Speculation. The Universe is God. God is the only thing that there is. We are all parts of God. Maximize the good that you do to God. Read more 04 Mar 2025 - less than 1 minute read
Updated: 22/2/25 Ref: “Undergraduate Algebra” by Lang. The goal is to study the structure of finite abelian groups. Let ${ (A, +) }$ be a finite abelian group. We can ask ourselves: Can we express ${ A }$ in terms of its subgroups? Note that for every integer ${ n \in \mathbb{Z} _{> 0} }$ there is a group homomorphism ${ \varphi _n : A \t... Read more 05 Feb 2025 - 11 minute read
Link to previous part ROUGH NOTES (!) Regular points; Manifolds; Tangents and Normals Back to top. \[{ \underline{\textbf{Regular points}} }\] The goal is to study the zero sets of ${ C ^P }$ maps ${ f : U (\subseteq \mathbb{R} ^n \text{ open}) \to \mathbb{R} ^m . }$ Obs [Implicit function theorem for ${ C ^p }$ maps ${ \mathbb{R} ^n \t... Read more 11 Oct 2024 - 78 minute read
Link to previous part ROUGH NOTES (!) Second derivative; Third derivative; Higher derivatives; Taylor’s theorem; Inverse function theorem; Implicit function theorem Back to top. \[{ \underline{\textbf{Second derivative}} }\] Def [${ C ^2 }$ maps]: Let ${ E, F }$ be complete normed spaces, and ${ f : U (\subseteq E \text{ open}) \to F .}$... Read more 31 May 2024 - 84 minute read
Link to previous part ROUGH NOTES (!) Integration of maps ${ f : [a, b] \to E }$; Mean value theorem Back to top. \[{ \underline{\textbf{Integration of maps } f : [a, b] \to E} }\] The goal is to study integration of maps ${ f : [a,b] \to E }$. Recall if a map is uniformly continuous and into a complete space, the domain of definition c... Read more 31 May 2024 - 51 minute read