Certain algebras possess an important classifying property known as a grading. This essentially amounts to being able to express these algebras as a special kind of direct sum that respects multiplication.
Definition[Graded Algebra]: An algebra over a field is said to be graded if, as a vector space over , can be written as the direct sum of a family of subspaces – i.e.
and such that multiplication behaves according to
, for all
In classical mechanics, one deals with smooth manifolds. To understand what these are we need some definitions.
Definition[Chart]: Let be a manifold of dimension . Then a chart is a pair , where is open and is a homeomorphism to some open .
The collection of all charts such that each belongs to some chart is called an atlas. An atlas characterizes a manifold. More formally,
Definition[Atlas]: An atlas on a topological space is a collection of charts where the are open sets that cover , and for each index
is a homeomorphism of onto an open subset of -dimensional Euclidean space.
Definition[Smooth Atlas]: An atlas is called smooth if for all we have that
is a diffeomorphism.
Definition[Diffeomorphism]: Given two manifolds and , a differentiable map is called a diffeomorphism if it is a bijection, and its inverse is also differentiable.
Now, a manifold may not be generated by a unique atlas. To account for this, we seek a preferred atlas known as a maximal atlas. In defining such a thing, we need to know what it means for atlases to be compatible.
Definition[Compatibility]: Let and be two smooth atlases. Then and are called compatible if and only if is again a smooth atlas.
This notion of compatibility is actually an equivalence relation. The union over an equivalence class of atlases gives us our maximal atlas. If we denote an equivalence class by , then we have
Definition[Maximal Atlas]: A maximal atlas for a manifold is the union of all smooth atlases in an equivalence class – i.e.
This maximal atlas is said to be the differentiable structure of the manifold. We are now ready to understand the idea of a smooth manifold.
Definition[Smooth Manifold]: A smooth manifold is a pair where is a manifold and is a differentiable structure of .
Smooth manifolds are important in classical mechanics because they allow us to do calculus on them. Now, to return to the original topic, we are also interested in the graded algebra of smooth differentiable forms on a smooth manifold with respect to the wedge product, which is denoted by
– Differential Forms –
For now, let’s restrict our focus to what are called differential -forms, or just -forms. According to one source, “Informally, a differential form is what can be integrated along a path.”
Starting in , let be some open subset of and let be two real-valued functions defined on . Then an expression of the form
is called a differential -form on . A particularly important example of a differential -form is the total differential of a real-valued function defined on some open subset .
Definition[Total Differential]: Let by a real-valued function defined on an open subset of . Then its total differential is defined by
This can also be expressed as a kind of dot product of vectors, namely , where is the gradient of . Note that in this case we have and .
-forms can be defined similarly for higher dimensions. For instance, a differential -form on an open subset of is an expression of the form
where and are real-valued functions on . Again, if is a function defined on , then the total differential
is an example.
In general, a -form on an open subset of is an expression of the form
or simply . If we let , then we can write a -form even more succinctly as
So, if is a real-valued function defined on , the the general total differential is
Note that each is a -form. These are defined by the property that for each vector
Clearly, these form a basis for the -forms on .
Definition[Smooth -Form]: A smooth -form on is a real-valued function on the set of all tangent vectors to , i.e. –
with the properties that
- is linear on the tangent space for each – i.e. if and ,then .
- For any smooth vector field , the function is smooth.
A smooth -form on an -dimensional manifold is defined similarly.
Note: Another source simply says that a smooth -form on an open subset of is given by an expression
To better understand the graded algebra of smooth differential forms on with respect to the wedge product we need to understand that the -forms on are part of an algebra called the algebra of differential forms on . Multiplication in this algebra is known as the wedge product and is denoted by the symbol ““. It has the property of being skew-symmetric or anti-commutative:
From this it follows that , which implies that (provided we are not working over a field with characteristic 2). This means that the wedge product is alternating.
From here we can build up -forms on . If each summand of a differential form contains ‘s, then is called a -form.
Note: Functions are considered to be -forms.
is a basis for the -forms on . Thus, every -form can be expressed in the form
where ranges over all multi-indecies of length .
The Exterior Derivative
The exterior derivative is an operation that sends a -form to a -form.
Definition[Exterior Derivative (-Form)]: Let be a -form (function). Then its exterior derivative is the -form
Note that this corresponds to the total differential of .
Definition[Exterior Derivative (-form)]: Let be a -form. Then its exterior derivative is the -form obtained from by applying $lated d$ to each of the functions involved in .
Definition[Directional Derivative]: Let be a differentiable function defined on a region in . Let be a vector based at the point . Then the derivative of along , denoted , is defined as follows. Let . Then
Theorem: Let be a differentiable function defined on a region of . Let be defined on tangent vectors to points of by
Let be a vector based at a point of with coordinates . Let have coordinates . Let be the th unit vector. Then
Next, let’s evaluate where we recall that . So, really, we should write . Now, is actually a composition of two functions, , where . If we now apply the chain rule, then we get
But , so . Therefore,
This finishes the proof.
The exterior derivative obeys both the Leibniz rule and the chain rule:
- Leibniz Rule
- Chain Rule
If is a -form and is a -form, then the Leibniz rule takes the form
Theorem: For any differential form ,
(Or more succinctly, ).
First, let be a function (i.e. -form). Then
since mixed partials commute. Now, since actually means , where is the -th coordinate function, we have that . Let
[By the Leibniz Rule]
Per the reasoning above with functions, this is clearly .
Much of the above has been described in terms of differential forms on . Most of it transfers fairly seamlessly to general manifolds (i.e. the concepts are defined similarly). So, to tie this all back to the graded algebra given above, we should understand to be (i.e. the -forms on ) and to be the algebra of smooth differential -forms on . We’ll look more at graded algebras in the next post.