Analysis on Manifolds IV: Integration
Part IV of a five-part lecture series on differential forms and the generalised Stokes theorem.
This lecture includes interactive SymPy cells that verify key results symbolically. SymPy is a Python library for symbolic mathematics. Loading it fetches a WebAssembly Python runtime (approx. 15 MB, cached after first load). You can also load it on demand from any code cell below.
0. From the exterior derivative to integration
Parts I through III built the analytic and algebraic machinery that makes integration on manifolds possible. Part I established the calculus of : the inverse and implicit function theorems, the definition of the derivative as a linear map, and the theory of smooth maps. Part II built the algebra of a single vector space: the dual, alternating tensors, the wedge product, and the pullback of a linear map. Part III glued the algebra onto the smooth structure, producing differential -forms, the exterior derivative , and the key identity . Part III closed by defining closed and exact forms and previewing de Rham cohomology.
Part IV uses all of that to integrate. The central insight is that a -form is precisely the kind of object that can be integrated over an oriented -dimensional region: the alternating sign encodes orientation, and the pullback formula that is already built into the form's definition automatically handles changes of coordinates. Integration requires no new constructions beyond what Parts I-III developed.
This post has two acts. Act I (Sections 1-6) sets up the integration machine: what it means for a form to be a derivative or antiderivative (Section 1), what orientation means for manifolds (Section 2), what the boundary of a manifold is (Section 3), how to patch local definitions globally with partitions of unity (Section 4), the definition of the integral of a -form (Section 5), and the Riemannian volume form that lets one integrate ordinary functions (Section 6). Act II (Sections 7-10) is the payoff: periods and the winding number (Section 7), the Mayer-Vietoris long exact sequence (Section 8), cohomology computations for a zoo of surfaces (Section 9), and Poincaré duality (Section 10). Section 11 previews Part V.
A note on notation. Part I worked entirely in , where points and vectors are the same object: every point is simultaneously a vector in the vector space . That identification breaks down on a general manifold. A point lives in the manifold; a tangent vector lives in the tangent space at , a separate vector space. Throughout Part IV we write points in roman italic () and tangent vectors in boldface (), maintaining the distinction that collapses only when .
We follow Bott-Tu [1], Lee [2], Tu [3], and Bredon [4].
Act I: Integration
1. Forms as derivatives and antiderivatives
Before defining the integral we ask: what makes a form the right object to integrate? The answer is that forms already encode the infinitesimal change-of-variable formula. Unpacking this reveals that the exterior derivative is the correct notion of derivative for forms and that antiderivatives (primitives) of forms are exactly exact forms.
1.1. The ratio
Let be smooth. In Part III we defined by , where is the standard (constant) 1-form on . The symbol is a genuine differential 1-form; it pairs with a tangent vector to give .
The classical derivative is the coefficient of in the 1-form . That is:
The equality is a literal equality of 1-forms, rearranged: dividing both sides by the 1-form (a non-vanishing section of the cotangent bundle on ) gives the smooth function . In dimension one the cotangent bundle is a line bundle, so the ratio of two 1-forms is a well-defined smooth function at each point. This is the conceptual origin of Leibniz notation.
On with coordinates , for :
The coefficient of is the partial derivative . The 1-form packages all partial derivatives into a single coordinate-free object (its value on a tangent vector is the directional derivative ). The ratio interpretation generalises: .
1.2. Antiderivatives as exact forms
A -form is a primitive (or antiderivative) of if . Equivalently, is exact.
On , a 1-form is exact iff has a smooth antiderivative (i.e. ), in which case . This is precisely the classical notion. The key point is that primitives of forms are forms of one degree lower, and the operator taking primitives is the inverse of (when it exists).
On , a 1-form is exact iff for some smooth , which requires for all (the curl-free condition). This is a necessary condition for exactness because implies for any exact form. The Poincaré lemma (Part III, §10.3) shows this is also sufficient locally. Globally, the failure is measured by .
On , define the 1-form
A direct computation gives (closed). In polar coordinates where , , one finds . Locally, is a perfectly good smooth antiderivative. Globally, jumps by as one circles the origin, so no single-valued smooth function serves as a primitive on all of . The form is closed with no global primitive: it generates .
1.3. The fundamental theorem as baby Stokes
The exterior derivative assigns to a smooth function the 1-form . Integrating this 1-form over the interval should recover at the endpoints. That is precisely the content of the fundamental theorem of calculus.
Let be smooth. Then
Proof
Define the accumulation function by
Step 1: . For and small ,
Since is continuous, for every there exists such that whenever . For ,
Hence for all , with the same argument applying one-sidedly at the endpoints.
Step 2: is constant. Since on and is connected, the mean value theorem implies is constant.
Step 3: Evaluate. At , , so the constant equals . Therefore for all , and setting gives
The theorem takes on additional meaning in the language of forms. Since is the exterior derivative of the 0-form , and since the interval is a 1-manifold with boundary (two points carrying the induced orientation, positive at and negative at ), the theorem reads
Equation [eq:baby-stokes] is the generalised Stokes theorem in dimension 1: the integral of the exterior derivative of a form over a manifold equals the integral of the form over the boundary. Every classical integral theorem—Green's theorem in the plane, the divergence theorem in space, the classical Stokes theorem for surfaces—is the same identity with replaced by a higher-dimensional manifold with boundary. Part V proves the full theorem; equation [eq:baby-stokes] is the baby case that motivates the entire structure.
A differential -form is exactly the right kind of object to integrate over an oriented -manifold: its pullback transformation law already encodes the change-of-variables formula, and its exterior derivative is the object whose integral over a region equals the integral of the original form over the boundary.
2. Orientation
Integration requires a choice of sign: which way is "positive"? This is orientation. On a vector space, orientation amounts to choosing a preferred basis up to positive-determinant changes. On a manifold, orientation amounts to making this choice consistently across all tangent spaces.
2.1. Oriented vector spaces
Let be a real vector space of dimension . Two ordered bases and are equivalent if the transition matrix defined by has positive determinant. This is an equivalence relation with exactly two classes; an orientation of is a choice of one class.
An orientation of is equivalently a choice of nonzero element of up to positive scalar: a nonzero -form is positively oriented if for a positively oriented basis .
has a standard orientation: the standard basis is positive. The standard -form is .
2.2. Oriented manifolds
A smooth -manifold is orientable if there exists a nowhere-vanishing -form . An orientation is a choice of such an up to positive smooth function; two choices are equivalent if their ratio is everywhere positive. An oriented manifold is a pair .
A smooth -manifold is orientable if and only if it has an atlas whose transition maps have everywhere positive Jacobian determinant. Such an atlas is called oriented.
Proof
Given a non-vanishing -form , choose charts such that in local coordinates with (replacing by its composition with if needed). The transition map then has Jacobian determinant equal to .
Conversely, given an oriented atlas, define a local -form in each chart, and patch these with a partition of unity subordinate to the cover . The positivity of the Jacobians ensures the patches glue to a globally non-vanishing -form.
- : orientable, with the standard orientation .
- : orientable for all . The outward unit normal to the embedding gives a volume form via .
- : orientable, with volume form (where are the angle coordinates).
- : orientable iff is odd.
- Möbius band: not orientable. As one traverses the core circle, the fibre orientation is reversed, preventing a consistent choice.
- Klein bottle: not orientable.
2.3. Orientation-reversing maps
A diffeomorphism between oriented manifolds is orientation-preserving if its Jacobian has positive determinant everywhere, and orientation-reversing otherwise. The reflection defined by is orientation-reversing (). The antipodal map on is orientation-preserving iff is odd.
3. Manifolds with boundary
Integration by Stokes' theorem relates integrals over a manifold to integrals over its boundary. We need to give a precise definition.
3.1. The definition
A smooth -manifold with boundary is a second-countable Hausdorff topological space together with a maximal smooth atlas in which each chart maps to either an open subset of or an open subset of the upper half-space .
The boundary is the set of points that map to the hyperplane under some (hence every) chart whose domain contains . The interior is .
If is a smooth -manifold with boundary, then is a smooth -manifold without boundary (i.e., ).
Proof
Each boundary chart restricts to a homeomorphism from to the open set in (identifying with ). The transition maps of are the restrictions of those of to the boundary hyperplane; they remain smooth (since the defining maps are smooth up to the boundary). The chart images lie in the open subsets of , so no second boundary stratum arises.
- The closed interval : boundary , both points.
- The closed disc : boundary .
- A compact surface (genus- surface) with circular holes cut out: boundary is disjoint copies of .
- itself: boundary .
The Möbius band is a smooth 2-manifold with boundary. Its boundary is a single circle that goes around the band twice (not two separate circles). This reflects the non-orientability: the boundary circle is doubly covered by the boundary of the rectangle used to construct .
3.2. Induced orientation on the boundary
If is an oriented -manifold with boundary, the boundary inherits a canonical orientation.
Let be an oriented -manifold with boundary. At each , let denote an outward-pointing tangent vector (one whose image in any half-space chart points in the direction of decreasing ). The induced orientation on is defined by: a basis of is positively oriented iff is positively oriented in .
This is the "outward normal first" convention. For the interval : the orientation of is the positive direction; the outward normal at points right (positive), at points left (negative). The induced orientation assigns to and to , giving . This recovers the sign in the fundamental theorem.
4. Partitions of unity
A differential form is a global object, but we can only compute with it locally in charts. Partitions of unity allow us to break a global computation into local pieces and reassemble them.
4.1. Smooth bump functions
The key analytic fact is that (unlike the complex plane) supports non-trivial smooth functions with compact support.
There exists a smooth function with on a closed ball and for any . Such a function is called a smooth bump function.
Proof
Define for and for ; this is smooth on . Set ; then is smooth, for , and for . Finally, (after reparametrisation) gives a smooth radially symmetric bump supported in and equal to 1 on .
4.2. Existence of partitions of unity
Let be a smooth manifold and an open cover of . There exists a smooth partition of unity subordinate to : a collection of smooth functions such that:
- for all .
- The collection is locally finite (every has a neighbourhood meeting only finitely many supports).
- for all (the sum is finite at each point by local finiteness).
Proof sketch
Since is second-countable (part of the definition of a smooth manifold), every open cover has a countable locally finite refinement (by paracompactness, which follows from second-countability). For a countable locally finite refinement , choose a bump function supported in and positive on a compact set that covers all points whose only cover is . Set ; local finiteness makes the denominator a finite sum at each point, so it is smooth and positive.
The analytic fact that smooth bump functions exist distinguishes smooth manifolds from real-analytic or complex-analytic manifolds. On a connected complex manifold there are no compactly supported holomorphic functions (other than zero), so the same patching argument fails. Smooth partitions of unity are why many global results (integration, Riemannian metrics, connections) are easier to construct in the smooth category than in the holomorphic one.
5. Integration of differential forms
5.1. Integration in a single chart
Let be an open set and an -form with compact support. Write . Define
where the right-hand side is the standard Lebesgue (or Riemann) integral in .
This definition depends on the orientation: reversing the order of two wedge factors negates and negates the integral.
5.2. The change-of-variables formula
Let be an orientation-preserving diffeomorphism of open subsets of , and . Then
If is orientation-reversing, the equality becomes .
Proof
Write in coordinates on , with . Then
where we used the fact that (Part II, wedge product and determinants). Therefore
where in the middle step we used the classical change-of-variables theorem and the fact that (orientation-preserving). The orientation-reversing case introduces a factor of since .
This result is the key: the pullback formula that is built into the definition of differential forms (Part II) is exactly the change-of-variables formula for integrals. The two notions are the same thing.
5.3. Global definition via partition of unity
Let be a compact oriented -manifold (possibly with boundary) and . Choose an atlas compatible with the orientation and a subordinate partition of unity . Define
Each term is an integral of a compactly supported -form on an open subset of , as in §5.1.
The integral is independent of the choice of oriented atlas and subordinate partition of unity.
Proof
Let be a second partition of unity. On the overlap , the change-of-variables proposition above guarantees that the two chart computations agree (the transition map is orientation-preserving and has positive Jacobian). The sum over all pairs therefore gives the same total regardless of which system is used.
Let be a compact oriented -manifold and .
- Linearity: for .
- Orientation reversal: if denotes with the opposite orientation, then .
- Positivity: if is a volume form compatible with the orientation, then .
- Pullback: if is an orientation-preserving diffeomorphism, then .
- Disjoint union: if with having measure zero, then .
5.4. Integration over submanifolds
More generally, one integrates a -form over an oriented -dimensional submanifold :
where is the inclusion and is the pullback (restriction) to . This is the integral of a -form over a -manifold, reduced to the case above.
A 1-form on integrates over an oriented curve as
This is the classical line integral. The form is the object that, when given the velocity tangent vector , returns the integrand at time .
A 2-form on integrates over an oriented surface with parametrisation as the classical flux integral . The wedge product encodes the cross product; the orientation determines the sign.
6. The Riemannian volume form
A differential form is the natural object to integrate. But in applications one often wants to integrate an ordinary function (a density, a charge distribution, a probability). A Riemannian metric provides the missing ingredient: a canonical volume -form that converts functions into -forms.
6.1. The metric volume form
Let be an oriented Riemannian -manifold. In any positively oriented local chart with coordinates , the Riemannian volume form is
where is the metric tensor in local coordinates.
The form is independent of the choice of positively oriented chart.
Proof
Under a change of positively oriented coordinates with positive Jacobian , the metric transforms as , so . Thus . Meanwhile . The product gives , so is the same -form in both charts.
The integral of a function against the Riemannian measure is
computed via a partition of unity as in §5.3.
The standard round metric on in spherical coordinates , where , , and is the polar angle from the north pole, gives metric components , , . Thus on and
The flat metric has , so and .
Act II: de Rham Cohomology
7. Period integrals and closed forms
Part III defined closed and exact forms and hinted that their ratio, , is a topological invariant. Now that we can integrate, we can make this precise: closed forms integrate to give periods, which are real numbers that depend only on the cohomology class of the form. Periods are the bridge between the analytic definition of de Rham cohomology and its topological content.
7.1. Periods over cycles
Let be a closed form and a compact oriented -dimensional submanifold of without boundary (a -cycle). The period of over is
If and are cohomologous closed forms, then for every cycle . Similarly, if and differ by a boundary (i.e., for some -chain ), then .
Proof
For the first claim: , since (Z has no boundary). For the second claim: , since is closed. (Both applications use the Stokes theorem, proved in Part V; we use them here as motivation.)
The conclusion is that periods define a bilinear pairing
The de Rham theorem (Part III, §10.2) implies this pairing is non-degenerate: cohomology classes are completely determined by their periods against all cycles.
7.2. The winding number as a period
The angular 1-form on ,
is closed (as computed in Part III). Its period over the unit circle (parametrised as ) is
For an arbitrary closed loop (with ), the winding number around the origin is
Since is not exact, the period can be nonzero. The winding number measures how many times wraps around the origin; it is always an integer because changes by integer multiples of around a closed loop.
The interactive figure below lets you drag four control points to deform a closed Catmull-Rom loop around the origin. The winding number is computed in real time as the period .
The winding number [eq:winding-num] is the generator of : any closed 1-form on is of the form for some and , and is exactly the period over the unit circle. The integer-valuedness reflects that (equation [eq:angular-form]) generates the -lattice inside corresponding to singular cohomology .
8. The Mayer-Vietoris sequence
Computing de Rham cohomology directly from the definition (quotient of closed forms by exact forms) is usually intractable. The Mayer-Vietoris sequence is the main inductive tool: it expresses in terms of , , and when .
8.1. The restriction maps
The direct sum of two vector spaces is their Cartesian product made into a vector space by componentwise addition and scalar multiplication. Concretely, if and , elements of are column vectors with an upper block from and a lower block from :
A linear map is captured by a block row . For infinite-dimensional spaces such as and , the direct sum carries the same componentwise structure: a pair with , .
Let with open. The inclusions and and the inclusions , induce restriction maps on forms:
where restricts to each open set, and takes the difference of the two restrictions to the overlap. This sequence is exact (a partition of unity argument proves surjectivity of ).
8.2. The long exact sequence
Let with open. There is a long exact sequence
The connecting homomorphism is defined as follows: given a closed form representing a class, write for forms and (which exist by the short exact sequence above). Then on , so they glue to a global closed -form on (with ). Set .
Proof
The short exact sequence [eq:mv-ses] of cochain complexes (each map commuting with ) induces a long exact sequence in cohomology by the standard algebraic snake lemma. The connecting homomorphism is exactly the one described above. Exactness of the short sequence at each degree uses a partition of unity subordinate to for the surjectivity of .
8.3. Computation:
Decompose as the union of two open arcs:
where and are the north and south poles. Each of and is diffeomorphic to (contractible), so and for . The intersection has two connected components (two open arcs), so .
Applying [eq:mv-les] in low degrees:
Substituting the known groups: . The map is (restriction of constants to each arc). Its kernel is (the image of the map from ) and its image has rank 1 (the diagonal). Thus has dimension 1, and by exactness , giving .
Result: , , for .
The Mayer-Vietoris decomposition of (and the dependence of on the gap size) is illustrated in the interactive figure below.
The key algebraic step is that has two components when the gaps at and are small. The two components of are forced by the topology of : every decomposition of into two contractible open sets produces a disconnected intersection. The rank-1 cokernel of then forces .
8.4. Computation:
Decompose as and , both diffeomorphic to (via stereographic projection). The intersection has the homotopy type of a circle, so and .
The Mayer-Vietoris sequence in degree 1:
Since and are contractible, . The sequence becomes , giving .
Result: , .
8.5. Computation:
The torus can be handled by Mayer-Vietoris applied to where and are open annular strips. Alternatively, the Künneth formula for de Rham cohomology gives the cohomology of a product directly.
The binary direct sum generalises to an iterated direct sum indexed by a condition on degrees. Writing means: take one copy of for each pair of non-negative integers with , and sum them all. For instance, with and spaces indexed by :
Each summand contributes a block: if and , the full direct sum has dimension . This is the graded analogue of how polynomial multiplication combines terms of matching total degree.
For smooth manifolds and ,
Applying this to with :
The generators of are the two angular 1-forms and (one for each circle factor). The generator of is the area form .
9. Computing cohomology
With Mayer-Vietoris and Künneth in hand, we can compute the cohomology of a wide range of spaces. We organise by example.
9.1. Spheres
Proof by induction via Mayer-Vietoris
Base case (): proved in §8.3 above.
Inductive step: Assume the result for . Decompose with and , so (contractible) and (an equatorial sphere, of dimension ). The Mayer-Vietoris sequence in degree reads:
So is an isomorphism for .
By induction, and for , giving and for . The and cases are handled separately by the degree-0 part of MV (connectedness) and the degree-1 part (which gives for since and the difference map has rank 1, kernel , cokernel 0, so ).
9.2. Tori
By the Künneth formula applied inductively to ,
The generators of are the wedge products for . The full cohomology ring is the exterior algebra:
which is exactly the Part II algebra at the level of global cohomology. The Betti numbers are the binomial coefficients: .
The interactive figure shows the three harmonic forms on :
9.3. Real projective spaces
With real coefficients:
Proof sketch
The antipodal map is a covering map with deck group . On , the antipodal map acts by on the generator of (since is orientation-preserving iff is odd) and by on . Forms on correspond to -invariant forms on . The generator of is -invariant iff , i.e. is odd. All other cohomology vanishes by the same argument (no invariant representatives).
In particular, with real coefficients (the torsion in integral cohomology disappears upon tensoring with ).
9.4. The Möbius band
The Möbius band deformation-retracts onto its core circle (the centre line). Therefore :
The generator of is the pullback of along the retraction. With real coefficients, the de Rham cohomology sees only the homotopy type: and the orientable annulus share the same cohomology groups, even though they are distinct as smooth manifolds (one orientable, one not).
9.5. The Klein bottle
The Klein bottle is a compact non-orientable surface without boundary. It can be constructed from two Möbius bands by gluing their boundaries. A Mayer-Vietoris argument gives:
The vanishing of reflects non-orientability: a compact manifold without boundary has iff it is oriented (which fails for ). The torsion in integral cohomology () disappears when tensoring with , leaving only in degree 1.
9.6. Euler characteristic
The Euler characteristic of a compact manifold is
Examples:
| Manifold | ||||
|---|---|---|---|---|
| 1 | 0 | 1 | 2 | |
| 1 | 2 | 1 | 0 | |
| 1 | 0 | 0 | 1 | |
| Klein bottle | 1 | 1 | 0 | 0 |
| Genus- surface | 1 | 1 |
The Euler characteristic satisfies (from Künneth) and (from Mayer-Vietoris).
10. Poincaré duality
For a compact oriented manifold there is a deep symmetry between cohomology in complementary degrees: and are dual to each other, paired by integration.
10.1. The Poincaré pairing
Let be a compact oriented smooth -manifold without boundary. The bilinear pairing
is non-degenerate. Consequently , and since all groups are finite-dimensional, .
Proof sketch
That the pairing is well-defined (independent of representative) follows from Stokes' theorem (Part V): if , then , using (closedness) and on a closed manifold (Stokes).
Non-degeneracy is proved using Hodge theory: each cohomology class has a unique harmonic representative (a form in the kernel of the Laplacian ), and the Hodge star (Part II) maps harmonic -forms to harmonic -forms. For a harmonic , taking gives , proving non-degeneracy.
If is a compact oriented -manifold, then for all .
This is visible in all the examples: ; ; the Euler characteristic of a compact oriented odd-dimensional manifold is 0 (since and gives cancellation for odd).
10.2. Examples
. The pairing [eq:poincare-pair] for , reduces to (product of a function and a 2-form). Non-degeneracy: take and to get .
with generators and . The pairing [eq:poincare-pair] gives and (since ). In the basis , the pairing matrix is (skew-symmetric, as expected for a 2-manifold: since ). This matrix has determinant 1, confirming non-degeneracy.
Poincaré duality with real coefficients fails for non-orientable manifolds. For : but , so . The correct statement for non-orientable manifolds requires twisted (local system) coefficients; this is outside our scope here.
11. Looking ahead: Stokes' theorem
The machinery assembled across Parts I-IV now stands complete. Parts I-II provided the algebraic and analytic foundations. Part III gave the exterior derivative. Part IV gave integration and, as a payoff, the de Rham cohomology of a variety of spaces.
Part V is the culmination: the generalised Stokes theorem
valid for any compact oriented -manifold with boundary and any -form . Equation [eq:baby-stokes] is the case of [eq:stokes-gen] with ; the two are the same identity at different dimensions. Specialised further:
- , a smooth function: the fundamental theorem (Theorem [ftc]).
- , a 1-form: Green's theorem.
- , a 2-form on a region in : the divergence theorem.
- , a 1-form on an oriented surface: the classical Stokes theorem for surface integrals.
Part V will prove the theorem, recover all classical cases, and use it to give a clean proof that the de Rham cohomology is indeed a topological invariant: a manifold's topology, not its specific smooth structure, is what determines . The series ends with the identification of the period pairing as the natural map
and the statement that this pairing is the de Rham isomorphism: the analytic and topological descriptions of the same cohomology.
References
- R. Bott and L.W. Tu, Differential Forms in Algebraic Topology, Springer, 1982. The primary reference for de Rham cohomology, Mayer-Vietoris, and Poincaré duality. DOI
- J.M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. Chapters 15-17 (integration, Stokes). DOI
- L.W. Tu, An Introduction to Manifolds, 2nd ed., Springer, 2011. Chapters 5-8 (integration and de Rham theory). DOI
- G.E. Bredon, Topology and Geometry, Springer, 1993. Chapter V (cohomology). DOI
- M. Spivak, Calculus on Manifolds, W.A. Benjamin, 1965. The classic short treatment of integration on manifolds. DOI
- W. Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, 1976. Chapter 10 (classical change-of-variables).