Differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, volumes, 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 is an example of a -form, and can be integrated over an interval contained in the domain of :
Similarly, the expression is a -form that can be integrated over a surface :
The symbol denotes the exterior product, sometimes called the wedge product, of two differential forms. Likewise, a -form represents a volume element that can be integrated over a region of space. In general, a -form is an object that may be integrated over a -dimensional manifold, and is homogeneous of degree in the coordinate differentials
On an -dimensional manifold, a top-dimensional form is called a volume form.
The differential forms form an alternating algebra. This implies that and This alternating property reflects the orientation of the domain of integration.
The exterior derivative is an operation on differential forms that, given a -form, produces a -form This operation extends the differential of a function. 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 -forms are naturally dual to vector fields on a differentiable manifold, and the pairing between vector fields and -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. Some aspects of the exterior algebra of differential forms appears in Hermann Grassmann's 1844 work, Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik .Concept
Differential forms provide an approach to multivariable calculus that is independent of coordinates.Integration and orientation
A differential -form can be integrated over an oriented manifold of dimension. A differential -form can be thought of as measuring an infinitesimal oriented length, or 1-dimensional oriented density. A differential -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, and intervals can be given an orientation: they are positively oriented if, and negatively oriented otherwise. If then the integral of the differential -form over the interval is
which is the negative of the integral of the same differential form over the same interval, when equipped with the opposite orientation. That is:
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, the increment is negative in the direction of integration.
More generally, an -form is an oriented density that can be integrated over an -dimensional oriented manifold. If is an oriented -dimensional manifold, and is the same manifold with opposite orientation and is an -form, then one has:
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 with respect to a measure and integrates over a subset, without any notion of orientation; one writes to indicate integration over a subset. 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,,..., can be used as a basis for all -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 -form is a linear combination of these differentials at every point on the manifold:
where the are functions of all the coordinates. A differential -form is integrated along an oriented curve as a line integral.
The expressions, where can be used as a basis at every point on the manifold for all -forms. This may be thought of as an infinitesimal oriented square parallel to the –-plane. A general -form is a linear combination of these at every point on the manifold: and it is integrated just like a surface integral.
A fundamental operation defined on differential forms is the exterior product. This is similar to the cross product from vector calculus, in that it is an alternating product. For instance,
because the square whose first side is and second side is is to be regarded as having the opposite orientation as the square whose first side is and whose second side is. This is why we only need to sum over expressions, with ; for example:. 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, 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, if any two of the indices,..., 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 -forms is so called multi-index notation: in an -dimensional context, for we define Another useful notation is obtained by defining the set of all strictly increasing multi-indices of length, in a space of dimension, denoted Then locally, spans the space of differential -forms in a manifold of dimension, when viewed as a module over the ring of smooth functions on. By calculating the size of combinatorially, the module of -forms on an -dimensional manifold, and in general space of -covectors on an -dimensional vector space, is choose : 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. The exterior derivative of a differential form is a generalization of the differential of a function, in the sense that the exterior derivative of is exactly the differential of. When generalized to higher forms, if is a simple -form, then its exterior derivative is a -form defined by taking the differential of the coefficient functions:with extension to general -forms through linearity: if then its exterior derivative is
In, 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 Stokes' theorem, which is a central result in the theory of integration on manifolds.
Differential calculus
Let be an open set in. A differential -form is defined to be a smooth function on – the set of which is denoted. If is any vector in, then has a directional derivative, which is another function on whose value at a point is the rate of change of in the direction:In particular, if is the th coordinate vector then is the partial derivative of with respect to the th coordinate vector, i.e.,, where,,..., are the coordinate vectors in. By their very definition, partial derivatives depend upon the choice of coordinates: if new coordinates,,..., are introduced, then
The first idea leading to differential forms is the observation that is a linear function of :
for any vectors, and any real number. At each point p, this linear map from to is denoted and called the derivative or differential of at. Thus. Extended over the whole set, the object can be viewed as a function that takes a vector field on, and returns a real-valued function whose value at each point is the derivative along the vector field of the function. Note that at each, the differential is not a real number, but a linear functional on tangent vectors, and a prototypical example of a differential -form.
Since any vector is a linear combination of its components, is uniquely determined by for each and each, which are just the partial derivatives of on. Thus provides a way of encoding the partial derivatives of. It can be decoded by noticing that the coordinates,,..., are themselves functions on, and so define differential -forms,,...,. Let. Since, the Kronecker delta function, it follows that
The meaning of this expression is given by evaluating both sides at an arbitrary point : on the right hand side, the sum is defined "pointwise", so that
Applying both sides to, the result on each side is the th partial derivative of at. Since and were arbitrary, this proves the formula.
More generally, for any smooth functions and on, we define the differential -form pointwise by
for each. Any differential -form arises this way, and by using it follows that any differential -form on may be expressed in coordinates as
for some smooth functions on.
The second idea leading to differential forms arises from the following question: given a differential -form on, when does there exist a function on such that ? The above expansion reduces this question to the search for a function whose partial derivatives are equal to given functions. For, such a function does not always exist: any smooth function satisfies
so it will be impossible to find such an unless
for all and.
The skew-symmetry of the left hand side in and suggests introducing an antisymmetric product on differential -forms, the exterior product, so that these equations can be combined into a single condition
where is defined so that:
This is an example of a differential -form. This -form is called the exterior derivative of. It is given by
To summarize: is a necessary condition for the existence of a function with.
Differential -forms, -forms, and -forms are special cases of differential forms. For each, there is a space of differential -forms, which can be expressed in terms of the coordinates as
for a collection of functions. Antisymmetry, which was already present for -forms, makes it possible to restrict the sum to those sets of indices for which.
Differential forms can be multiplied together using the exterior product, and for any differential -form, there is a differential -form called the exterior derivative of.
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. One way to do this is cover with coordinate charts and define a differential -form on to be a family of differential -forms on each chart which agree on the overlaps. However, there are more intrinsic definitions which make the independence of coordinates manifest.