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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
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.
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.