diff --git a/docs/math/abstract-algebra/16-rings.md b/docs/math/abstract-algebra/16-rings.md index 2e50d19..8e20e3f 100644 --- a/docs/math/abstract-algebra/16-rings.md +++ b/docs/math/abstract-algebra/16-rings.md @@ -73,12 +73,12 @@ $$ **Definition**. Proposition 16.22: Let $\phi: R \rightarrow S$ be a ring homomorphism. Then, -1. If $R$ is a commutative ring, then $\phi(R) \subset S$ is a commutative ring. +1. If $R$ is a commutative ring, then $\phi(R) \subseteq S$ is a commutative ring. 2. $\phi(0_R) = 0_S$ 3. Let $1_R$ and $1_S$ be the identities in $R$ and $S$. If $\phi$ is onto, then $\phi(1_R) = 1_S$ -4. If $R$ is a field an $\phi(R) \neq \{0\}$, then $\phi(R) \subset S$ is a field. +4. If $R$ is a field an $\phi(R) \neq \{0\}$, then $\phi(R) \subseteq S$ is a field. -**Definition**. A subring $I \subset R$ is asn *ideal* of $R$ if, when given $a \in I, r \in R$, then $ar$ and $ra$ are both in $I$. That is, $rI \subset I$ and $Ir \subset I$. +**Definition**. A subring $I \subseteq R$ is asn *ideal* of $R$ if, when given $a \in I, r \in R$, then $ar$ and $ra$ are both in $I$. That is, $rI \subseteq I$ and $Ir \subseteq I$. **Definition**. Given a commutative ring $R$ with identity, and $r \in R$, the set @@ -120,13 +120,13 @@ $$ ## Section 16.4 - Maximal and Prime Ideals -**Definition**. Consider ring $R$ and proper ideal $M \subset R$. Then, $M$ is a *maximal ideal* of $R$ if the ideal $M$ is not a subset of any ideal except $R$ itself. That is, given any ideal $I$ properly containing $M$, $I = R$. +**Definition**. Consider ring $R$ and proper ideal $M \subseteq R$. Then, $M$ is a *maximal ideal* of $R$ if the ideal $M$ is not a subset of any ideal except $R$ itself. That is, given any ideal $I$ properly containing $M$, $I = R$. **Theorem**. 16.35: Given a commutative ring with identity $R$, $M$ is a maximal ideal if and only if $R/M$ is a field. -**Definition**. Consider ring $R$ and proper ideal $P \subset R$. Then, $P$ is a *prime ideal* if given $ab \in P$, either $a \in P$ or $b \in P$. +**Definition**. Consider ring $R$ and proper ideal $P \subseteq R$. Then, $P$ is a *prime ideal* if given $ab \in P$, either $a \in P$ or $b \in P$. -**Theorem**. 16.38: Let $R$ be a commutative ring with identity $1$. Then, $P \subset R$ is a prime ideal of $R$ if and only if $R/P$ is a field. +**Theorem**. 16.38: Let $R$ be a commutative ring with identity $1$. Then, $P \subseteq R$ is a prime ideal of $R$ if and only if $R/P$ is a field. Let us assume that $P$ is an ideal in $R$ and $R/P$ is an integral domain. Take two elements $ab \in P$. Now, consider $a + P$ and $b + P$ in $R/P$ such that $(a+P)(b+P) = 0+P = P$. As $R/P$ is a field, either $a + P = 0 + P = P$ or $b + P = 0 + P = P$, meaning either $a \in P$ or $b \in P$. Thus, $P$ is as prime ideal. diff --git a/docs/math/abstract-algebra/18-integral-domains.md b/docs/math/abstract-algebra/18-integral-domains.md index 5a292d4..49d61b8 100644 --- a/docs/math/abstract-algebra/18-integral-domains.md +++ b/docs/math/abstract-algebra/18-integral-domains.md @@ -43,7 +43,7 @@ Additionally, $F_D$ is unique. That is, given field $E$ such that $E \supset D$, **Lemma**. 18.11: Let $D$ be an integral domain and $a, b \in D$. Then, -1. $a | b$ if and only if $\langle b \rangle \subset \langle a \rangle$ +1. $a | b$ if and only if $\langle b \rangle \subseteq \langle a \rangle$ 2. $a$ and $b$ are associates if and only if $\langle b \rangle = \langle a \rangle$ 3. $a$ is a unit in $D$ if and only if $\langle a \rangle = D$. @@ -51,7 +51,7 @@ Additionally, $F_D$ is unique. That is, given field $E$ such that $E \supset D$, **Collary**. 18.13: Let $D$ be a PID. For any $p \in D$, if $p$ is irreducible, then $p$ is prime. -**Lemma**. 18.14: Let $D$ be a PID. Let $I_1 \subset I_2 \subset \ldots$. Then, there exists some integer $N$ such that $I_n = I_N$ for all $n > N$. That is, any chain of ideals converges. +**Lemma**. 18.14: Let $D$ be a PID. Let $I_1 \subseteq I_2 \subseteq \ldots$. Then, there exists some integer $N$ such that $I_n = I_N$ for all $n > N$. That is, any chain of ideals converges. **Definition**. Any commutative ring that satisfies the above condition (the *ascending chain condition*), even if it's not a PID, is called a *Noetherien ring*. diff --git a/docs/math/real-analysis/2-reals.md b/docs/math/real-analysis/2-reals.md index 9d04547..6c2ecc3 100644 --- a/docs/math/real-analysis/2-reals.md +++ b/docs/math/real-analysis/2-reals.md @@ -1,4 +1,4 @@ -# Chapter 2 +# Chapter 2 - The Real Number Line ## Section 2.1 - The Algebraic and Order Properties of Real Numbers @@ -18,7 +18,7 @@ **Theorem**. In any ordered field $F$, the following hold 1. $1 \in F^+$ -2. $\mathbb{N} \subset F^+$ +2. $\mathbb{N} \subseteq F^+$ 3. If $a \in F^+$, then $\frac{1}{a} \in F^+$ **Definition** The order relation $a > b$ and $b < a$ is defined by $a - b \in F^+$. @@ -40,9 +40,9 @@ **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 $S \subseteq F$, 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 field $F$ and nonempty subset $S \subseteq F$, 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. @@ -62,6 +62,46 @@ **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 $x < -r$ +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 $x < -r$ + +--- + +**Definition**. The *standard distance function* or *metric* on the real numbers $\mathbb{R}$ given $a, b$ is $\abs{a - b}$. + +**Theorem**. For any real numbers $a, b, c$, + +1. $\abs{a - b} > 0$ if and only if $a \neq b$ and $\abs{a - b} = 0$ if and only if $a = b$ +2. $\abs{a - b} = \abs{b - a}$ +3. $\abs{a - c} \leq \abs{a - b} + \abs{b + c}$ + +**Definition** A set together with a function satisfying these three properties is known as a *metric space*. + +**Definition** The $\epsilon$-neighborhood of $a \in \mathbb{R}$, denoted $V_\epsilon(a)$ is the set of all real numbers $x \in \mathbb{R}$ such that $\abs{x - a} < \epsilon$. That is, + +$$ +V_\epsilon(a) = (a - \epsilon, a + \epsilon) +$$ + +--- + +**Decimals**. Let $x \in \mathbb{R}$ such that $x > 0$. By the archimedian property, there exists some $b_0 \in \mathbb{N} \cup {0}$ such that $b_0 < x < b_0 + 1$. We can repeat this to see + +$$ +x = b_0 + \frac{b_1}{10} + \frac{b_2}{100} + \ldots + \frac{b_n}{100^n} + \ldots +$$ + +**Definition**. The *decimal expansion* of $x$ is denoted $b_0.b_1 b_2 b_3 \ldots$. + +## Section 2.5 - Intervals + +**Definition**. A subset $I$ is an *interval* if and only if, given $a, b \in I$, then $[a, b] \subseteq I$. + +**Definition**. Intervals $I_1, I_2, \ldots, I_n, \ldots$ are *nested* if and only if $I_1 \subseteq I_2 \subseteq \ldots \subseteq I_n \subseteq \ldots$. + +**Theorem**. Nested Intervals Property. If $I_n = [a_n, b_n]$ is a set of nested intervals that are closed and bound, then there exists some number $z \in \mathbb{R}$ such that $z \in I_n$ for all $n$. + +**Theorem**. If $a < b$, then the interval $[a, b]$ is an uncountable set. + +**Collary**. $\mathbb{R}$ is uncountable. diff --git a/docs/physics/electrostatics/4-conductors.md b/docs/physics/electrostatics/4-conductors.md index c16b8c3..d28c4d4 100644 --- a/docs/physics/electrostatics/4-conductors.md +++ b/docs/physics/electrostatics/4-conductors.md @@ -22,7 +22,7 @@ Consider the surface of a conductor with surface charge density $\sigma_e$. A cy Consider a square with left and right potentials $V(0, y) = V(l, y) = V_1$ and $V(x, 0) = V(x, l) = V_2$. Since we are uniform in $z$, we can say that $V(x, y) = X(x)Y(y)$ and apply separation of variables. -In spherical polar coordinates, we see that with azimuthal symnetry, $V(r, \theta) = \sun_{l=0}^\infty a_l r^l P_l(cos\theta)$ where $P_l(x)$ are Legendre polynomials. +In spherical polar coordinates, we see that with azimuthal symnetry, $V(r, \theta) = \sum_{l=0}^\infty a_l r^l P_l(cos\theta)$ where $P_l(x)$ are Legendre polynomials. **Theorem**. 4.3.3: A Laplace equation's solution must be unique inside a volume $\Omega$ if $\int_{\dd{\Omega}}[\Phi(\vb{r})\grad{\Phi{\vb{r}}} \vdot \vu{n} \dd{S} = 0]$. With this, consider a surface $\dd{\Omega}$ that surrounds conductors. The integral vanishes if a) the potential is specified on each conductor or b) the total charge on each conductor is specified. diff --git a/requirements.txt b/requirements.txt index 39b0269..396d69f 100644 --- a/requirements.txt +++ b/requirements.txt @@ -2,7 +2,9 @@ babel==2.16.0 certifi==2024.8.30 charset-normalizer==3.3.2 click==8.1.7 +codespell==2.3.0 colorama==0.4.6 +editdistpy==0.1.5 ghp-import==2.1.0 gitdb==4.0.11 GitPython==3.1.43 @@ -17,6 +19,7 @@ mkdocs-git-authors-plugin==0.9.0 mkdocs-git-revision-date-localized-plugin==1.2.9 mkdocs-material==9.5.39 mkdocs-material-extensions==1.3.1 +mkdocs-spellcheck==1.1.0 packaging==24.1 paginate==0.5.7 pathspec==0.12.1 @@ -31,5 +34,6 @@ regex==2024.9.11 requests==2.32.3 six==1.16.0 smmap==5.0.1 +symspellpy==6.7.8 urllib3==2.2.3 watchdog==5.0.3