From 33e1ea68e7e63b24d72f67c97f27c1f6767c787d Mon Sep 17 00:00:00 2001 From: Nathan Nguyen <159226326+Indigo5684@users.noreply.github.com> Date: Wed, 23 Oct 2024 15:54:49 -0500 Subject: [PATCH] Real Analysis + Electrostatics HW --- docs/math/real-analysis/2-reals.md | 67 +++++++++++++++++++ docs/physics/electrostatics/Homework 5.rnote | Bin 0 -> 2251792 bytes docs/physics/electrostatics/Homework5.rnote | Bin 0 -> 1056082 bytes 3 files changed, 67 insertions(+) create mode 100644 docs/math/real-analysis/2-reals.md create mode 100644 docs/physics/electrostatics/Homework 5.rnote create mode 100644 docs/physics/electrostatics/Homework5.rnote diff --git a/docs/math/real-analysis/2-reals.md b/docs/math/real-analysis/2-reals.md new file mode 100644 index 0000000..9d04547 --- /dev/null +++ b/docs/math/real-analysis/2-reals.md @@ -0,0 +1,67 @@ +# Chapter 2 + +## Section 2.1 - The Algebraic and Order Properties of Real Numbers + +**Proposition**. 2.1.1: $\mathbb{R}$ is a field, with zero element $0$ and identity $1$. + +**Definition**. The rational numbers $\mathbb{Q}$ is the field of fractions of the natural numbers $\mathbb{N}$. + +**Theorem**. 2.1.4: There does not exist a rational number $r$ such that $r^2 = 2$. + +--- + +**Definition**. An *ordered field* is a field $F$ together with subset $F^+$ such that + +1. $F+$ is closed under addition and multiplication +2. If $a \in F$, then exclusively $a \in F^+$, $a = 0$, or $-a \in F^+$. + +**Theorem**. In any ordered field $F$, the following hold + +1. $1 \in F^+$ +2. $\mathbb{N} \subset F^+$ +3. If $a \in F^+$, then $\frac{1}{a} \in F^+$ + +**Definition** The order relation $a > b$ and $b < a$ is defined by $a - b \in F^+$. + +**Theorem**. If $a, b, c \in F$, then + +1. One of $a > b$, $a = b$, or $a < b$ hold (trichotomy) +2. If $a > b$ and $b > c$, then $a > c$ (transitivity) +3. If $a > b$, then $-a < -b$ +4. If $a > b$ and $c > 0$, then $ac > bc$ +5. If $a > b$ and $c < 0$, then $ac < bc$ +6. If $a > b > 0$, then $\frac{1}{b} > \frac{1}{a} > 0$ + +--- + +**Definition**. Let $S$ be a nonempty subset of ordered field $F$. Then, $S$ is *bounded above* if there exists some $u \in F$ such that $s \leq u$ for all $s \in S$. Then, said element $u$ is an *upper bound* of $S$. + +**Definition**. Let $S$ be a nonempty subset of ordered field $F$. Then, $S$ is *bounded below* if there exists some $u \in F$ such that $s \geq u$ for all $s \in S$. Then, said element $u$ is a *lower bound* of $S$. + +**Definition**. Let $S$ be a nonempty subset of ordered field $F$. Then, $S$ is *bounded* if it is bounded both above and below. + +**Definition**. Given field $F$ and nonempty subset $S \subset F$, an element $u \in F$ is a *supremum* or *least upper bound* of $S$ if $u$ is an upper bound of $S$, and given any other upper bound $v$, then $u < v$ + +**Definition**. Given field $F$ and nonempty subset $S \subset F$, an element $u \in F$ is an *infimum* or *greatest lower bound* of $S$ if $u$ is a lower bound of $S$, and given any other lower bound $v$, then $u > v$ + +**Definition**. Given an ordered field $F$, the field has the *supremum/infimum property* if given any nonempty subset $S$, if $S$ is bounded above/below, $S$ has a supremum/infimum. + +## Section 2.2 - Absolute Value and the Real Line + +**Definition**. Absolute value is defined as normal (piecewise). Multiline function in LaTeX are hard. + +**Theorem**. Given any $a, b \in \mathbb{R}$, we know that + +1. $|a| > 0$ for $a \neq 0$ +2. $|ab| = |a||b|$ +3. $|a + b| \leq |a| + |b|$ + +**Collary**. Givem $a, b \in \mathbb{R}$, then $\abs{\abs{a} - \abs{b}} \leq \abs{a - b}$. + +**Remark**. Every field has at least one absolute value function. + +**Theorem**. In an ordered field $F$, for any $r > 0$, we know that + +1. $\abs x = r$ if and only if $x = r$ or $x = -r$ +2. $\abs x < r$ if and only if $-r < x < r$ +3. $\abs x > r$ if either $x > r$ or $x < -r$ diff --git a/docs/physics/electrostatics/Homework 5.rnote b/docs/physics/electrostatics/Homework 5.rnote new file mode 100644 index 0000000000000000000000000000000000000000..76f8d678e1e6d1436c1d605e13d35079c8c74c45 GIT binary patch literal 2251792 zcmeFY=T{Tl6D>?fMF9&UQhh)W=^(v?A|L{SRFx8%^p12$s3L-Nr5mIdsnT2MMM~(Q zhaP%>5JCt^UVnG3_tX6s?tDAv+nF_M&z?Q|aK_yr`#(WC>~Kmi|63<@1-vjFS5#pL z?2Ij)w@vKaOpth~{x7Jv*s)MVp1q{A(|29tFyDUfK~6Br0yO zghdbu2O-sv#%f7pBqWq5ap-~n-whs;Ed4X6fV2`on6-=5QQA71bOp0}!LE1Wlk@V+CE{Kp zY54tB9C6yTqdj*snq<#_sCYh%=$$NorVtI>O}4#c=n*gNbDd%8c!m?!6ySfBm!n(3`}%5Kxx28 zcfdDLL~zNfr`)bbDoV$)Pj8;k?RqHUzbcB$>_u#*(6IqXP{K8-8d|^a(ZmgC*Rj-uj>3 zgG&Gj3s3=f_KKYOp>g1jF%`BFcsdaUKHT%01nM#qPNyN&q%#)~k@ngDi~m0ZAo>L5 zu!#RFHbkN~!h7CrhULv*1+h0%6v*oS$cxRlIMS)Mwn?x1M-EXkKg=zqb~W;#ErhSz 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