43 lines
1.4 KiB
Markdown
43 lines
1.4 KiB
Markdown
# Math 401, Fall 2025: Thesis notes, S4, Differential Forms
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This note aim to investigate What is homology and cohomology?
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To answer this question, it's natural to revisit some concepts we have in Calc III. Particularly, Stoke's Theorem and De Rham Theorem.
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Recall that the Stock's theorem states that:
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$$
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\int_c d\omega=\int_{\partial c} \omega
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$$
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Where $\partial c$ is a closed curve and $\omega$ is a 1-form.
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What is form means here?
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> This section is based on extension for conversation with Professor Feres on [11/12/2025].
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## Differential Forms and applications
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> Main reference: [Differential Forms and its applications](https://link.springer.com/book/10.1007/978-3-642-57951-6)
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### Differential Forms in our sweet home, $\mathbb{R}^n$
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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.
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A vector field is a map that associates to each point $p$ in $\mathbb{R}^n$ a vector $v(p)$ in $T_p\mathbb{R}^n$.
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That is
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$$
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v(p)=a_1(p)e_1+...+a_n(p)e_n
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$$
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where $e_1,...,e_n$ is the standard basis of $\mathbb{R}^n$, (in fact could be anything you like)
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And $a_i(p)$ is a function that maps $\mathbb{R}^n$ to $\mathbb{R}$.
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$v$ is differentiable at $p$ if the function $a_i$ is differentiable at $p$.
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This gives a vector field $v$ on $\mathbb{R}^n$.
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