Differential form

In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.

For instance, the expression <math>f(x) \, dx</math> is an example of a [[1-form|Template:Math-form]], and can be integrated over an interval <math>[a,b]</math> contained in the domain of <math>f</math>: <math display="block">\int_a^b f(x)\,dx.</math> Similarly, the expression <math display="block">f(x,y,z) \, dx \wedge dy + g(x,y,z) \, dz \wedge dx + h(x,y,z) \, dy \wedge dz</math> is a Template:Math-form that can be integrated over a surface <math>S</math>: <math display="block">\int_S \left(f(x,y,z) \, dx \wedge dy + g(x,y,z) \, dz \wedge dx + h(x,y,z) \, dy \wedge dz\right).</math> The symbol <math>\wedge</math> denotes the exterior product, sometimes called the wedge product, of two differential forms. Likewise, a Template:Math-form <math> f(x,y,z) \, dx \wedge dy \wedge dz</math> represents a volume element that can be integrated over a region of space. In general, a Template:Mvar-form is an object that may be integrated over a Template:Mvar-dimensional manifold, and is homogeneous of degree Template:Mvar in the coordinate differentials <math>dx, dy, \ldots.</math> On an Template:Math-dimensional manifold, a top-dimensional form (Template:Math-form) is called a volume form.

The differential forms form an alternating algebra. This implies that <math>dy \wedge dx = -dx \wedge dy</math> and <math>dx \wedge dx = 0.</math> This alternating property reflects the orientation of the domain of integration.

The exterior derivative is an operation on differential forms that, given a Template:Math-form <math>\varphi</math>, produces a Template:Math-form <math>d\varphi.</math> This operation extends the differential of a function (a function can be considered as a Template:Math-form, and its differential is <math>df(x) = f'(x) \, dx</math>). This allows expressing the fundamental theorem of calculus, the divergence theorem, Green's theorem, and Stokes' theorem as special cases of a single general result, the generalized Stokes theorem.

Differential Template:Math-forms are naturally dual to vector fields on a differentiable manifold, and the pairing between vector fields and Template:Math-forms is extended to arbitrary differential forms by the interior product. The algebra of differential forms along with the exterior derivative defined on it is preserved by the pullback under smooth functions between two manifolds. This feature allows geometrically invariant information to be moved from one space to another via the pullback, provided that the information is expressed in terms of differential forms. As an example, the change of variables formula for integration becomes a simple statement that an integral is preserved under pullback.

History

Differential forms are part of the field of differential geometry, influenced by linear algebra. Although the notion of a differential is quite old, the initial attempt at an algebraic organization of differential forms is usually credited to Élie Cartan with reference to his 1899 paper.<ref>Template:Citation</ref> Some aspects of the exterior algebra of differential forms appears in Hermann Grassmann's 1844 work, Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (The Theory of Linear Extension, a New Branch of Mathematics).

Concept

Differential forms provide an approach to multivariable calculus that is independent of coordinates.

Integration and orientation

A differential Template:Mvar-form can be integrated over an oriented manifold of dimension Template:Mvar. A differential Template:Math-form can be thought of as measuring an infinitesimal oriented length, or 1-dimensional oriented density. A differential Template:Math-form can be thought of as measuring an infinitesimal oriented area, or 2-dimensional oriented density. And so on.

Integration of differential forms is well-defined only on oriented manifolds. An example of a 1-dimensional manifold is an interval Template:Math, and intervals can be given an orientation: they are positively oriented if Template:Math, and negatively oriented otherwise. If Template:Math then the integral of the differential Template:Math-form Template:Math over the interval Template:Math (with its natural positive orientation) is <math display="block">\int_a^b f(x) \,dx</math> which is the negative of the integral of the same differential form over the same interval, when equipped with the opposite orientation. That is: <math display="block">\int_b^a f(x)\,dx = -\int_a^b f(x)\,dx.</math> This gives a geometrical context to the conventions for one-dimensional integrals, that the sign changes when the orientation of the interval is reversed. A standard explanation of this in one-variable integration theory is that, when the limits of integration are in the opposite order (Template:Math), the increment Template:Math is negative in the direction of integration.

More generally, an Template:Mvar-form is an oriented density that can be integrated over an Template:Mvar-dimensional oriented manifold. (For example, a Template:Math-form can be integrated over an oriented curve, a Template:Math-form can be integrated over an oriented surface, etc.) If Template:Mvar is an oriented Template:Mvar-dimensional manifold, and Template:Math is the same manifold with opposite orientation and Template:Mvar is an Template:Mvar-form, then one has: <math display="block">\int_M \omega = - \int_{M'} \omega \,.</math> These conventions correspond to interpreting the integrand as a differential form, integrated over a chain. In measure theory, by contrast, one interprets the integrand as a function Template:Mvar with respect to a measure Template:Mvar and integrates over a subset Template:Mvar, without any notion of orientation; one writes <math display="inline">\int_A f\,d\mu = \int_{[a,b]} f\,d\mu</math> to indicate integration over a subset Template:Mvar. This is a minor distinction in one dimension, but becomes subtler on higher-dimensional manifolds; see below for details.

Making the notion of an oriented density precise, and thus of a differential form, involves the exterior algebra. The differentials of a set of coordinates, Template:Math, ..., Template:Math can be used as a basis for all Template:Math-forms. Each of these represents a covector at each point on the manifold that may be thought of as measuring a small displacement in the corresponding coordinate direction. A general Template:Math-form is a linear combination of these differentials at every point on the manifold: <math display="block">f_1\,dx^1+\cdots+f_n\,dx^n ,</math> where the Template:Math are functions of all the coordinates. A differential Template:Math-form is integrated along an oriented curve as a line integral.

The expressions Template:Math, where Template:Math can be used as a basis at every point on the manifold for all Template:Math-forms. This may be thought of as an infinitesimal oriented square parallel to the Template:Math–Template:Math-plane. A general Template:Math-form is a linear combination of these at every point on the manifold: Template:Nowrap and it is integrated just like a surface integral.

A fundamental operation defined on differential forms is the exterior product (the symbol is the wedge Template:Math). This is similar to the cross product from vector calculus, in that it is an alternating product. For instance, <math display="block">dx^1\wedge dx^2=-dx^2\wedge dx^1</math>

because the square whose first side is Template:Math and second side is Template:Math is to be regarded as having the opposite orientation as the square whose first side is Template:Math and whose second side is Template:Math. This is why we only need to sum over expressions Template:Math, with Template:Math; for example: Template:Math. The exterior product allows higher-degree differential forms to be built out of lower-degree ones, in much the same way that the cross product in vector calculus allows one to compute the area vector of a parallelogram from vectors pointing up the two sides. Alternating also implies that Template:Math, in the same way that the cross product of parallel vectors, whose magnitude is the area of the parallelogram spanned by those vectors, is zero. In higher dimensions, Template:Math if any two of the indices Template:Math, ..., Template:Math are equal, in the same way that the "volume" enclosed by a parallelotope whose edge vectors are linearly dependent is zero.

Multi-index notation

A common notation for the wedge product of elementary Template:Mvar-forms is so called multi-index notation: in an Template:Mvar-dimensional context, for Template:Nowrap we define Template:Nowrap<ref>Template:Cite book</ref> Another useful notation is obtained by defining the set of all strictly increasing multi-indices of length Template:Mvar, in a space of dimension Template:Mvar, denoted Template:Nowrap Then locally (wherever the coordinates apply), <math>\{dx^I\}_{I \in \mathcal{J}_{k,n}}</math> spans the space of differential Template:Mvar-forms in a manifold Template:Mvar of dimension Template:Mvar, when viewed as a module over the ring Template:Math of smooth functions on Template:Mvar. By calculating the size of <math>\mathcal{J}_{k,n}</math> combinatorially, the module of Template:Mvar-forms on an Template:Mvar-dimensional manifold, and in general space of Template:Mvar-covectors on an Template:Mvar-dimensional vector space, is Template:Mvar choose Template:Mvar: Template:Nowrap This also demonstrates that there are no nonzero differential forms of degree greater than the dimension of the underlying manifold.

The exterior derivative

In addition to the exterior product, there is also the exterior derivative operator Template:Math. The exterior derivative of a differential form is a generalization of the differential of a function, in the sense that the exterior derivative of Template:Math is exactly the differential of Template:Mvar. When generalized to higher forms, if Template:Math is a simple Template:Mvar-form, then its exterior derivative Template:Math is a Template:Math-form defined by taking the differential of the coefficient functions: <math display="block">d\omega = \sum_{i=1}^n \frac{\partial f}{\partial x^i} \, dx^i \wedge dx^I.</math> with extension to general Template:Mvar-forms through linearity: if Template:Nowrap a_I \, dx^I \in \Omega^k(M)</math>,}} then its exterior derivative is <math display="block">d\tau = \sum_{I \in \mathcal{J}_{k,n}}\left(\sum_{j=1}^n \frac{\partial a_I}{\partial x^j} \, dx^j\right)\wedge dx^I \in \Omega^{k+1}(M)</math>

In Template:Math, with the Hodge star operator, the exterior derivative corresponds to gradient, curl, and divergence, although this correspondence, like the cross product, does not generalize to higher dimensions, and should be treated with some caution.

The exterior derivative itself applies in an arbitrary finite number of dimensions, and is a flexible and powerful tool with wide application in differential geometry, differential topology, and many areas in physics. Of note, although the above definition of the exterior derivative was defined with respect to local coordinates, it can be defined in an entirely coordinate-free manner, as an antiderivation of degree 1 on the exterior algebra of differential forms. The benefit of this more general approach is that it allows for a natural coordinate-free approach to integrate on manifolds. It also allows for a natural generalization of the fundamental theorem of calculus, called the (generalized) Stokes' theorem, which is a central result in the theory of integration on manifolds.

Differential calculus

Let Template:Mvar be an open set in Template:Math. A differential Template:Math-form ("zero-form") is defined to be a smooth function Template:Mvar on Template:Mvar – the set of which is denoted Template:Math. If Template:Math is any vector in Template:Math, then Template:Math has a directional derivative Template:Math, which is another function on Template:Mvar whose value at a point Template:Math is the rate of change (at Template:Mvar) of Template:Mvar in the Template:Math direction: <math display="block"> (\partial_\mathbf{v} f)(p) = \left. \frac{d}{dt} f(p+t\mathbf{v})\right|_{t=0} .</math> (This notion can be extended pointwise to the case that Template:Math is a vector field on Template:Mvar by evaluating Template:Math at the point Template:Mvar in the definition.)

In particular, if Template:Math is the Template:Mvarth coordinate vector then Template:Math is the partial derivative of Template:Mvar with respect to the Template:Mvarth coordinate vector, i.e., Template:Math, where Template:Math, Template:Math, ..., Template:Math are the coordinate vectors in Template:Mvar. By their very definition, partial derivatives depend upon the choice of coordinates: if new coordinates Template:Math, Template:Math, ..., Template:Math are introduced, then <math display="block">\frac{\partial f}{\partial x^j} = \sum_{i=1}^n\frac{\partial y^i}{\partial x^j}\frac{\partial f}{\partial y^i} .</math>

The first idea leading to differential forms is the observation that Template:Math is a linear function of Template:Math:

<math display="block">\begin{align}

 (\partial_{\mathbf{v} + \mathbf{w}} f)(p) &= (\partial_\mathbf{v} f)(p) + (\partial_\mathbf{w} f)(p) \\
   (\partial_{c \mathbf{v}} f)(p) &= c (\partial_\mathbf{v} f)(p)

\end{align}</math>

for any vectors Template:Math, Template:Math and any real number Template:Mvar. At each point p, this linear map from Template:Math to Template:Math is denoted Template:Math and called the derivative or differential of Template:Mvar at Template:Mvar. Thus Template:Math. Extended over the whole set, the object Template:Math can be viewed as a function that takes a vector field on Template:Mvar, and returns a real-valued function whose value at each point is the derivative along the vector field of the function Template:Mvar. Note that at each Template:Mvar, the differential Template:Math is not a real number, but a linear functional on tangent vectors, and a prototypical example of a differential [[1-form|Template:Math-form]].

Since any vector Template:Math is a linear combination Template:Math of its components, Template:Math is uniquely determined by Template:Math for each Template:Math and each Template:Math, which are just the partial derivatives of Template:Mvar on Template:Mvar. Thus Template:Math provides a way of encoding the partial derivatives of Template:Mvar. It can be decoded by noticing that the coordinates Template:Math, Template:Math, ..., Template:Math are themselves functions on Template:Mvar, and so define differential Template:Math-forms Template:Math, Template:Math, ..., Template:Math. Let Template:Math. Since Template:Math, the Kronecker delta function, it follows that

The meaning of this expression is given by evaluating both sides at an arbitrary point Template:Mvar: on the right hand side, the sum is defined "pointwise", so that <math display="block">df_p = \sum_{i=1}^n \frac{\partial f}{\partial x^i}(p) (dx^i)_p .</math> Applying both sides to Template:Math, the result on each side is the Template:Mvarth partial derivative of Template:Mvar at Template:Mvar. Since Template:Mvar and Template:Mvar were arbitrary, this proves the formula Template:EquationNote.

More generally, for any smooth functions Template:Math and Template:Math on Template:Mvar, we define the differential Template:Math-form Template:Math pointwise by <math display="block">\alpha_p = \sum_i g_i(p) (dh_i)_p</math> for each Template:Math. Any differential Template:Math-form arises this way, and by using Template:EquationNote it follows that any differential Template:Math-form Template:Mvar on Template:Mvar may be expressed in coordinates as <math display="block"> \alpha = \sum_{i=1}^n f_i\, dx^i</math> for some smooth functions Template:Math on Template:Mvar.

The second idea leading to differential forms arises from the following question: given a differential Template:Math-form Template:Mvar on Template:Mvar, when does there exist a function Template:Mvar on Template:Mvar such that Template:Math? The above expansion reduces this question to the search for a function Template:Mvar whose partial derivatives Template:Math are equal to Template:Mvar given functions Template:Math. For Template:Math, such a function does not always exist: any smooth function Template:Mvar satisfies <math display="block"> \frac{\partial^2 f}{\partial x^i \, \partial x^j} = \frac{\partial^2 f}{\partial x^j \, \partial x^i} ,</math> so it will be impossible to find such an Template:Mvar unless <math display="block"> \frac{\partial f_j}{\partial x^i} - \frac{\partial f_i}{\partial x^j} = 0</math> for all Template:Mvar and Template:Mvar.

The skew-symmetry of the left hand side in Template:Mvar and Template:Mvar suggests introducing an antisymmetric product Template:Math on differential Template:Math-forms, the exterior product, so that these equations can be combined into a single condition <math display="block"> \sum_{i,j=1}^n \frac{\partial f_j}{\partial x^i} \, dx^i \wedge dx^j = 0 ,</math> where Template:Math is defined so that: <math display="block"> dx^i \wedge dx^j = - dx^j \wedge dx^i. </math>

This is an example of a differential Template:Math-form. This Template:Math-form is called the exterior derivative Template:Math of Template:Math. It is given by <math display="block"> d\alpha = \sum_{j=1}^n df_j \wedge dx^j = \sum_{i,j=1}^n \frac{\partial f_j}{\partial x^i} \, dx^i \wedge dx^j .</math>

To summarize: Template:Math is a necessary condition for the existence of a function Template:Mvar with Template:Math.

Differential Template:Math-forms, Template:Math-forms, and Template:Math-forms are special cases of differential forms. For each Template:Mvar, there is a space of differential Template:Mvar-forms, which can be expressed in terms of the coordinates as <math display="block"> \sum_{i_1,i_2\ldots i_k=1}^n f_{i_1i_2\ldots i_k} \, dx^{i_1} \wedge dx^{i_2} \wedge\cdots \wedge dx^{i_k}</math> for a collection of functions Template:Math. Antisymmetry, which was already present for Template:Math-forms, makes it possible to restrict the sum to those sets of indices for which Template:Math.

Differential forms can be multiplied together using the exterior product, and for any differential Template:Mvar-form Template:Mvar, there is a differential Template:Math-form Template:Math called the exterior derivative of Template:Mvar.

Differential forms, the exterior product and the exterior derivative are independent of a choice of coordinates. Consequently, they may be defined on any smooth manifold Template:Mvar. One way to do this is cover Template:Mvar with coordinate charts and define a differential Template:Mvar-form on Template:Mvar to be a family of differential Template:Mvar-forms on each chart which agree on the overlaps. However, there are more intrinsic definitions which make the independence of coordinates manifest.

Intrinsic definitions

Let Template:Math be a smooth manifold. A smooth differential form of degree Template:Math is a smooth section of the Template:Mathth exterior power of the cotangent bundle of Template:Math. The set of all differential Template:Math-forms on a manifold Template:Math is a vector space, often denoted <math>\Omega^k(M)</math>.

The definition of a differential form may be restated as follows. At any point <math>p\in M</math>, a Template:Math-form <math>\beta</math> defines an element <math display="block"> \beta_p \in {\textstyle\bigwedge}^k T_p^* M,</math> where <math>T_pM</math> is the tangent space to Template:Math at Template:Math and <math>T^*_p(M)</math> is its dual space. This space is Template:Clarify to the fiber at Template:Math of the dual bundle of the Template:Mathth exterior power of the tangent bundle of Template:Math. That is, <math>\beta</math> is also a linear functional <math display="inline">\beta_p \colon {\textstyle\bigwedge}^k T_pM \to \mathbf{R}</math>, i.e. the dual of the Template:Mathth exterior power is isomorphic to the Template:Mathth exterior power of the dual: <math display="block">{\textstyle\bigwedge}^k T^*_p M \cong \Big({\textstyle\bigwedge}^k T_p M\Big)^*</math>

By the universal property of exterior powers, this is equivalently an alternating multilinear map: <math display="block">\beta_p\colon \bigoplus_{n=1}^k T_p M \to \mathbf{R}.</math> Consequently, a differential Template:Math-form may be evaluated against any Template:Math-tuple of tangent vectors to the same point Template:Math of Template:Math. For example, a differential Template:Math-form Template:Math assigns to each point <math>p\in M</math> a linear functional Template:Math on <math>T_pM</math>. In the presence of an inner product on <math>T_pM</math> (induced by a Riemannian metric on Template:Math), Template:Math may be represented as the inner product with a tangent vector <math>X_p</math>. Differential Template:Math-forms are sometimes called covariant vector fields, covector fields, or "dual vector fields", particularly within physics.

The exterior algebra may be embedded in the tensor algebra by means of the alternation map. The alternation map is defined as a mapping <math display="block">\operatorname{Alt} \colon {\bigotimes}^k T^*M \to {\bigotimes}^k T^*M.</math> For a tensor <math>\tau</math> at a point Template:Math, <math display="block">\operatorname{Alt}(\tau_p)(x_1, \dots, x_k) = \frac{1}{k!}\sum_{\sigma \in S_k} \sgn(\sigma) \tau_p(x_{\sigma(1)}, \dots, x_{\sigma(k)}),</math> where Template:Math is the symmetric group on Template:Math elements. The alternation map is constant on the cosets of the ideal in the tensor algebra generated by the symmetric 2-forms, and therefore descends to an embedding <math display="block">\operatorname{Alt} \colon {\textstyle\bigwedge}^k T^*M \to {\bigotimes}^k T^*M.</math>

This map exhibits <math>\beta</math> as a totally antisymmetric covariant tensor field of rank Template:Math. The differential forms on Template:Math are in one-to-one correspondence with such tensor fields.

Operations

As well as the addition and multiplication by scalar operations which arise from the vector space structure, there are several other standard operations defined on differential forms. The most important operations are the exterior product of two differential forms, the exterior derivative of a single differential form, the interior product of a differential form and a vector field, the Lie derivative of a differential form with respect to a vector field and the covariant derivative of a differential form with respect to a vector field on a manifold with a defined connection.

Exterior product

The exterior product of a Template:Math-form Template:Math and an Template:Math-form Template:Math, denoted Template:Math, is a (Template:Math)-form. At each point Template:Math of the manifold Template:Math, the forms Template:Math and Template:Math are elements of an exterior power of the cotangent space at Template:Math. When the exterior algebra is viewed as a quotient of the tensor algebra, the exterior product corresponds to the tensor product (modulo the equivalence relation defining the exterior algebra).

The antisymmetry inherent in the exterior algebra means that when Template:Math is viewed as a multilinear functional, it is alternating. However, when the exterior algebra is embedded as a subspace of the tensor algebra by means of the alternation map, the tensor product Template:Math is not alternating. There is an explicit formula which describes the exterior product in this situation. The exterior product is <math display="block">\alpha \wedge \beta = \operatorname{Alt}(\alpha \otimes \beta).</math> If the embedding of <math>{\textstyle\bigwedge}^n T^*M</math> into <math>{\bigotimes}^n T^*M</math> is done via the map <math>n!\operatorname{Alt}</math> instead of <math>\operatorname{Alt}</math>, the exterior product is <math display="block">\alpha \wedge \beta = \frac{(k + \ell)!}{k!\ell!}\operatorname{Alt}(\alpha \otimes \beta).</math> This description is useful for explicit computations. For example, if Template:Math, then Template:Math is the Template:Math-form whose value at a point Template:Math is the alternating bilinear form defined by <math display="block"> (\alpha\wedge\beta)_p(v,w)=\alpha_p(v)\beta_p(w) - \alpha_p(w)\beta_p(v)</math> for Template:Math.

The exterior product is bilinear: If Template:Math, Template:Math, and Template:Math are any differential forms, and if Template:Math is any smooth function, then <math display="block">\alpha \wedge (\beta + \gamma) = \alpha \wedge \beta + \alpha \wedge \gamma,</math> <math display="block">\alpha \wedge (f \cdot \beta) = f \cdot (\alpha \wedge \beta).</math>

It is skew commutative (also known as graded commutative), meaning that it satisfies a variant of anticommutativity that depends on the degrees of the forms: if Template:Math is a Template:Math-form and Template:Math is an Template:Math-form, then <math display="block">\alpha \wedge \beta = (-1)^{k\ell} \beta \wedge \alpha .</math>

One also has the graded Leibniz rule:

<math>d(\alpha\wedge\beta)=d\alpha\wedge\beta + (-1)^{k}\alpha\wedge d\beta.</math>

Riemannian manifold

On a Riemannian manifold, or more generally a pseudo-Riemannian manifold, the metric defines a fibre-wise isomorphism of the tangent and cotangent bundles. This makes it possible to convert vector fields to covector fields and vice versa. It also enables the definition of additional operations such as the Hodge star operator <math>\star \colon \Omega^k(M)\ \stackrel{\sim}{\to}\ \Omega^{n-k}(M)</math> and the codifferential <math>\delta\colon \Omega^k(M)\rightarrow \Omega^{k-1}(M)</math>, which has degree Template:Math and is adjoint to the exterior differential Template:Math.

Vector field structures

On a pseudo-Riemannian manifold, Template:Math-forms can be identified with vector fields; vector fields have additional distinct algebraic structures, which are listed here for context and to avoid confusion.

Firstly, each (co)tangent space generates a Clifford algebra, where the product of a (co)vector with itself is given by the value of a quadratic form – in this case, the natural one induced by the metric. This algebra is distinct from the exterior algebra of differential forms, which can be viewed as a Clifford algebra where the quadratic form vanishes (since the exterior product of any vector with itself is zero). Clifford algebras are thus non-anticommutative ("quantum") deformations of the exterior algebra. They are studied in geometric algebra.

Another alternative is to consider vector fields as derivations. The (noncommutative) algebra of differential operators they generate is the Weyl algebra and is a noncommutative ("quantum") deformation of the symmetric algebra in the vector fields.

Exterior differential complex

One important property of the exterior derivative is that Template:Math. This means that the exterior derivative defines a cochain complex: <math display="block">0\ \to\ \Omega^0(M)\ \stackrel{d}{\to}\ \Omega^1(M)\ \stackrel{d}{\to}\ \Omega^2(M)\ \stackrel{d}{\to}\ \Omega^3(M)\ \to\ \cdots \ \to\ \Omega^n(M)\ \to \ 0.</math>

This complex is called the de Rham complex, and its cohomology is by definition the de Rham cohomology of Template:Math. By the Poincaré lemma, the de Rham complex is locally exact except at Template:Math. The kernel at Template:Math is the space of locally constant functions on Template:Math. Therefore, the complex is a resolution of the constant sheaf Template:Math, which in turn implies a form of de Rham's theorem: de Rham cohomology computes the sheaf cohomology of Template:Math.

Pullback

Suppose that Template:Math is smooth. The differential of Template:Math is a smooth map Template:Math between the tangent bundles of Template:Math and Template:Math. This map is also denoted Template:Math and called the pushforward. For any point Template:Math and any tangent vector Template:Math, there is a well-defined pushforward vector Template:Math in Template:Math. However, the same is not true of a vector field. If Template:Math is not injective, say because Template:Math has two or more preimages, then the vector field may determine two or more distinct vectors in Template:Math. If Template:Math is not surjective, then there will be a point Template:Math at which Template:Math does not determine any tangent vector at all. Since a vector field on Template:Math determines, by definition, a unique tangent vector at every point of Template:Math, the pushforward of a vector field does not always exist.

By contrast, it is always possible to pull back a differential form. A differential form on Template:Math may be viewed as a linear functional on each tangent space. Precomposing this functional with the differential Template:Math defines a linear functional on each tangent space of Template:Math and therefore a differential form on Template:Math. The existence of pullbacks is one of the key features of the theory of differential forms. It leads to the existence of pullback maps in other situations, such as pullback homomorphisms in de Rham cohomology.

Formally, let Template:Math be smooth, and let Template:Math be a smooth Template:Math-form on Template:Math. Then there is a differential form Template:Math on Template:Math, called the pullback of Template:Math, which captures the behavior of Template:Math as seen relative to Template:Math. To define the pullback, fix a point Template:Math of Template:Math and tangent vectors Template:Math, ..., Template:Math to Template:Math at Template:Math. The pullback of Template:Math is defined by the formula <math display="block">(f^*\omega)_p(v_1, \ldots, v_k) = \omega_{f(p)}(f_*v_1, \ldots, f_*v_k).</math>

There are several more abstract ways to view this definition. If Template:Math is a Template:Math-form on Template:Math, then it may be viewed as a section of the cotangent bundle Template:Math of Template:Math. Using Template:I sup to denote a dual map, the dual to the differential of Template:Math is Template:Math. The pullback of Template:Math may be defined to be the composite <math display="block">M\ \stackrel{f}{\to}\ N\ \stackrel{\omega}{\to}\ T^*N\ \stackrel{(df)^*}{\longrightarrow}\ T^*M.</math> This is a section of the cotangent bundle of Template:Math and hence a differential Template:Math-form on Template:Math. In full generality, let <math display="inline">\bigwedge^k (df)^*</math> denote the Template:Mathth exterior power of the dual map to the differential. Then the pullback of a Template:Math-form Template:Math is the composite <math display="block">M\ \stackrel{f}{\to}\ N\ \stackrel{\omega}{\to}\ {\textstyle\bigwedge}^k T^*N\ \stackrel{{\bigwedge}^k (df)^*}{\longrightarrow}\ {\textstyle\bigwedge}^k T^*M.</math>

Another abstract way to view the pullback comes from viewing a Template:Math-form Template:Math as a linear functional on tangent spaces. From this point of view, Template:Math is a morphism of vector bundles <math display="block">{\textstyle\bigwedge}^k TN\ \stackrel{\omega}{\to}\ N \times \mathbf{R},</math> where Template:Math is the trivial rank one bundle on Template:Math. The composite map <math display="block">{\textstyle\bigwedge}^k TM\ \stackrel{{\bigwedge}^k df}{\longrightarrow}\ {\textstyle\bigwedge}^k TN\ \stackrel{\omega}{\to}\ N \times \mathbf{R}</math> defines a linear functional on each tangent space of Template:Math, and therefore it factors through the trivial bundle Template:Math. The vector bundle morphism <math display="inline">{\textstyle\bigwedge}^k TM \to M \times \mathbf{R}</math> defined in this way is Template:Math.

Pullback respects all of the basic operations on forms. If Template:Math and Template:Math are forms and Template:Math is a real number, then <math display="block">\begin{align}

            f^*(c\omega) &= c(f^*\omega), \\
      f^*(\omega + \eta) &= f^*\omega + f^*\eta, \\
 f^*(\omega \wedge \eta) &= f^*\omega \wedge f^*\eta, \\
            f^*(d\omega) &= d(f^*\omega).

\end{align}</math>

The pullback of a form can also be written in coordinates. Assume that Template:Math, ..., Template:Math are coordinates on Template:Math, that Template:Math, ..., Template:Math are coordinates on Template:Math, and that these coordinate systems are related by the formulas Template:Math for all Template:Math. Locally on Template:Math, Template:Math can be written as <math display="block">\omega = \sum_{i_1 < \cdots < i_k} \omega_{i_1\cdots i_k} \, dy^{i_1} \wedge \cdots \wedge dy^{i_k},</math> where, for each choice of Template:Math, ..., Template:Math, Template:Math is a real-valued function of Template:Math, ..., Template:Math. Using the linearity of pullback and its compatibility with exterior product, the pullback of Template:Math has the formula <math display="block">f^*\omega = \sum_{i_1 < \cdots < i_k} (\omega_{i_1\cdots i_k}\circ f) \, df_{i_1} \wedge \cdots \wedge df_{i_k}.</math>

Each exterior derivative Template:Math can be expanded in terms of Template:Math, ..., Template:Math. The resulting Template:Math-form can be written using Jacobian matrices: <math display="block"> f^*\omega = \sum_{i_1 < \cdots < i_k} \sum_{j_1 < \cdots < j_k} (\omega_{i_1\cdots i_k}\circ f)\frac{\partial(f_{i_1}, \ldots, f_{i_k})}{\partial(x^{j_1}, \ldots, x^{j_k})} \, dx^{j_1} \wedge \cdots \wedge dx^{j_k}.</math>

Here, <math display="inline>\frac{\partial(f_{i_1}, \ldots, f_{i_k})}{\partial(x^{j_1}, \ldots, x^{j_k})}</math> denotes the determinant of the matrix whose entries are <math display="inline">\frac{\partial f_{i_m}}{\partial x^{j_n}}</math>, <math>1\leq m,n\leq k</math>.

Integration

A differential Template:Math-form can be integrated over an oriented Template:Math-dimensional manifold. When the Template:Math-form is defined on an Template:Math-dimensional manifold with Template:Math, then the Template:Math-form can be integrated over oriented Template:Math-dimensional submanifolds. If Template:Math, integration over oriented 0-dimensional submanifolds is just the summation of the integrand evaluated at points, according to the orientation of those points. Other values of Template:Math correspond to line integrals, surface integrals, volume integrals, and so on. There are several equivalent ways to formally define the integral of a differential form, all of which depend on reducing to the case of Euclidean space.

Integration on Euclidean space

Let Template:Math be an open subset of Template:Math. Give Template:Math its standard orientation and Template:Math the restriction of that orientation. Every smooth Template:Math-form Template:Math on Template:Math has the form <math display="block">\omega = f(x)\,dx^1 \wedge \cdots \wedge dx^n</math> for some smooth function Template:Math. Such a function has an integral in the usual Riemann or Lebesgue sense. This allows us to define the integral of Template:Math to be the integral of Template:Math: <math display="block">\int_U \omega\ \stackrel{\text{def}}{=} \int_U f(x)\,dx^1 \cdots dx^n.</math> Fixing an orientation is necessary for this to be well-defined. The skew-symmetry of differential forms means that the integral of, say, Template:Math must be the negative of the integral of Template:Math. Riemann and Lebesgue integrals cannot see this dependence on the ordering of the coordinates, so they leave the sign of the integral undetermined. The orientation resolves this ambiguity.

Integration over chains

Let Template:Math be an Template:Math-manifold and Template:Math an Template:Math-form on Template:Math. First, assume that there is a parametrization of Template:Math by an open subset of Euclidean space. That is, assume that there exists a diffeomorphism <math display="block">\varphi \colon D \to M</math> where Template:Math. Give Template:Math the orientation induced by Template:Math. Then Template:Harv defines the integral of Template:Math over Template:Math to be the integral of Template:Math over Template:Math. In coordinates, this has the following expression. Fix an embedding of Template:Math in Template:Math with coordinates Template:Math. Then <math display="block">\omega = \sum_{i_1 < \cdots < i_n} a_{i_1,\ldots,i_n}({\mathbf x})\,dx^{i_1} \wedge \cdots \wedge dx^{i_n}.</math> Suppose that Template:Math is defined by <math display="block">\varphi({\mathbf u}) = (x^1({\mathbf u}),\ldots,x^I({\mathbf u})).</math> Then the integral may be written in coordinates as <math display="block">\int_M \omega = \int_D \sum_{i_1 < \cdots < i_n} a_{i_1,\ldots,i_n}(\varphi({\mathbf u})) \frac{\partial(x^{i_1},\ldots,x^{i_n})}{\partial(u^{1},\dots,u^{n})}\,du^1 \cdots du^n,</math> where <math display="block">\frac{\partial(x^{i_1},\ldots,x^{i_n})}{\partial(u^{1},\ldots,u^{n})}</math> is the determinant of the Jacobian. The Jacobian exists because Template:Math is differentiable.

In general, an Template:Math-manifold cannot be parametrized by an open subset of Template:Math. But such a parametrization is always possible locally, so it is possible to define integrals over arbitrary manifolds by defining them as sums of integrals over collections of local parametrizations. Moreover, it is also possible to define parametrizations of Template:Math-dimensional subsets for Template:Math, and this makes it possible to define integrals of Template:Math-forms. To make this precise, it is convenient to fix a standard domain Template:Math in Template:Math, usually a cube or a simplex. A Template:Math-chain is a formal sum of smooth embeddings Template:Math. That is, it is a collection of smooth embeddings, each of which is assigned an integer multiplicity. Each smooth embedding determines a Template:Math-dimensional submanifold of Template:Math. If the chain is <math display="block">c = \sum_{i=1}^r m_i \varphi_i,</math> then the integral of a Template:Math-form Template:Math over Template:Math is defined to be the sum of the integrals over the terms of Template:Math: <math display="block">\int_c \omega = \sum_{i=1}^r m_i \int_D \varphi_i^*\omega.</math>

This approach to defining integration does not assign a direct meaning to integration over the whole manifold Template:Math. However, it is still possible to assign such a meaning indirectly because every smooth manifold may be smoothly triangulated in an essentially unique way, and the integral over Template:Math may be defined to be the integral over the chain determined by a triangulation.

Integration using partitions of unity

There is another approach, expounded in Template:Harv, which does directly assign a meaning to integration over Template:Math, but this approach requires fixing an orientation of Template:Math. The integral of an Template:Math-form Template:Math on an Template:Math-dimensional manifold is defined by working in charts. Suppose first that Template:Math is supported on a single positively oriented chart. On this chart, it may be pulled back to an Template:Math-form on an open subset of Template:Math. Here, the form has a well-defined Riemann or Lebesgue integral as before. The change of variables formula and the assumption that the chart is positively oriented together ensure that the integral of Template:Math is independent of the chosen chart. In the general case, use a partition of unity to write Template:Math as a sum of Template:Math-forms, each of which is supported in a single positively oriented chart, and define the integral of Template:Math to be the sum of the integrals of each term in the partition of unity.

It is also possible to integrate Template:Math-forms on oriented Template:Math-dimensional submanifolds using this more intrinsic approach. The form is pulled back to the submanifold, where the integral is defined using charts as before. For example, given a path Template:Math, integrating a Template:Math-form on the path is simply pulling back the form to a form Template:Math on Template:Math, and this integral is the integral of the function Template:Math on the interval.

Integration along fibers

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

Fubini's theorem states that the integral over a set that is a product may be computed as an iterated integral over the two factors in the product. This suggests that the integral of a differential form over a product ought to be computable as an iterated integral as well. The geometric flexibility of differential forms ensures that this is possible not just for products, but in more general situations as well. Under some hypotheses, it is possible to integrate along the fibers of a smooth map, and the analog of Fubini's theorem is the case where this map is the projection from a product to one of its factors.

Because integrating a differential form over a submanifold requires fixing an orientation, a prerequisite to integration along fibers is the existence of a well-defined orientation on those fibers. Let Template:Math and Template:Math be two orientable manifolds of pure dimensions Template:Math and Template:Math, respectively. Suppose that Template:Math is a surjective submersion. This implies that each fiber Template:Math is Template:Math-dimensional and that, around each point of Template:Math, there is a chart on which Template:Math looks like the projection from a product onto one of its factors. Fix Template:Math and set Template:Math. Suppose that <math display="block">\begin{align} \omega_x &\in {\textstyle\bigwedge}^m T_x^*M, \\[2pt] \eta_y &\in {\textstyle\bigwedge}^n T_y^*N, \end{align}</math> and that Template:Math does not vanish. Following Template:Harv, there is a unique <math display="block">\sigma_x \in {\textstyle\bigwedge}^{m-n} T_x^*(f^{-1}(y))</math> which may be thought of as the fibral part of Template:Math with respect to Template:Math. More precisely, define Template:Math to be the inclusion. Then Template:Math is defined by the property that <math display="block">\omega_x = (f^*\eta_y)_x \wedge \sigma'_x \in {\textstyle\bigwedge}^m T_x^*M,</math> where <math display="block">\sigma'_x \in {\textstyle\bigwedge}^{m-n} T_x^*M</math> is any Template:Math-covector for which <math display="block">\sigma_x = j^*\sigma'_x.</math> The form Template:Math may also be notated Template:Math.

Moreover, for fixed Template:Math, Template:Math varies smoothly with respect to Template:Math. That is, suppose that <math display="block">\omega \colon f^{-1}(y) \to T^*M</math> is a smooth section of the projection map; we say that Template:Math is a smooth differential Template:Math-form on Template:Math along Template:Math. Then there is a smooth differential Template:Math-form Template:Math on Template:Math such that, at each Template:Math, <math display="block">\sigma_x = \omega_x / \eta_y.</math> This form is denoted Template:Math. The same construction works if Template:Math is an Template:Math-form in a neighborhood of the fiber, and the same notation is used. A consequence is that each fiber Template:Math is orientable. In particular, a choice of orientation forms on Template:Math and Template:Math defines an orientation of every fiber of Template:Math.

The analog of Fubini's theorem is as follows. As before, Template:Math and Template:Math are two orientable manifolds of pure dimensions Template:Math and Template:Math, and Template:Math is a surjective submersion. Fix orientations of Template:Math and Template:Math, and give each fiber of Template:Math the induced orientation. Let Template:Math be an Template:Math-form on Template:Math, and let Template:Math be an Template:Math-form on Template:Math that is almost everywhere positive with respect to the orientation of Template:Math. Then, for almost every Template:Math, the form Template:Math is a well-defined integrable Template:Math form on Template:Math. Moreover, there is an integrable Template:Math-form on Template:Math defined by <math display="block">y \mapsto \bigg(\int_{f^{-1}(y)} \omega / \eta_y\bigg)\,\eta_y.</math> Denote this form by <math display="block">\bigg(\int_{f^{-1}(y)} \omega / \eta\bigg)\,\eta.</math> Then Template:Harv proves the generalized Fubini formula <math display="block">\int_M \omega = \int_N \bigg(\int_{f^{-1}(y)} \omega / \eta\bigg)\,\eta.</math>

It is also possible to integrate forms of other degrees along the fibers of a submersion. Assume the same hypotheses as before, and let Template:Math be a compactly supported Template:Math-form on Template:Math. Then there is a Template:Math-form Template:Math on Template:Math which is the result of integrating Template:Math along the fibers of Template:Math. The form Template:Math is defined by specifying, at each Template:Math, how Template:Math pairs with each Template:Math-vector Template:Math at Template:Math, and the value of that pairing is an integral over Template:Math that depends only on Template:Math, Template:Math, and the orientations of Template:Math and Template:Math. More precisely, at each Template:Math, there is an isomorphism <math display="block">{\textstyle\bigwedge}^k T_yN \to {\textstyle\bigwedge}^{n-k} T_y^*N</math> defined by the interior product <math display="block">\mathbf{v} \mapsto \mathbf{v}\,\lrcorner\,\zeta_y,</math> for any choice of volume form Template:Math in the orientation of Template:Math. If Template:Math, then a Template:Math-vector Template:Math at Template:Math determines an Template:Math-covector at Template:Math by pullback: <math display="block">f^*(\mathbf{v}\,\lrcorner\,\zeta_y) \in {\textstyle\bigwedge}^{n-k} T_x^*M.</math> Each of these covectors has an exterior product against Template:Math, so there is an Template:Math-form Template:Math on Template:Math along Template:Math defined by <math display="block">(\beta_{\mathbf{v}})_x = \left(\alpha_x \wedge f^*(\mathbf{v}\,\lrcorner\,\zeta_y)\right) \big/ \zeta_y \in {\textstyle\bigwedge}^{m-n} T_x^*M.</math> This form depends on the orientation of Template:Math but not the choice of Template:Math. Then the Template:Math-form Template:Math is uniquely defined by the property <math display="block">\langle\gamma_y, \mathbf{v}\rangle = \int_{f^{-1}(y)} \beta_{\mathbf{v}},</math> and Template:Math is smooth Template:Harv. This form also denoted Template:Math and called the integral of Template:Math along the fibers of Template:Math. Integration along fibers is important for the construction of Gysin maps in de Rham cohomology.

Integration along fibers satisfies the projection formula Template:Harv. If Template:Math is any Template:Math-form on Template:Math, then <math display="block">\alpha^\flat \wedge \lambda = (\alpha \wedge f^*\lambda)^\flat.</math>

Stokes's theorem

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The fundamental relationship between the exterior derivative and integration is given by the Stokes' theorem: If Template:Math is an (Template:Math)-form with compact support on Template:Math and Template:Math denotes the boundary of Template:Math with its induced orientation, then <math display="block">\int_M d\omega = \int_{\partial M} \omega.</math>

A key consequence of this is that "the integral of a closed form over homologous chains is equal": If Template:Math is a closed Template:Math-form and Template:Math and Template:Math are Template:Math-chains that are homologous (such that Template:Math is the boundary of a Template:Math-chain Template:Math), then <math>\textstyle{\int_M \omega = \int_N \omega}</math>, since the difference is the integral <math>\textstyle\int_W d\omega = \int_W 0 = 0</math>.

For example, if Template:Math is the derivative of a potential function on the plane or Template:Math, then the integral of Template:Math over a path from Template:Math to Template:Math does not depend on the choice of path (the integral is Template:Math), since different paths with given endpoints are homotopic, hence homologous (a weaker condition). This case is called the gradient theorem, and generalizes the fundamental theorem of calculus. This path independence is very useful in contour integration.

This theorem also underlies the duality between de Rham cohomology and the homology of chains.

Relation with measures

On a general differentiable manifold (without additional structure), differential forms cannot be integrated over subsets of the manifold; this distinction is key to the distinction between differential forms, which are integrated over chains or oriented submanifolds, and measures, which are integrated over subsets. The simplest example is attempting to integrate the Template:Math-form Template:Math over the interval Template:Math. Assuming the usual distance (and thus measure) on the real line, this integral is either Template:Math or Template:Math, depending on orientation: Template:Nowrap while Template:Nowrap By contrast, the integral of the measure Template:Math on the interval is unambiguously Template:Math (i.e. the integral of the constant function Template:Math with respect to this measure is Template:Math). Similarly, under a change of coordinates a differential Template:Math-form changes by the Jacobian determinant Template:Math, while a measure changes by the absolute value of the Jacobian determinant, Template:Math, which further reflects the issue of orientation. For example, under the map Template:Math on the line, the differential form Template:Math pulls back to Template:Math; orientation has reversed; while the Lebesgue measure, which here we denote Template:Math, pulls back to Template:Math; it does not change.

In the presence of the additional data of an orientation, it is possible to integrate Template:Math-forms (top-dimensional forms) over the entire manifold or over compact subsets; integration over the entire manifold corresponds to integrating the form over the fundamental class of the manifold, Template:Math. Formally, in the presence of an orientation, one may identify Template:Math-forms with densities on a manifold; densities in turn define a measure, and thus can be integrated Template:Harv.

On an orientable but not oriented manifold, there are two choices of orientation; either choice allows one to integrate Template:Math-forms over compact subsets, with the two choices differing by a sign. On a non-orientable manifold, Template:Math-forms and densities cannot be identified —notably, any top-dimensional form must vanish somewhere (there are no volume forms on non-orientable manifolds), but there are nowhere-vanishing densities— thus while one can integrate densities over compact subsets, one cannot integrate Template:Math-forms. One can instead identify densities with top-dimensional pseudoforms.

Even in the presence of an orientation, there is in general no meaningful way to integrate Template:Math-forms over subsets for Template:Math because there is no consistent way to use the ambient orientation to orient Template:Math-dimensional subsets. Geometrically, a Template:Math-dimensional subset can be turned around in place, yielding the same subset with the opposite orientation; for example, the horizontal axis in a plane can be rotated by 180 degrees. Compare the Gram determinant of a set of Template:Math vectors in an Template:Math-dimensional space, which, unlike the determinant of Template:Math vectors, is always positive, corresponding to a squared number. An orientation of a Template:Math-submanifold is therefore extra data not derivable from the ambient manifold.

On a Riemannian manifold, one may define a Template:Math-dimensional Hausdorff measure for any Template:Math (integer or real), which may be integrated over Template:Math-dimensional subsets of the manifold. A function times this Hausdorff measure can then be integrated over Template:Math-dimensional subsets, providing a measure-theoretic analog to integration of Template:Math-forms. The Template:Math-dimensional Hausdorff measure yields a density, as above.

Currents

The differential form analog of a distribution or generalized function is called a current. The space of Template:Math-currents on Template:Math is the dual space to an appropriate space of differential Template:Math-forms. Currents play the role of generalized domains of integration, similar to but even more flexible than chains.

Applications in physics

Differential forms arise in some important physical contexts. For example, in Maxwell's theory of electromagnetism, the Faraday 2-form, or electromagnetic field strength, is <math display="block">\mathbf{F} = \frac 1 2 f_{ab}\, dx^a \wedge dx^b\,,</math> where the Template:Math are formed from the electromagnetic fields <math>\vec E</math> and <math>\vec B</math>; e.g., Template:Math, Template:Math, or equivalent definitions.

This form is a special case of the curvature form on the Template:Math principal bundle on which both electromagnetism and general gauge theories may be described. The connection form for the principal bundle is the vector potential, typically denoted by Template:Math, when represented in some gauge. One then has <math display="block">\mathbf{F} = d\mathbf{A}.</math>

The current Template:Math-form is <math display="block"> \mathbf{J} = \frac 1 6 j^a\, \varepsilon_{abcd}\, dx^b \wedge dx^c \wedge dx^d\,,</math> where Template:Math are the four components of the current density. (Here it is a matter of convention to write Template:Math instead of Template:Math, i.e. to use capital letters, and to write Template:Math instead of Template:Math. However, the vector rsp. tensor components and the above-mentioned forms have different physical dimensions. Moreover, by decision of an international commission of the International Union of Pure and Applied Physics, the magnetic polarization vector has been called <math>\vec J</math> for several decades, and by some publishers Template:Math; i.e., the same name is used for different quantities.)

Using the above-mentioned definitions, Maxwell's equations can be written very compactly in geometrized units as <math display="block">\begin{align}

       d {\mathbf{F}} &= \mathbf{0} \\
 d {\star \mathbf{F}} &= \mathbf{J},

\end{align}</math> where <math>\star</math> denotes the Hodge star operator. Similar considerations describe the geometry of gauge theories in general.

The Template:Math-form <math>{\star} \mathbf{F}</math>, which is dual to the Faraday form, is also called Maxwell 2-form.

Electromagnetism is an example of a Template:Math gauge theory. Here the Lie group is Template:Math, the one-dimensional unitary group, which is in particular abelian. There are gauge theories, such as Yang–Mills theory, in which the Lie group is not abelian. In that case, one gets relations which are similar to those described here. The analog of the field Template:Math in such theories is the curvature form of the connection, which is represented in a gauge by a Lie algebra-valued one-form Template:Math. The Yang–Mills field Template:Math is then defined by <math display="block">\mathbf{F} = d\mathbf{A} + \mathbf{A}\wedge\mathbf{A}.</math>

In the abelian case, such as electromagnetism, Template:Math, but this does not hold in general. Likewise the field equations are modified by additional terms involving exterior products of Template:Math and Template:Math, owing to the structure equations of the gauge group.

Applications in geometric measure theory

Numerous minimality results for complex analytic manifolds are based on the Wirtinger inequality for 2-forms. A succinct proof may be found in Herbert Federer's classic text Geometric Measure Theory. The Wirtinger inequality is also a key ingredient in Gromov's inequality for complex projective space in systolic geometry.

See also

• Closed and exact differential forms
• Complex differential form
• Vector-valued differential form
• Equivariant differential form
• Calculus on Manifolds
• Multilinear form
• Polynomial differential form
• Presymplectic form

References

• Template:Citation
• Template:Citation
• Template:Citation—Translation of Formes différentielles (1967)
• Template:Citation
• Template:Citation
• Template:Citation provides a brief discussion of integration on manifolds from the point of view of measure theory in the last section.
• Template:Citation
• Template:Citation This textbook in multivariate calculus introduces the exterior algebra of differential forms at the college calculus level.
• Template:Citation
• Template:Citation
• Template:Citation standard introductory text.
• Template:Citation
• Template:Citation

External links

• Template:Mathworld
• Template:Citation, a course taught at Cornell University.
• Template:Citation, an undergraduate text.
• Needham, Tristan. Visual differential geometry and forms: a mathematical drama in five acts. Princeton University Press, 2021.