayer-Vietoris sequence. Part IV closed with Poincaré duality and a preview of this conclusion.

Part V proves the theorem for which all of this was preparation. The generalised Stokes theorem

holds for any compact oriented smooth -manifold with boundary and any -form . Every classical integral theorem—the fundamental theorem of calculus, Green's theorem, the divergence theorem, the classical Stokes theorem for surface integrals—is the same identity at a different value of and a different choice of . Equation [eq:stokes-main] is one theorem that subsumes them all.

After the proof and its classical special cases, this part develops the analytic depth behind de Rham cohomology. Hodge theory reinterprets each cohomology class as containing a unique harmonic representative, a smooth form in the kernel of the Laplacian . The Hodge decomposition theorem, proved here via Sobolev spaces and elliptic regularity, is the analytic engine that makes this identification rigorous. It implies that the Betti numbers equal the dimension of the space of harmonic forms, connecting topology to analysis in the deepest possible way.

The series ends with the de Rham isomorphism: the identification of with via integration, proved here for and and stated in full generality.

Notation. We follow the conventions of Parts I–IV: roman italic for points (), boldface for tangent vectors (). All manifolds are smooth, oriented, and compact unless stated otherwise. We follow Bott-Tu [1], Lee [2], Warner [3], Griffiths-Harris [4], Hodge [5], and Evans [6].

1. Stokes' theorem

1.1. Statement

Equation [eq:stokes] is the same as [eq:stokes-main]; we give it a second label for local cross-references. The theorem is proved in two steps: first on the half-space (the core computation), then globally via a partition of unity.

1.2. Proof on the half-space

Let be a compactly supported -form on the half-space with boundary . Write

where the hat denotes omission. Then

Integrating over using Fubini and the fundamental theorem of calculus in each variable:

For : the inner integral is by compact support. For : the inner integral is (again by compact support, the upper limit vanishes). Therefore

The boundary with the induced orientation (outward normal first, then ) contributes the sign . With this sign, the right-hand side of [eq:half-stokes-rhs] equals exactly . This proves [stokes] on .

1.3. Proof via partition of unity

The proof is essentially just bookkeeping: the half-space computation contains the entire mathematical content, and the partition of unity assembles local pieces into a global result.

2. Classical theorems recovered

[stokes] has four classical names. Each is the same identity [eq:stokes] with a specific manifold and a specific form . We record them as corollaries with explicit proofs identifying the pieces.

2.1. Fundamental theorem of calculus

from sympy import *
x = symbols('x')

# Verify FTC: int_a^b F'(x) dx = F(b) - F(a)
F = x**3 - 2*x
a, b = 0, 2

lhs = integrate(diff(F, x), (x, a, b))
rhs = F.subs(x, b) - F.subs(x, a)

print(f"F(x)          = {F}")
print(f"F'(x)         = {diff(F, x)}")
print(f"integral      = {lhs}")
print(f"F(2) - F(0)   = {rhs}")
print(f"FTC verified  : {lhs == rhs}")

2.2. Green's theorem

from sympy import *
r, t = symbols('r t', real=True)

# Verify Green's theorem: P = -y, Q = x on the unit disk
# Interior: integrate (Q_x - P_y) = 2 over the disk in polar
interior = integrate(integrate(2*r, (r, 0, 1)), (t, 0, 2*pi))

# Boundary: parametrise unit circle x = cos t, y = sin t
x_c, y_c = cos(t), sin(t)
boundary = integrate(-y_c * diff(x_c, t) + x_c * diff(y_c, t), (t, 0, 2*pi))

print(f"Interior  iint_D (Q_x - P_y) dA = {interior}")
print(f"Boundary  oint   P dx + Q dy    = {boundary}")
print(f"Green's theorem verified: {interior == boundary}")
print(f"(Both equal 2*pi = {float(interior):.6f})")

2.3. Divergence theorem

2.4. Classical Stokes theorem

The four theorems are now seen as one: in each case the left side integrates the exterior derivative of over an oriented manifold , and the right side integrates over the boundary . The geometry of and the choice of vary; the identity does not.

3. Stokes' theorem and de Rham cohomology

[stokes] is not merely a formula for computing integrals. It is the key that unlocks the topological content of de Rham cohomology: exact forms cannot be detected by integration over cycles, and the cohomology groups are diffeomorphism invariants.

3.1. Exact forms on closed manifolds

If is a closed manifold and , then is not exact; it carries non-trivial cohomological information.

3.2. Pullback functoriality

3.3. Diffeomorphism invariance of Betti numbers

The cohomology computations of Part IV—spheres, tori, projective planes, the Möbius band—are therefore topological facts: they depend only on the topology of the space, not on the specific smooth structure chosen.

4. Hodge theory

The de Rham cohomology of Part IV was defined purely algebraically: . This section gives it an analytic description. The key insight, due to Hodge [5], is that each cohomology class contains a distinguished smooth representative—the harmonic form—characterised by being killed by both and its adjoint . The proof that such representatives exist requires functional analysis: Sobolev spaces and elliptic regularity for the Laplacian.

4.1. The L² inner product on forms

Fix a compact oriented Riemannian -manifold without boundary. Recall from Part II §8 the Hodge star , defined by .

4.2. The co-differential

from sympy import *
theta, phi = symbols('theta phi', real=True)
a, b, c, d = symbols('a b c d', real=True)

# On T^2 with flat metric, the L^2 inner product of 1-forms is
#   <alpha, beta>_L2 = int_0^{2pi} int_0^{2pi} (a*c + b*d) dtheta dphi
# Verify adjointness: <d*f, omega>_L2 = <f, d* omega>_L2
# For f = function, omega = 1-form, d*omega is a 0-form (function)
# d*(a dtheta + b dphi) = -(a_theta + b_phi) on flat T^2

# Take f = cos(theta), omega = sin(phi) dtheta + cos(theta) dphi
f = cos(theta)
a_val = sin(phi)
b_val = cos(theta)

# d f = -sin(theta) dtheta  (exterior derivative of 0-form)
df_a = diff(f, theta)   # coefficient of dtheta
df_b = diff(f, phi)     # coefficient of dphi  (= 0)

# d* omega = -(d/dtheta)(a_val) - (d/dphi)(b_val)  [flat T^2 formula]
d_star_omega = -(diff(a_val, theta) + diff(b_val, phi))

# <df, omega>_L2
lhs = integrate(integrate(df_a * a_val + df_b * b_val, (theta, 0, 2*pi)), (phi, 0, 2*pi))

# <f, d* omega>_L2
rhs = integrate(integrate(f * d_star_omega, (theta, 0, 2*pi)), (phi, 0, 2*pi))

print(f"f              = {f}")
print(f"omega          = sin(phi) dtheta + cos(theta) dphi")
print(f"d*omega        = {d_star_omega}")
print(f"<df, omega>    = {lhs}")
print(f"<f, d*omega>   = {rhs}")
print(f"Adjointness verified: {simplify(lhs - rhs) == 0}")

4.3. The Hodge Laplacian and harmonic forms

4.4. Hodge decomposition theorem

The full proof goes through Sobolev spaces and elliptic regularity. We sketch the main steps.

The proof has two moves: Gårding gives control of the Sobolev norm from the Laplacian, and Rellich turns that into compactness of the resolvent. Together they force the spectrum of to be discrete and the harmonic space to be finite-dimensional.

5. Harmonic representatives

The Hodge decomposition makes de Rham cohomology concrete: every class has a canonical smooth representative, determined uniquely by the Riemannian structure.

5.1. Every class has a unique harmonic representative

5.2. Betti numbers as harmonic dimensions

The Betti number —a topological invariant—equals the dimension of a space of smooth differential equations: the solutions to . Topology and analysis are measuring the same thing.

from sympy import binomial, Integer

# Betti numbers of spheres and tori
print("Betti numbers b_k(S^n)  [b_0 = b_n = 1, all others 0]:")
for n in range(5):
  row = [1 if (k == 0 or k == n) else 0 for k in range(n + 1)]
  print(f"  S^{n}: {row}  (chi = {sum((-1)**k * v for k,v in enumerate(row))})")

print()
print("Betti numbers b_k(T^n)  [b_k = C(n,k)]:")
for n in range(5):
  row = [int(binomial(n, k)) for k in range(n + 1)]
  chi = sum((-1)**k * v for k, v in enumerate(row))
  print(f"  T^{n}: {row}  (sum = 2^{n} = {2**n}, chi = {chi})")

5.3. Poincaré duality via the Hodge star

6. The de Rham isomorphism

The series has two descriptions of the same cohomological data: the analytic one (de Rham cohomology, built from differential forms) and the topological one (singular homology, built from continuous maps of simplices). The de Rham theorem says they carry identical information, connected by integration.

6.1. The period map

6.2. Worked cases

from sympy import *
t = symbols('t', real=True)
c, a, b = symbols('c a b', real=True)

# Period of c*dtheta over [S^1]
# Parametrise S^1 as theta = t in [0, 2*pi], dtheta evaluates to dt
period_S1 = integrate(c, (t, 0, 2*pi))
print(f"Per([c dtheta])([S^1]) = int_S1 c dtheta = {period_S1}")
print(f"                      = 2*pi*c  (= {period_S1})")

print()

# Period matrix entries for omega = a*dtheta + b*dphi on T^2
# alpha-cycle: theta varies, phi fixed => dtheta = dt, dphi = 0
per_alpha_dtheta = integrate(a, (t, 0, 2*pi))   # int_alpha a dtheta
per_alpha_dphi   = integrate(0, (t, 0, 2*pi))   # int_alpha b dphi = 0

# beta-cycle: phi varies, theta fixed => dtheta = 0, dphi = dt
per_beta_dtheta  = integrate(0, (t, 0, 2*pi))   # int_beta a dtheta = 0
per_beta_dphi    = integrate(b, (t, 0, 2*pi))   # int_beta b dphi

P = Matrix([[per_alpha_dtheta, per_alpha_dphi],
          [per_beta_dtheta,  per_beta_dphi]])
print("Period matrix P for T^2:")
print(P)
print(f"P / (2*pi) = {P / (2*pi)}")
print(f"det(P) = {P.det()} (non-zero => de Rham isomorphism)")

6.3. The de Rham theorem

The isomorphism [eq:de-rham-eq] is the culmination of the entire series. Every integral theorem proved here is a shadow of this single algebraic fact: integration is the natural pairing between differential forms and cycles, and it is non-degenerate.

Proof strategy (full proof: Bott-Tu [1] §I.15). Induction on the number of open sets in a good cover of (every finite intersection is contractible). Base case: a single contractible set; Poincaré lemma gives for , matching singular cohomology. Inductive step: Mayer-Vietoris on both de Rham and singular cohomology, connected by period maps; the five-lemma propagates the isomorphism.

7. Epilogue

Five parts. One idea. A linear map that commutes with the boundary.

The derivative of Part I is the linear map. The exterior derivative of Part III is the same map, acting on forms. The pullback of Part III is the transpose. The partition-of-unity integration of Part IV assembles local linear maps into a global pairing. And Stokes' theorem says: integration commutes with the boundary. The whole series is one idea, rephrased five times at increasing levels of abstraction.

This mathematics is not a closed chapter. Gauge theory rewrites Maxwell's equations as and — the exterior derivative on a principal bundle. General relativity is the statement that the Riemann curvature 2-form satisfies the Bianchi identity, the gravitational analogue of . The Atiyah-Singer index theorem—the deepest result in 20th-century differential geometry—generalises the Hodge theorem: for an elliptic operator on a vector bundle over a compact manifold,

where the right side is an integral of characteristic forms. The Euler characteristic, the signature, the Dirac index; each is a special case of [eq:atiyah-singer], obtained by choosing appropriately. The Stokes theorem you just proved is the case.

Three natural sequels follow. A series on connections and curvature takes the exterior derivative on functions and extends it to sections of vector bundles, the language of gauge theory and general relativity. A series on advanced PDE theory develops the Sobolev spaces and elliptic regularity used in §4 to their natural depth: spectral theory, heat kernels, the Atiyah-Singer index theorem. And a series on algebraic topology gives the complete proof of the de Rham theorem and develops singular cohomology alongside de Rham, with cup products, cap products, and Poincaré-Lefschetz duality.

The exterior algebra you built in Part II sits at the foundation of all of it.

References

1. R. Bott and L.W. Tu, Differential Forms in Algebraic Topology, Springer, 1982. Primary reference for de Rham cohomology, Mayer-Vietoris, and the de Rham theorem. DOI
2. J.M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. Chapter 16 (Stokes' theorem), Chapter 17 (de Rham cohomology). DOI
3. F.W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer, 1983. Chapter 6 (Hodge theory). DOI
4. P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley Classics Library, 1994. Chapter 0 (Hodge theory on Kähler manifolds). (ISBN 978-0-471-05059-9)
5. W.V.D. Hodge, The Theory and Applications of Harmonic Integrals, Cambridge University Press, 1941 (repr. 1989). The original source. (ISBN 978-0-521-35881-1)
6. L.C. Evans, Partial Differential Equations, 2nd ed., AMS, 2010. §6.3 (Gårding inequality), §6.6 (elliptic regularity), Appendix D (Sobolev spaces). (AMS GSM 19, ISBN 978-0-8218-4974-3)