NoteNextra-origin/content/Math401/Extending_thesis/Math401_S5.md

Math 401, Fall 2025: Thesis notes, S4, Differential Forms

This note aim to investigate What is homology and cohomology?

To answer this question, it's natural to revisit some concepts we have in Calc III. Particularly, Stoke's Theorem and De Rham Theorem.

Recall that the Stock's theorem states that:

\int_c d\omega=\int_{\partial c} \omega

Where \partial c is a closed curve and \omega is a 1-form.

What is form means here?

This section is based on extension for conversation with Professor Feres on [11/12/2025].

Differential Forms and applications

Main reference: Differential Forms and its applications

Differential Forms in our sweet home, \mathbb{R}^n

Note

I'm a bit deviated form the notation we used in the book, in the actual text, they use \mathbb{R}^n_p to represent the tangent space of \mathbb{R}^n at p. But to help you link those concepts as we see in smooth manifolds, T_pM, we will use T_p\mathbb{R}^n to represent the tangent space of \mathbb{R}^n at p.

Let p be a point in \mathbb{R}^n. The tangent space of \mathbb{R}^n at p is denoted by T_p\mathbb{R}^n, is the set of all vectors in \mathbb{R}^n that use p as origin.

Definition of a vector field

A vector field is a map that associates to each point p in \mathbb{R}^n a vector v(p) in T_p\mathbb{R}^n.

That is

v(p)=a_1(p)e_1+...+a_n(p)e_n

where e_1,...,e_n is the standard basis of \mathbb{R}^n, (in fact could be anything you like)

And a_i(p) is a function that maps \mathbb{R}^n to \mathbb{R}.

v is differentiable at p if the function a_i is differentiable at p.

This gives a vector field v on \mathbb{R}^n.

Definition of dual space of tangent space

To each tangent space T_p\mathbb{R}^n we can associate the dual space (T_p\mathbb{R}^n)^*, the set of all linear maps from T_p\mathbb{R}^n to \mathbb{R}. (\varphi:T_p\mathbb{R}^n\to \mathbb{R})

The basis for (T_p\mathbb{R}^n)^* is obtained by taking (dx_i)_p for i=1,...,n.

This is the dual basis for \{(e_i)_p\} since.

(dx_i)_p(e_j)=\frac{\partial x_i}{\partial x_j}=\begin{cases}0 \text{ if } i\neq j\\
1 \text{ if } i=j
\end{cases}

Definition of a 1-form

A 1-form is a linear map from (T_p\mathbb{R}^n)^* to \mathbb{R}.

\omega(p)=a_1(p)(dx_1)_p+...+a_n(p)(dx_n)_p

where a_i(p) is a function that maps \mathbb{R}^n to \mathbb{R}.

Generalization of 1-form is $k$-form defined as follows:

Definition of a $k$-form

We can define the set of linear map \Lambda^2(\mathbb{R}^n_p)^* where \varphi maps from (T_p\mathbb{R}^n)^*\times ... \times (T_p\mathbb{R}^n)^* to \mathbb{R}, that are bilinear and alternate (\varphi(v_1,v_2)=-\varphi(v_2,v_1).

when \varphi_1 and \varphi_2 are linear maps from (T_p\mathbb{R}^n)^* to \mathbb{R}, then \varphi_1\wedge \varphi_2 is a bilinear map from (T_p\mathbb{R}^n)^*\times (T_p\mathbb{R}^n)^* to \mathbb{R} by setting

(\varphi_1\wedge \varphi_2)(v_1,v_2)=\varphi_1(v_1)\varphi_2(v_2)-\varphi_1(v_2)\varphi_2(v_1)=\det(\varphi_i(v_j))

where i,j=1,\ldots,k, k is the degree of the exterior form

More generally, (\varphi_1\wedge \varphi_2\wedge\dots \wedge \varphi_k)(v_1,v_2,\dots,v_k)=\det(\varphi_i(v_j)).

And \{(dx_i\wedge dx_j)_p,i<j\} forms a basis for \Lambda^2(\mathbb{R}^n_p)^*.

• (dx_i\wedge dx_j)_p=-(dx_j\wedge dx_i)_p
• (dx_i\wedge dx_i)_p=0

An exterior fom of degree 2 in \mathbb{R}^n is a correspondence \omega that associates to each point p in \mathbb{R}^n an element \omega(p)\in \Lambda^2(\mathbb{R}^n_p)^*.

That is

\omega(p)=a_{12}(p)(dx_1\wedge dx_2)_p+a_{13}(p)(dx_1\wedge dx_3)_p+a_{23}(p)(dx_2\wedge dx_3)_p

In the case of \mathbb{R}^3.

Example for real space 4 product

0-forms: functino in \mathbb{R}^4

1-forms: a_1(p)(dx_1)_p+a_2(p)(dx_2)_p+a_3(p)(dx_3)_p+a_4(p)(dx_4)_p

2-forms: a_{12}(p)(dx_1\wedge dx_2)_p+a_{13}(p)(dx_1\wedge dx_3)_p+a_{14}(p)(dx_1\wedge dx_4)_p+a_{23}(p)(dx_2\wedge dx_3)_p+a_{24}(p)(dx_2\wedge dx_4)_p+a_{34}(p)(dx_3\wedge dx_4)_p

3-forms: a_{123}(p)(dx_1\wedge dx_2\wedge dx_3)_p+a_{124}(p)(dx_1\wedge dx_2\wedge dx_4)_p+a_{134}(p)(dx_1\wedge dx_3\wedge dx_4)_p+a_{234}(p)(dx_2\wedge dx_3\wedge dx_4)_p

4-forms: a_{1234}(p)(dx_1\wedge dx_2\wedge dx_3\wedge dx_4)_p

Exterior product of forms

Let \omega=\sum a_{I}dx_I be a k form where I=(i_1,i_2,\ldots,i_k) and i_1<i_2<\cdots<i_k.

\varphi\wedge \omega=\sum b_jdx_j be a s form where j=(j_1,j_2,\ldots,j_s) and j_1<j_2<\cdots<j_s.

The exterior product is defined as

(\varphi\wedge \omega)(v_1,\ldots,v_k)=\sum_{IJ}a_I b_J dx_I\wedge dx_J

Example for exterior product of forms

Let \omega=x_1dx_1+x_2dx_2+x_3dx_3 be a 1-form in \mathbb{R}^3 and \varphi=x_1dx_1\wedge dx_1\wedge dx_3 be a 2-form in \mathbb{R}^3.

Then

\begin{aligned}
\omega\wedge \varphi&=x_2 dx_2\wedge dx_1\wedge dx_3+x_3x_1 dx_3\wedge dx_1\wedge dx_2\\
&=(x_1x_3-x_2)dx_1\wedge dx_2\wedge dx_3
\end{aligned}

Note dx_1\wedge dx_1=0 therefore dx_1\wedge dx_1\wedge dx_3=0

Additional properties of exterior product

Let \omega be a k form, \varphi be a s form, and \theta be an r form.

• (\omega\wedge\varphi)\wedge\theta=\omega\wedge(\varphi\wedge\theta)
• (\omega\wedge\varphi)=(-1)^{k+s}(\varphi\wedge\omega)
• \omega\wedge(\varphi+\theta)=\omega\wedge\varphi+\omega\wedge\theta

Important implications with differential maps

Let f:\mathbb{R}^n\to \mathbb{R}^m be a differentiable map. Then f induces a map f^* from k-forms in \mathbb{R}^n to k-forms in \mathbb{R}^m.

That is

(f^*\omega)(p)(v_1,\ldots,v_k)=\omega(f(p))(df(p)_1v_1,\ldots,df(p)_kv_k)

Here p\in \mathbb{R}^n, v_1,\ldots,v_k\in T_p\mathbb{R}^n, and df(p):T_p\mathbb{R}^n\to T_{f(p)}\mathbb{R}^m.

If g is a 0-form, we have

f^*(g)=g\circ f

Additional properties for differential maps

Let f:\mathbb{R}^n\to \mathbb{R}^m be a differentiable map, \omega,\varphi be k-forms on \mathbb{R}^m and g:\mathbb{R}^m\to \mathbb{R} be a 0-form on \mathbb{R}^m. Then:

• f^*(\omega+\varphi)=f^*\omega+f^*\varphi
• f^*(g\omega)=f^*(g)f^*\omega
• If \varphi_1,\dots,\varphi_k are 1-forms in \mathbb{R}^m, f^*(\varphi_1\wedge\cdots\wedge\varphi_k)=f^*\varphi_1\wedge\cdots\wedge f^*\varphi_k

If g:\mathbb{R}^p\to \mathbb{R}^n is a differential map and \varphi,\omega are any two-forms in \mathbb{R}^m.

• f^*(\omega\wedge\varphi)=f^*\omega\wedge f^*\varphi
• (f\circ g)^*omega=g^*(f^*\omega)

Exterior Differential

Let \omega=\sum a_{I}dx_I be a k form in mathbb{R}^n. The exterior differential d\omega of \omega is defined by

d\omega=\sum da_{I}\wedge dx_I

Additional properties of exterior differential

• d(\omega_1+\omega_2)=d\omega_1+d\omega_2 where \omega_1,\omega_2 are k-forms
• d(\omega\wedge\varphi)=d\omega\wedge\varphi+(-1)^kw\wedge d\varphi where \omega is a k-form and \varphi is a s-form
• d(d\omega)=d^2\omega=0
• d(f^*\omega)=f^*d\omega where f is a differentiable map and \omega is a k-form

Differentiable manifolds

A different flavor of differential manifolds

Definition of differentiable manifold

An $n$-dimensional differentiable manifold is a set M together with a family of of injective maps f_\alpha:U_\alpha\subseteq \mathbb{R}^n\to M of open sets U_\alpha in \mathbb{R}^n in to M such that:

• \bigcup_\alpha f_\alpha(U_\alpha)=M
• For each pair \alpha,\beta, with f_\alpha(U_\alpha)\cap f_\beta(U_\beta)=W\neq \emptyset, the sets f_\alpha^{-1}(W) and f_\beta^{-1}(W) are open sets in \mathbb{R}^n and the maps f_\beta^{-1}\circ f_\alpha and f_\alpha^{-1}\circ f_\beta are differentiable.
• The family \{(U_\alpha,f_\alpha)\} is the maximal relative to the two properties above.

This condition is weaker than smooth manifold, in smooth manifold, we require the function to be class of C^\infty (continuous differentiable of all order), now we only needs it to be differentiable.

Definition of differentiable map between differentiable manifolds

Let M_1^n and M_2^M be differentiable manifolds. A map \varphi:M_1\to M_2 is a differentiable at a point p\in M_1 if given a parameterization g:V\subset \mathbb{R}^m\to M_2 around \varphi(p), there exists a parameterization f:U\subseteq \mathbb{R}^n\to M_1 around p such that:

\varphi(f(U))\subset g(V) and the map

g^{-1}\circ \varphi\circ f: U\subset \mathbb{R}^n\to \mathbb{R}^m

is differentiable at f^{-1}(p).

It is differentiable in an open set of M_1 if it is differentiable at all points in such set.

The map g^{-1}\circ \varphi\circ f is the expression of parameterization of f and g. (Since the change of parameterization is differentiable, the property that f is differentiable does not depends on the choice of parameterization.)

Tangent vector over differentiable curve

Let \alpha: I\to M be a differentiable curve on a differentiable manifold M, with \alpha(0)=p\in M, and let D be the set of functions of M which are differentiable at p. then tangent vector to the curve \alpha at p is the map \alpha'(0):D\to \mathbb{R} given by

\alpha'(0)\varphi=\frac{d}{dt}(\varphi\cdot \alpha(t))|_{t=0}

A tangent vector at p\in M is the