Differential forms provide a modern, coordinate-free framework for calculus on manifolds, unifying classical vector analysis with the language of algebraic topology. Rather than working with gradient, curl, and divergence as separate operations on vector fields, this course develops a single geometric object—the differential form—whose behavior under integration, differentiation, and pullback captures all of multivariable calculus in a unified system. The course explores how forms encode information about smooth spaces and reveals deep connections between the local analytic properties of a manifold (captured by the [exterior derivative](/theorems/1525)) and its global topological structure.

The chapters progress systematically from foundations to applications. We begin with exterior algebra, the algebraic machinery underlying forms, then introduce differential forms concretely on open subsets of ℝⁿ where calculus intuition is strongest. The [exterior derivative](/theorems/1525) emerges as the natural generalization of classical differentiation, and the pullback operation shows how forms behave under smooth maps. Once these tools are mastered, we extend the theory to smooth manifolds—abstract curved spaces where these ideas become essential—and develop integration via orientation and the generalised Stokes theorem. The Poincaré lemma establishes when closed forms are exact, leading naturally to de Rham cohomology: a way of extracting topological invariants from the algebra of forms.

De Rham's theorem, the course's central result, reveals that the cohomology computed from differential forms coincides exactly with singular cohomology, a fundamental topological invariant. This identification shows that smooth and purely topological information are intimately related, and the final chapters explore how this perspective illuminates problems across differential geometry, physics, and topology. Throughout, the framework develops the conceptual tools that professional mathematicians and physicists use daily to understand smooth manifolds.

# Introduction

This opening chapter sets the perspective for the course. Differential forms give a single language for line integrals, surface flux, volume integration, change of variables, and the boundary terms that appear in Stokes-type theorems. The first aim is to replace several coordinate-dependent vector calculus constructions by one functorial object. The second aim is to understand why the equation $d\omega = 0$ carries topological information about the space on which $\omega$ lives.

The course starts from linear algebra on finite-dimensional real vector spaces and then moves to smooth manifolds. Smooth manifolds will be used as spaces that locally look like open subsets of $\mathbb R^n$, so the early chapters develop every construction first on open sets. Later chapters globalise these constructions by checking that they behave correctly under change of coordinates.

## Why Forms Replace Vector Calculus Notation

What goes wrong if line integrals, surface integrals, and volume integrals are treated as unrelated operations? The formulas of vector calculus depend heavily on the coordinates chosen to write them down. A line integral such as $P\,dx + Q\,dy$ does not transform like a vector field; it transforms by pullback along parametrisations. Differential forms are designed so that the integrand already contains the transformation rule needed for integration.

[definition: Differential Form On An Open Set] Let $U \subset \mathbb R^n$ be open. A differential $k$-form on $U$ is a smooth assignment $\omega$ sending each point $x \in U$ to an alternating $k$-[linear map](/page/Linear%20Map) \begin{align*} \omega_x : (\mathbb R^n)^k \to \mathbb R. \end{align*} The [vector space](/page/Vector%20Space) of smooth $k$-forms on $U$ is denoted $\Omega^k(U)$. For $k=0$, set $\Omega^0(U)=C^\infty(U)$. [/definition]

The word alternating means that the value changes sign when two input vectors are exchanged and is zero when two input vectors are equal. Thus a $1$-form eats one tangent vector, a $2$-form eats an ordered pair of tangent vectors, and a top-degree $n$-form eats an ordered basis of $\mathbb R^n$.

[example: Line Integral As A One Form] Let $U\subset \mathbb R^2$ be open, and let \begin{align*} \omega=P\,dx+Q\,dy\in\Omega^1(U), \end{align*} where $P,Q\in C^\infty(U)$. Let $\gamma:[a,b]\to U$ be a smooth curve, and write \begin{align*} \gamma(t)=(\gamma_1(t),\gamma_2(t)),\qquad \dot\gamma(t)=(\dot\gamma_1(t),\dot\gamma_2(t)). \end{align*} For $p\in U$ and $v=(v_1,v_2)\in\mathbb R^2$, the coordinate covectors satisfy \begin{align*} dx_p(v)=v_1,\qquad dy_p(v)=v_2. \end{align*} The integral of $\omega$ along $\gamma$ is \begin{align*} \int_\gamma\omega =\int_a^b\left(P(\gamma(t))\dot\gamma_1(t)+Q(\gamma(t))\dot\gamma_2(t)\right)\,dt. \end{align*} By the definition of integrating a $1$-form over a parametrised curve, \begin{align*} \int_\gamma\omega =\int_a^b \omega_{\gamma(t)}(\dot\gamma(t))\,dt. \end{align*} For each $t\in[a,b]$, evaluate the covector $\omega_{\gamma(t)}$ on the velocity vector: \begin{align*} \omega_{\gamma(t)}(\dot\gamma(t)) &=\left(P(\gamma(t))\,dx_{\gamma(t)}+Q(\gamma(t))\,dy_{\gamma(t)}\right)(\dot\gamma(t)) \\ &=P(\gamma(t))\,dx_{\gamma(t)}(\dot\gamma(t)) +Q(\gamma(t))\,dy_{\gamma(t)}(\dot\gamma(t)) \\ &=P(\gamma(t))\,dx_{\gamma(t)}(\dot\gamma_1(t),\dot\gamma_2(t)) +Q(\gamma(t))\,dy_{\gamma(t)}(\dot\gamma_1(t),\dot\gamma_2(t)) \\ &=P(\gamma(t))\dot\gamma_1(t)+Q(\gamma(t))\dot\gamma_2(t). \end{align*} Substituting this pointwise identity into the defining integral gives \begin{align*} \int_\gamma\omega &=\int_a^b \omega_{\gamma(t)}(\dot\gamma(t))\,dt \\ &=\int_a^b\left(P(\gamma(t))\dot\gamma_1(t)+Q(\gamma(t))\dot\gamma_2(t)\right)\,dt. \end{align*} The parametrised curve contributes the velocity vector $\dot\gamma(t)$, and the $1$-form contributes the covector $P(\gamma(t))\,dx_{\gamma(t)}+Q(\gamma(t))\,dy_{\gamma(t)}$. Feeding the velocity into this covector produces exactly the coordinate integrand $P(\gamma(t))\dot\gamma_1(t)+Q(\gamma(t))\dot\gamma_2(t)$, so the ordinary planar line integral is naturally the integral of a $1$-form. [/example]

Coordinate notation is useful because it gives a basis for all forms on an open subset of Euclidean space. If $x=(x_1,\dots,x_n)$ are the standard coordinates, then $dx_i$ denotes the $i$-th coordinate covector.

[example: Coordinate Basis For Forms] Let $U\subset \mathbb R^n$ be open, and let \begin{align*} \omega\in\Omega^k(U). \end{align*} Write $e_1,\dots,e_n$ for the standard basis of $\mathbb R^n$, and write $dx_1,\dots,dx_n$ for the dual coordinate covectors, so \begin{align*} dx_i(e_j)= \begin{cases} 1,& i=j,\\ 0,& i\ne j. \end{cases} \end{align*} For a strictly increasing multi-index $I=(i_1,\dots,i_k)$, set \begin{align*} dx_I:=dx_{i_1}\wedge\cdots\wedge dx_{i_k}. \end{align*} There are unique smooth functions $a_I\in C^\infty(U)$ such that \begin{align*} \omega=\sum_{1\le i_1<\cdots<i_k\le n} a_{i_1\cdots i_k}\,dx_{i_1}\wedge\cdots\wedge dx_{i_k}. \end{align*} For $n=3$ and $k=2$, this becomes \begin{align*} \omega=A\,dy\wedge dz+B\,dz\wedge dx+C\,dx\wedge dy. \end{align*} Fix $x\in U$. Since $\omega_x$ is an alternating $k$-[linear map](/page/Linear%20Map), its values on arbitrary vectors are determined by its values on ordered $k$-tuples of basis vectors. Define \begin{align*} a_{i_1\cdots i_k}(x):=\omega_x(e_{i_1},\dots,e_{i_k}) \end{align*} for every $1\le i_1<\cdots<i_k\le n$. Now evaluate the coordinate wedge $dx_I$ on an ordered basis tuple $(e_{j_1},\dots,e_{j_k})$. By the defining determinant formula for the wedge of coordinate covectors, \begin{align*} dx_I(e_{j_1},\dots,e_{j_k}) &=(dx_{i_1}\wedge\cdots\wedge dx_{i_k})(e_{j_1},\dots,e_{j_k})\\ &=\det\left(dx_{i_p}(e_{j_q})\right)_{1\le p,q\le k}. \end{align*} If $(j_1,\dots,j_k)=(i_1,\dots,i_k)$, the matrix is the identity matrix, so \begin{align*} dx_I(e_{i_1},\dots,e_{i_k})=\det(I_k)=1. \end{align*} If some $j_q$ is not one of $i_1,\dots,i_k$, then the $q$-th column of the matrix has all entries $0$, so \begin{align*} dx_I(e_{j_1},\dots,e_{j_k})=0. \end{align*} If $(j_1,\dots,j_k)$ is a permutation of $(i_1,\dots,i_k)$, then the matrix is the corresponding permutation matrix, so \begin{align*} dx_I(e_{j_1},\dots,e_{j_k})=\operatorname{sgn}(j_1,\dots,j_k). \end{align*} Set \begin{align*} \widetilde\omega_x =\sum_{1\le i_1<\cdots<i_k\le n} a_{i_1\cdots i_k}(x)\,dx_{i_1}\wedge\cdots\wedge dx_{i_k}. \end{align*} For every strictly increasing tuple $J=(j_1,\dots,j_k)$, \begin{align*} \widetilde\omega_x(e_{j_1},\dots,e_{j_k}) &=\sum_I a_I(x)\,dx_I(e_{j_1},\dots,e_{j_k})\\ &=a_J(x)\\ &=\omega_x(e_{j_1},\dots,e_{j_k}). \end{align*} Both $\widetilde\omega_x$ and $\omega_x$ are alternating and $k$-linear, so equality on the ordered basis tuples gives equality on all $k$ input vectors. Hence \begin{align*} \omega_x=\sum_{1\le i_1<\cdots<i_k\le n} a_{i_1\cdots i_k}(x)\,dx_{i_1}\wedge\cdots\wedge dx_{i_k}. \end{align*} Since this holds for every $x\in U$, \begin{align*} \omega=\sum_{1\le i_1<\cdots<i_k\le n} a_{i_1\cdots i_k}\,dx_{i_1}\wedge\cdots\wedge dx_{i_k}. \end{align*} The functions $a_I$ are smooth because they are the coordinate coefficient functions of the smooth form $\omega$. For uniqueness, suppose also that \begin{align*} \omega=\sum_I b_I\,dx_I. \end{align*} Evaluating both sides at $x$ on $(e_{i_1},\dots,e_{i_k})$ gives \begin{align*} \omega_x(e_{i_1},\dots,e_{i_k}) &=\sum_J b_J(x)\,dx_J(e_{i_1},\dots,e_{i_k})\\ &=b_{i_1\cdots i_k}(x). \end{align*} But by the definition of $a_{i_1\cdots i_k}$, \begin{align*} \omega_x(e_{i_1},\dots,e_{i_k})=a_{i_1\cdots i_k}(x). \end{align*} Therefore \begin{align*} a_{i_1\cdots i_k}(x)=b_{i_1\cdots i_k}(x) \end{align*} for every $x\in U$ and every strictly increasing multi-index, so the coefficients are unique. For $n=3$ and $k=2$, the strictly increasing pairs are \begin{align*} (1,2),\qquad (1,3),\qquad (2,3). \end{align*} Thus every $2$-form on $U\subset\mathbb R^3$ has the form \begin{align*} \omega=a_{12}\,dx\wedge dy+a_{13}\,dx\wedge dz+a_{23}\,dy\wedge dz. \end{align*} Using $dx\wedge dz=-dz\wedge dx$, this can be rewritten as \begin{align*} \omega &=a_{23}\,dy\wedge dz-a_{13}\,dz\wedge dx+a_{12}\,dx\wedge dy. \end{align*} If we set \begin{align*} A=a_{23},\qquad B=-a_{13},\qquad C=a_{12}, \end{align*} then \begin{align*} \omega=A\,dy\wedge dz+B\,dz\wedge dx+C\,dx\wedge dy. \end{align*} Coordinate wedge products form the natural coordinate basis for differential forms. The coefficients are not arbitrary decorations: each coefficient is recovered by feeding $\omega_x$ the corresponding ordered coordinate vectors. In dimension $3$, the three basic $2$-forms $dy\wedge dz$, $dz\wedge dx$, and $dx\wedge dy$ record the three oriented coordinate planes. [/example]

## The Operations This Course Builds

Which operations on integrands survive change of coordinates and still remember orientation, dimension, and boundary? The course develops three fundamental constructions: the wedge product, the [exterior derivative](/theorems/1525), and pullback. Together they replace dot products, cross products, gradients, curls, divergences, and Jacobian determinants by coordinate-free operations.

[definition: Wedge Product] Let $U \subset \mathbb R^n$ be open. The wedge product is a bilinear operation \begin{align*} \wedge : \Omega^p(U)\times \Omega^q(U)\to \Omega^{p+q}(U) \end{align*} whose value at each point is the exterior product of alternating covectors. On coordinate $1$-forms it is determined by multilinearity and the relations \begin{align*} dx_i\wedge dx_j=-dx_j\wedge dx_i,\qquad dx_i\wedge dx_i=0. \end{align*} [/definition]

The wedge product encodes signed area, signed volume, and higher-dimensional oriented content. The relation $dx_i\wedge dx_i=0$ is the algebraic expression of the fact that a parallelogram with two equal directions has zero area.

To do calculus with forms, one also needs an operation that differentiates their coefficient functions while raising degree by one. The exterior derivative is designed to generalize gradient, curl, and divergence in a single coordinate-independent formalism.

[definition: Exterior Derivative] Let $U \subset \mathbb R^n$ be open. The [exterior derivative](/theorems/1525) is the family of linear maps \begin{align*} d: \Omega^k(U)\to \Omega^{k+1}(U) \end{align*} defined in coordinates by \begin{align*} d\left(\sum_I a_I\,dx_{i_1}\wedge\cdots\wedge dx_{i_k}\right) =\sum_I\sum_{j=1}^n \frac{\partial a_I}{\partial x_j}\,dx_j\wedge dx_{i_1}\wedge\cdots\wedge dx_{i_k}, \end{align*} where $I=(i_1,\dots,i_k)$ ranges over strictly increasing multi-indices. [/definition]

For a function $f\in C^\infty(U)$, this gives $df=\sum_i (\partial f/\partial x_i)\,dx_i$. For a $1$-form on $\mathbb R^3$, it packages the curl; for a $2$-form on $\mathbb R^3$, it packages the divergence.