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content/Math401/Extending_thesis/Math401_S5.md

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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](https://link.springer.com/book/10.1007/978-3-642-57951-6)
+
+### 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 $\mathbb{R}^n$.
+