We have studied a variety of generalisations of the fundamental theorem of calculus:
- the fundamental theorem of line integrals;
- Green’s theorem;
- Stokes’ theorem;
- Gauss’ Divergence theorem;
Now we show how all of these formulae concisely fit into one approach. Differential forms provide the underlying theory to present all formulae in one framework. This also allows us to generalise the theorems above to arbitrary dimensions.
Consider an open set (more generally one could also consider open subsets of ). To start, a differential form is just a function
A differential -form is an expression of the form
where We have seen such expressions previously when we studied line integrals. If and the curve is parameterised by then the line integral of the vector field over the curve can be written as
As an example, let be continuously differentiable. Then
is a differential -form. It is called the total differential of If are functions of then
Hence by cancelling in this formula one obtains the total differential of
Using differential forms, this can be written as
Notice the similarity between this notation and the one dimensional integral (here we write which just stands for ).
Let be a differential -form. If there is a function such that then the differential form is called exact.
Let be a differential -form. Then we can associate with it a vector field Then, is conservative if there is a function such that
Hence, a differential -form is exact if and only if the associated vector field is conservative. If the region where is defined is simply connected, then we have seen that is conservative if and only if Hence we also get a criterion to check whether a differential -form is exact and a method of calculating a function such that The details are left as an exercise.
Exact differential forms also appear in ordinary differential equations (ode). Consider the ode
By multiplying with and defining the differential the ode can be written as
If is an exact differential, then there is a function such that Hence, an exact ode can be written as
where is a function defined on a subset of By integrating on both sides we get a solution where is a constant.
Differential -forms are useful for describing line integrals and gradients. But for double integrals, surface integrals, Green’s theorem and Stokes’ theorem we need a more general concept.
Bivectors and differential -forms
We now consider differential -forms, which can partly be understood geometrically.
Recall that, under certain conditions, Green’s theorem in tangential form states that
Leaving out the integral signs, then the left-hand side is again just a differential -form. The aim now is to investigate, given a differential -form, how can we obtain the formula on the right-hand side in Green’s theorem? Once we know the underlying principles, then we can obtain analogous results in other situations.
We have seen this principle already above. If is a differential -form, then we can `apply ‘ to obtain a differential -form Similarly, given a differential -form we would again want to `apply ‘ (what this means will be made more precise below) to obtain a differential -form such that we can express Green’s theorem as
where is a suitable region and is its boundary.
Before we can describe how the operator works on differential forms, let us study differential -forms in more detail.
Above we associated a differential -form with a vector. Differential -forms can be associated with bivectors (see also multivectors). For we define the bivector (the symbol is pronounced `wedge’ and we call the wedge product of and ). Such a bivector can be associated with the cross product Geometrically, the vectors and span a parallelogram in space. This parallelogram lies in a plane (defined by the vectors ) and the length of the cross product is the area of the parallelogram spanned by and Additionally, the cross product also has a direction, which is determined by the right-hand rule. For bivectors, we can associate this direction with a direction (or orientation) of the boundary curve of the parallelogram. Hence we can represent a bivector by the following picture:
Bivectors can be manipulated in the following ways. For a scalar we define the bivector whose associated parallelogram has area times If then the parallelogram has the same orientation and if then the orientation is reversed. The following statements can be interpreted using the parallelogram analogy:
- changing the order of the vectors changes the orientation of the parallelogram, hence:
- the area of a parallelogram spanned by and is zero, hence:
- times the area of the parallelogram spanned by and is equal to the area of the parallelogram spanned by and and is equal to the area of the parallelogram spanned by the vectors and hence:
- there is also a distributive law:
(Notice that the cross product satisfies the same properties.)
Differential -forms can now be understood in the same way as bivectors, with the difference that differential -forms are built from differential -forms instead of vectors. Hence, differential -forms on are expressions of the form
Let and be differential -forms and let be a differential -form. Then the following rules apply:
- where stands for the nullform
Let and be differential -forms. Then, using the rules for we obtain
Again, we can associate a vector field to differential -form. We obtain this association in the following way. First let us use the associations of differential -forms with vectors as above:
Then the wedge products of the differential -forms are associated with the cross products of the vectors, that is
By changing the order in the last wedge product we see that is associated with Hence, we associate a vector field over with a differential -form in the following way:
The Hodge star operator maps differential -forms to differential -forms and back in the following way:
We consider now surface integrals and show how they can be written using differential -forms. Let us consider a special case first where the surface is a suitable region in Let be a differential form where Then we define the integral of the differential -form by
(Notice that we write rather than since, in general, we can have differential -forms for which we would need integral signs. Since in the general case this becomes too cumbersome one just writes only one integral sign.)
Assume that now the surface is parameterised by where is the domain of Note that
See Chapter 4, Section 1 for the definition of
Verify the properties 1., 2., 3., 4. of the wedge product defined above for and
Let Then we have
Let a differential -form be given. Let be the associated vector field. Then, by the definition above, we have
To show the result see Chapter 4, Section 3.
Notice that the last equation also shows that the ordering of the sum in is the most natural form.
To define general differential forms, we introduce one more rule. Let and be differential forms.
- Distributive law:
Differential -forms over are expressions of the form
where In general, differential -forms defined over are expressions of the form
(Note that the cross product of vectors does not generalise to arbitrary dimensions, see here. Hence we do not use the analogy with the cross product of vectors anymore. Instead, one uses alternating multilinear forms.)
We call a differential form continuously differentiable if are continuously differentiable (analogously we define twice continuously differentiable differential forms, and so on).
- For each there is a zero differential -form such that and for any differential -form and any differential -form We call this form the nullform.
- A differential -form is different from the nullform For each the nullform where the sum is over all subsets of consisting of elements, is a differential -form.
- When adding differential forms and then both must be differential -forms defined on (or a subset of ) for some and For instance, the expression is not permitted (does not make sense).
- On the other hand the expression where is a differential -form and is a differential -form, both defined over (or an open subset of ), is well defined. For instance, we have
- Let be a differential -form and be a differential -form, both defined over . If then the differential -form is the nullform. The proof of this result is left as an exercise.
In particular, for we have the nullforms for all which is a differential -form, the nullform which is a differential -form, the nullform which is a differential -form and the nullform which is a differential -form.
Let be differential forms. Then the wedge product satisfies the following properties.
- For each there exists a nullform which is a differential -form. The nullform satisfies for all differential -forms and and for all differential -forms
- For any differential -form and any differential -form we have
- For differential -forms we have
- Let be a differential -form. Then
Let be a differential -form. Then we have already seen that the derivative of this differential form is the differential -form
If is a differential -form, then the derivative of , denoted by is a differential -form. The operator is called the exterior derivative.
The exterior derivative satisfies the following properties:
- Let be a continuously differentiable differential -form. Then
- If and are differential -forms, then
- For a continuously differentiable -form and a differential -form where we define
Laws for the exterior derivative
Let be continuously differentiable differential forms.
- if and are both differential -forms;
- for all
- where is a differential -form;
See also the de Rham cohomology.
We have already seen how the fundamental theorem of line integrals can be written in a concise form using differentials. We now consider Green’s theorem, the Divergence theorem and Stokes’ theorem.
- Green’s theorem
We now show how the formula in Green’s theorem can be obtained using differentials. Let be continuously differentiable and let Then
- Divergence theorem
- Stokes’ Theorem
Let the functions be continuously differentiable and let Then
Let be a smooth and bounded surface and let be its boundary which we assume to be oriented positively. We have
Hence we can write Stokes’ theorem as
Let be a suitable domain (see here for more information) and let denote the boundary curve oriented counterclockwise. Let be a continuously differentiable differential -form defined on Then Green’s theorem states that
Let the functions be continuously differentiable and let Then
Let be a closed, bounded and smooth surface and let be the region enclosed by Then we can write the divergence theorem as
We have now seen that all the main theorems can be written in a single form. It now becomes obvious how to generalise the results above to arbitrary dimension and manifolds. This theorem is called Stokes’ theorem (named after the same person as the theorem considered in Chapter 5, Section 2).
Let be an oriented smooth manifold of dimension Let be a differential -form with compact support on and let denote the boundary of with its induced orientation, then
Let be an oriented smooth manifold of dimension Let be a differential -form. Write down Stokes’ theorem in explicit form.
A differential form for which (the nullform) is called closed.
Let be twice continuously differentiable. Define the vector field and the associated differential -form Show that if and only if is closed.
We have previously defined conservative vector fields , see Chapter 3, Section 3. A similar concept applies to differentials.
A differential form is called exact if there is a differential form such that
Let be continuous. Define the vector field and the associated differential -form Show that is conservative if and only if is exact.
If is a contractible open subset of , any smooth closed differential -form defined on is exact, for any integer (this has content only when ).
In Chapter 2, Section 2 we have seen that and For differential forms we have the following result.
Let be a twice continuously differentiable form defined on an open set of Then
The proof is left as exercise. (Hint Use Clairaut’s theorem.)
Notice that there is an analogue to this proposition for the regions of integration. Namely, let be compact and assume that has a smooth boundary. Then the boundary of the boundary of is empty, that is,
Cauchy’s integral theorem is an important result in complex analysis. It can also be expressed using differential forms (since it is somewhat related to Green’s theorem.) Let denote the set of complex numbers.
Let A complex function can be written as
where It can be shown that the function is continuous if both and are continuous. We now consider complex derivatives. We set
where is a complex number. Hence for the derivative to be well defined we need to demand that the limit is the same for each way approaches In particular, if we restrict to real numbers, we get
On the other hand, if we restrict to imaginary numbers, we get
Since the limits must coincide we get
These equations are called the Cauchy-Riemann equations. The converse also holds, that is, if the are differentiable and the Cauchy-Riemann equations hold, then has a complex derivative. If has a complex derivative at some point then we say that is analytic at
A complex differential form is an expression of the form where and are differential -forms for some We define the integral of a complex differential form by
where can be parameterised by where are piecewise continuously differentiable.
Let and Then
If is analytic, then we obtain
Hence, by Green’s theorem, we obtain the following result.
Cauchy’s integral theorem
Let be a simple closed curve parameterised by a piecewise continuously differentiable function Then for any analytic function where is the region enclosed by we have
Prove Cauchy’s integral theorem using Green’s theorem (without the use of differentials).