From 9e3784cfd61b01da84ddb57594e5901dcae12007 Mon Sep 17 00:00:00 2001 From: Indigo5684 <159226326+Indigo5684@users.noreply.github.com> Date: Tue, 30 Sep 2025 13:19:29 -0500 Subject: [PATCH] Spellchecker --- .vscode/settings.json | 25 ++++++++++++++++++ docs/index.md | 3 ++- docs/math/abstract-algebra/16-rings.md | 18 ++++++------- .../abstract-algebra/17-polynomial-rings.md | 4 +-- .../abstract-algebra/18-integral-domains.md | 26 +++++++++---------- docs/math/abstract-algebra/DF-10-modules.md | 22 ++++++++-------- .../abstract-algebra/DF-12-modules-pids.md | 8 +++--- docs/math/diffeq/1-intro.md | 6 ++--- docs/math/diffeq/2-1st-order.md | 4 +-- docs/math/diffeq/3-2nd-order.md | 24 ++++++++--------- docs/math/diffeq/4-laplace.md | 4 +-- docs/math/real-analysis/2-reals.md | 4 +-- docs/math/real-analysis/3-sequences-series.md | 6 ++--- docs/math/real-analysis/4-limits.md | 16 ++++++------ docs/math/real-analysis/5-continuity.md | 20 +++++++------- docs/physics/electrostatics/2-coulomb.md | 6 ++--- .../3-electro-magnetic-potentials.md | 14 +++++----- docs/physics/electrostatics/4-conductors.md | 14 +++++----- .../electrostatics/5-moving-charges.md | 8 +++--- docs/recipes/cookies.md | 4 +-- docs/recipes/cupcakes.md | 4 +-- docs/recipes/meals.md | 2 +- docs/recipes/pies.md | 6 ++--- docs/recipes/snacks.md | 4 +-- 24 files changed, 139 insertions(+), 113 deletions(-) create mode 100644 .vscode/settings.json diff --git a/.vscode/settings.json b/.vscode/settings.json new file mode 100644 index 0000000..bfd816c --- /dev/null +++ b/.vscode/settings.json @@ -0,0 +1,25 @@ +{ + "cSpell.words": [ + // TeX + "infty", + "mathbb", + "mathcal", + "stackrel", + "vmatrix", + + // Vocab - Math + "subring", + "subrings", + "monic", + "Noetherian", + "abelian", + "indeterminates", + "infimum", + "supremum", + "Wronskian", + + // Vocab - Physics + "Magnetostatic", + "Magnetostatics" + ] +} \ No newline at end of file diff --git a/docs/index.md b/docs/index.md index 3fded34..070b2fd 100644 --- a/docs/index.md +++ b/docs/index.md @@ -1,4 +1,5 @@ # Personal Notes Collection + ## Textbook Reference -Abstract Algebra: [Abstract Algebra, Theory and Applications](http://abstract.ups.edu/download/aata-20220728.pdf) \ No newline at end of file +Abstract Algebra: [Abstract Algebra, Theory and Applications](http://abstract.ups.edu/download/aata-20220728.pdf) diff --git a/docs/math/abstract-algebra/16-rings.md b/docs/math/abstract-algebra/16-rings.md index a64af98..2faf456 100644 --- a/docs/math/abstract-algebra/16-rings.md +++ b/docs/math/abstract-algebra/16-rings.md @@ -2,7 +2,7 @@ ## Section 16.1 - Rings -**Definition**. A nonempty set $S$ is a *ring* if, with two binary operations called addition and multipllication, the following are satisfied: +**Definition**. A nonempty set $S$ is a *ring* if, with two binary operations called addition and multiplication, the following are satisfied: 1. Addition is commutative. $a + b = b + a$ for $a, b \in R$ 2. Addition is associative. $(a + b) + c = a + (b + c)$ for $a, b, c \in R$ @@ -46,17 +46,17 @@ Note that some books impose the condition that $1 \neq 0$. If $1 = 0$, we can sh **Definition**. For any non-negative integer $n \in \mathbb{N}$ and $r \in R$, we say that $nr = r + \ldots + r \text{(n times)}$. -**Definition**. The *charactaristic* of a ring is the leat possible $n \in \mathbb{N}$ such that $nr = 0$ for all $r \in R$. +**Definition**. The *characteristic* of a ring is the least possible $n \in \mathbb{N}$ such that $nr = 0$ for all $r \in R$. -**Example**. For every prime number $p$, $\mathbb{N}_p$ is a field of charactaristic $p$. +**Example**. For every prime number $p$, $\mathbb{N}_p$ is a field of characteristic $p$. -**Lemma**. 16.18: Given $R$ is a ring with identity, the charactaristic of $1$ is the charactartistic of the field. +**Lemma**. 16.18: Given $R$ is a ring with identity, the characteristic of $1$ is the characteristic of the field. -**Theorem**. 16.19: The charactaristic of an integral domain is prime or zero. +**Theorem**. 16.19: The characteristic of an integral domain is prime or zero. ## Section 16.3 - Ring Homomorphisms and Ideals -**Definition** Given rins $R$ and $S$, and a mapping $\varphi: R \rightarrow S$, we say that $\varphi$ is a *ring homomorphism* if the following are satisfied for all elements of $R$: +**Definition** Given rings $R$ and $S$, and a mapping $\varphi: R \rightarrow S$, we say that $\varphi$ is a *ring homomorphism* if the following are satisfied for all elements of $R$: $$ \begin{align} @@ -92,7 +92,7 @@ is an ideal in $R$. Specifically, $\langle a \rangle$ is a *principal ideal*. **Example**. Theorem 16.25. Every ideal in $\mathbb{Z}$ is a principal ideal. -**Examplee**. With $\varphi: R \rightarrow S$, $\ker \varphi$ is an ideal of $R$. +**Example**. With $\varphi: R \rightarrow S$, $\ker \varphi$ is an ideal of $R$. **Remark**. 16.28: We are working with *two-sided ideals*. If rings are not commutative, we may deal with *left ideals* and *right ideals*. @@ -118,7 +118,7 @@ $$ R/I \cong \frac{R/J}{I/J} $$ -**Theorem**. 16.34, *Correspondence Theorem*. Let $I$ be an ideal of $R$. Then, $S \mapsto S/I$ is a one-to-one correspeondence between the set of subrings $S$ containing $I$ (that is, $I \in S$) and the set of subrings of $R/I$. Furthermore, the ideals of $R$ containing $I$ correspond to the ideals of $R/I$. +**Theorem**. 16.34, *Correspondence Theorem*. Let $I$ be an ideal of $R$. Then, $S \mapsto S/I$ is a one-to-one correspondence between the set of subrings $S$ containing $I$ (that is, $I \in S$) and the set of subrings of $R/I$. Furthermore, the ideals of $R$ containing $I$ correspond to the ideals of $R/I$. ## Section 16.4 - Maximal and Prime Ideals @@ -140,6 +140,6 @@ $$ (a + P)(b + P) = ab + P = 0 + P = P $$ -Thus, $ab \in P$. By symnetry, assume $a \notin P$. Thus, $b \in P$ by the devinition of a prime ideal, so $b + P = 0 + P$, meaning $R/P$ is an integral domain. +Thus, $ab \in P$. By symnetry, assume $a \notin P$. Thus, $b \in P$ by the definition of a prime ideal, so $b + P = 0 + P$, meaning $R/P$ is an integral domain. **Theorem**. 16.40: In a commutative ring with identity, every maximal ideal is also a prime ideal. diff --git a/docs/math/abstract-algebra/17-polynomial-rings.md b/docs/math/abstract-algebra/17-polynomial-rings.md index 135e768..b27492e 100644 --- a/docs/math/abstract-algebra/17-polynomial-rings.md +++ b/docs/math/abstract-algebra/17-polynomial-rings.md @@ -14,7 +14,7 @@ where $a_i \in R$ and $a_n \neq 0$ is called a *polynomial over $R$* with *indet **Definition**. A polynomial is known as *monic* if the leading coefficient is equal to $1$. -**Definition**. The *degree* of $f$ is the largest nonnegative number such that $a_n \neq 0$, written as $\deg f(x) = n$. If no such mumber exists, that is, $f(x) = 0$, we say the degree of $f$ is $-\infty%. +**Definition**. The *degree* of $f$ is the largest nonnegative number such that $a_n \neq 0$, written as $\deg f(x) = n$. If no such number exists, that is, $f(x) = 0$, we say the degree of $f$ is $-\infty%. **Definition**. We denote the set of all polynomials with coefficients in $R$ as $R[x]$. @@ -46,7 +46,7 @@ where either $\deg r(x) < \deg g(x)$ or $r(x)$ is the zero polynomial. ## Section 17.3 Irreducible Polynomials -**Definition** A nonconstant polynomial $f(x) \ in F[x]$ is *irreducible* over a field $F$ if it cannot be expressed as the product of two non-identity polynomials $g(x)$ and $h(x)$ in $F[x]$, with the degree of both polynomials strictly less than the degree of $f(x)$. +**Definition** A non-constant polynomial $f(x) \ in F[x]$ is *irreducible* over a field $F$ if it cannot be expressed as the product of two non-identity polynomials $g(x)$ and $h(x)$ in $F[x]$, with the degree of both polynomials strictly less than the degree of $f(x)$. **Lemma**. Let $p(x) \in \mathbb{Q}[x]$. Then, with $r, s \in \mathbb{Z}, a(x) \in \mathbb{N}[x]$, we can write $p(x) = \frac{r}{s} a(x)$. diff --git a/docs/math/abstract-algebra/18-integral-domains.md b/docs/math/abstract-algebra/18-integral-domains.md index 3456c7c..6cf75aa 100644 --- a/docs/math/abstract-algebra/18-integral-domains.md +++ b/docs/math/abstract-algebra/18-integral-domains.md @@ -2,7 +2,7 @@ ## Section 18.1 - Fields of Fractions -**Definition**. Given an integral domain $D$, we can construct a field $F$ containing $D$ by stating that any $p/q \in F$, annd that any two elements $a/b = c/d$ if and only if $ad = bc$. We can consider this akin o a set of ordered pairs +**Definition**. Given an integral domain $D$, we can construct a field $F$ containing $D$ by stating that any $p/q \in F$, and that any two elements $a/b = c/d$ if and only if $ad = bc$. We can consider this akin o a set of ordered pairs $$ S = \{(a, b) : a, b \in D \text{ and } b \neq 0 \} @@ -18,13 +18,13 @@ $$ Additionally, $F_D$ is unique. That is, given field $E$ such that $E \supset D$, there exists a map $\psi: F_D \rightarrow D$ giving an isomorphism such that $\psi(a) = a$ for all $a \in D$. -**Corollary**. 18.6: Let $F$ be a field of charactaristic $0$. Then, $F$ contains a subfield isomorphic to $\mathbb{Q}$. +**Corollary**. 18.6: Let $F$ be a field of characteristic $0$. Then, $F$ contains a subfield isomorphic to $\mathbb{Q}$. -**Corollary**. 18.6: Let $F$ be a field of charactaristic $p$. Then, $F$ contains a subfield isomorphic to $\mathbb{Z}_p$. +**Corollary**. 18.6: Let $F$ be a field of characteristic $p$. Then, $F$ contains a subfield isomorphic to $\mathbb{Z}_p$. ## Section 18.2 - Factorization in Integral Domains -**Definition**. Let $R$ be a commutative ring with identity, and $a, b \in R$. We say that $a$ *divideds* $b$, that is, $a | b$, if there exists some $c \in R$ such that $b = ac$. +**Definition**. Let $R$ be a commutative ring with identity, and $a, b \in R$. We say that $a$ *divides* $b$, that is, $a | b$, if there exists some $c \in R$ such that $b = ac$. **Definition**. A *unit* element is any element that has a multiplicative inverse. @@ -37,7 +37,7 @@ Additionally, $F_D$ is unique. That is, given field $E$ such that $E \supset D$, **Definition**. Given integral domain $D$, we say that $D$ is a *Unique Factorization Domain (UFD)* if it satisfies the following criteria: 1. Given $a \in D, a \neq 0$, and $a$ is not a unit, $a$ can be written as a product of irreducible elements in $D$. -2. Let $a = p_1 \ldots p_r = q_1 \ldots q_s$, where $p_i$ and $q_i$ are all irreducible. Then, $r = s$, and there exists some fuction $\pi \in S_r$ such that $p_i$ and $q_{\pi(j)}$ are associates for $j = 1, \ldots, r$. +2. Let $a = p_1 \ldots p_r = q_1 \ldots q_s$, where $p_i$ and $q_i$ are all irreducible. Then, $r = s$, and there exists some function $\pi \in S_r$ such that $p_i$ and $q_{\pi(j)}$ are associates for $j = 1, \ldots, r$. **Definition**. A ring $R$ is a *principal ideal domain (PID)* if every ideal of $R$ is principal. @@ -61,24 +61,24 @@ Additionally, $F_D$ is unique. That is, given field $E$ such that $E \supset D$, --- -**Definition**. Any integral domain $D$ is a *euclidian domain* with a *euclidian function* $nu: D \\ \{0\} \rightarrow \mathbb{N}$ that satisfies the following: +**Definition**. Any integral domain $D$ is a *Euclidean domain* with a *Euclidean function* $nu: D \\ \{0\} \rightarrow \mathbb{N}$ that satisfies the following: 1. Given $a, b \neq 0$, then $\nu(a) \leq \nu(ab)$. 2. Given, $a, b \in D$ and $b \neq 0$, there exists some $q, r \in D$ such that $a = bq + r$ and either $r = 0$ or $\nu(r) < \nu(b)$. -**Example**. Absolute value on $\mathbb{Z}$ is a Euclidian validation. +**Example**. Absolute value on $\mathbb{Z}$ is a Euclidean validation. -**Example**. Degree on $F[x]$ is a Euclidian validation. +**Example**. Degree on $F[x]$ is a Euclidean validation. -**Example**. $\nu(a + bi) = a^2 + b^2$ is a Euclidian validation over $\mathbb{Z}[i]$. +**Example**. $\nu(a + bi) = a^2 + b^2$ is a Euclidean validation over $\mathbb{Z}[i]$. -**Theorem**. 18.21: Every Euclidian domain is a PID. +**Theorem**. 18.21: Every Euclidean domain is a PID. -**Corollary**. Every Euclidian domain is a UFD. +**Corollary**. Every Euclidean domain is a UFD. --- -**Definition**. Given a polynomial $p(x) \in D$, with $D$ bein an integer domain, we say that the *content* of $p(x)$ is the greatest common divisor of its coefficients. Additionally, if the content is $1$, we say that $p(x)$ is *primitive*. +**Definition**. Given a polynomial $p(x) \in D$, with $D$ being an integer domain, we say that the *content* of $p(x)$ is the greatest common divisor of its coefficients. Additionally, if the content is $1$, we say that $p(x)$ is *primitive*. **Theorem**. 18.24: Let $D$ be a UFD, and $f(x), g(x) \in D[x]$ be primitive. Then, $f(x)g(x)$ is primitive. @@ -94,7 +94,7 @@ As a direct consequence, we see the following. **Theorem**. If $D$ is as UFD, then $D[x]$ is a UFD. -**Corollary**. This theorem has several collaries: +**Corollary**. This theorem has several corollaries: 1. Given a field $F$, since $F$ is a PID, it is also a UFD. Thus, $F[x]$ is a UFD. 2. The ring of polynomials over integers, $\mathbb{Z}[x]$ is a UFD. diff --git a/docs/math/abstract-algebra/DF-10-modules.md b/docs/math/abstract-algebra/DF-10-modules.md index 705c2e2..8891779 100644 --- a/docs/math/abstract-algebra/DF-10-modules.md +++ b/docs/math/abstract-algebra/DF-10-modules.md @@ -15,13 +15,13 @@ **Remark**. Modules over a field $F$ and vector spaces over $F$ are identical. -**Definition** An *R-submodule* is a subset$N \subseteq M$ which is closed under the action taken forall $r \in R$. That is, given $r \in R, n \in N$, tthen $rn \in N$. Every module has at least two submodules: itself and the trivial (empty) submodule. +**Definition** An *R-submodule* is a subset$N \subseteq M$ which is closed under the action taken forall $r \in R$. That is, given $r \in R, n \in N$, then $rn \in N$. Every module has at least two submodules: itself and the trivial (empty) submodule. -**Remark**. If $F$ is a field, submodules are equivilent to subspaces. +**Remark**. If $F$ is a field, submodules are equivalent to subspaces. --- -**Example**. Let $F$ be a field and $F[x]$ a polynomial ring. Then, let $V$ be a vector space of $F$, and $T$ be a linear transformaion from $V$ to itself. That is, $V: T \rightarrow T$. We know that $V$ is an $F$-module. We will want to show that $V$ can be written as an $F[x]$-module for some choice of $T$. That is, we want an action $F[x] \cross V \rightarrow V$. +**Example**. Let $F$ be a field and $F[x]$ a polynomial ring. Then, let $V$ be a vector space of $F$, and $T$ be a linear transformation from $V$ to itself. That is, $V: T \rightarrow T$. We know that $V$ is an $F$-module. We will want to show that $V$ can be written as an $F[x]$-module for some choice of $T$. That is, we want an action $F[x] \cross V \rightarrow V$. Now, for a given linear transformation $T$, consider some polynomial $p(x) = a_n x^n + \ldots + a_0$ and some $v \in V$. We define $p(x) \cross v$ by$ @@ -42,17 +42,17 @@ with $T^n$ being defined as applying $T$ a total of $n$ times. **Recall**. The center of a ring $A$ is the subring $A'$ such that for all $x, y \in R'$, then $xy = yx$. In other words, it is the commutative subring of $A$. -**Definition**. Given two $R$-algebras $A, B$, an *$R$-algebra homomorphhism$ is a ring homomorphism $\varphi: A \rightarrow B$ that maps $1_A \rightarrow 1_B$ such that $\varphi(ra) = r\varphi(a)$. +**Definition**. Given two $R$-algebras $A, B$, an *$R$-algebra homomorphism$ is a ring homomorphism $\varphi: A \rightarrow B$ that maps $1_A \rightarrow 1_B$ such that $\varphi(ra) = r\varphi(a)$. ## Section 10.2 - Quotient Modules and Module Homomorphisms -**Definition**. Let $R$ be a ring and $M, N$ be $R$-modules. then a ring homomorphhism $\varphi: M \rightarrow N$ is an *$R$-module homomorphism* if for all $r \in R$, $\varphi(rx) = r\varphi(x)$. +**Definition**. Let $R$ be a ring and $M, N$ be $R$-modules. then a ring homomorphism $\varphi: M \rightarrow N$ is an *$R$-module homomorphism* if for all $r \in R$, $\varphi(rx) = r\varphi(x)$. **Theorem**. An $R$-module homomorphism is an *isomorphism* if it is 1-1 and onto, and said modules are *isomorphic*. **Definition**. Let $M, N$ be $R$-modules. The set $\Hom_R(M, N)$ is the set of all homomorphisms from $M$ to $N$. -**Promposition**. Let $M$, $N$, and $L$ be $R$-modules. Then, +**Proposition**. Let $M$, $N$, and $L$ be $R$-modules. Then, 1. A function $\varphi: M \rightarrow N$ is an $R$-module homomorphism if and only if $\varphi(rx + y) = r\varphi(x) + \varphi(y)$ for all $x, y \in M$ and $r \in R$. 2. Let $\varphi, \psi \in \Hom_R(M, N)$. Then, define $\varphi + \psi$ as @@ -85,13 +85,13 @@ $$ This is the smallest submodule that contains both $A$ and $B$. -**Theorem**. First Isomorphism Theorem. Let $M, N$ be $R$-modules, and $\varphi: M \rightarrow N$ be an $R$-module homomorphhiism. Then, $\ker \varphi$ is a submodule of $M$, and $M / \ker \varphi \cong \varphi(M)$. +**Theorem**. First Isomorphism Theorem. Let $M, N$ be $R$-modules, and $\varphi: M \rightarrow N$ be an $R$-module homomorphism. Then, $\ker \varphi$ is a submodule of $M$, and $M / \ker \varphi \cong \varphi(M)$. **Theorem**. Second Isomorphism Theorem. Let $A, B$ be submodules of the $R$-module $M$. Then, $(A + B)/B \cong A/(A \cap B)$. **Theorem**. Third Isomorphism Theorem. Let $M$ be an $R$-module, and $A \subseteq B$ be submodules of $M$. Then, $\frac{M/A}{B/A} \cong M/B$. -**Theorem**. Lattice Isomorphism Theorem. Let $N$ be a submodule of the $R$-module $M$. Then, there is a bijection between submoudles of $M$ containing $N$ and submodules of $M/N$. This is given by $A \leftrightarrow A/N$, for $A \supseteq N$. +**Theorem**. Lattice Isomorphism Theorem. Let $N$ be a submodule of the $R$-module $M$. Then, there is a bijection between submodules of $M$ containing $N$ and submodules of $M/N$. This is given by $A \leftrightarrow A/N$, for $A \supseteq N$. ## Section 10.3 - Generation of Modules, Direct Sums, and Free Modules @@ -99,7 +99,7 @@ This is the smallest submodule that contains both $A$ and $B$. 1. The *sum* of $N_1, \ldots, N_n$ is the set of all finite sums of elements from the sets $N_i$. That is, $N_1, \ldots, N_n := \{a_1 + a_2 + \ldots + a_n | a_i \in N_i\}$ 2. For any subset $A$ of $M$, let $RA = \{r_1 a_1 + r_2 a_2 + \ldots + r_m a_m | r_i \in R, a_i \in A\}$. If $N$ is a submodule of $M$ such that $N = RA$, then $A$ is called the *generating set* for $N$. -3. A submodule $N$ of $M$ is *finitely generaated* if there is some finite subset $A$ of $M$ such that $N = RA$. That is, $N$ is generated by some finite subset. +3. A submodule $N$ of $M$ is *finitely generated* if there is some finite subset $A$ of $M$ such that $N = RA$. That is, $N$ is generated by some finite subset. 4. A submodule of $M$ (up to equality) is $cyclic$ if there exists some element $a \in M$ such that $N = Ra = \{ra | r \in R\}$. **Definition**. Let $M_1, \ldots, M_k$ be a collection of $R$-modules. Then, the *direct product* is defined as @@ -118,13 +118,13 @@ This direct product is in itself an $R$-module. **Definition**. An $R$-module $F$ is said to be *free* on the subset $A$ of $F$ if for every nonzero $x \in F$, there exists nonzero elements $r_1, \ldots, r_n$ of $R$ and unique $a_1, \ldots, a_n$ such that $x = r_1 a_1 + \ldots + r_n a_n$ for some $n \in \mathbb{Z}^+$. That is, $A$ is a *basis* or *set of free generators* of $F$. -**Theorem**. For any set $A$, there is a free $R$-module $F(A)$ on $A$ such that $F(A)$ satisfies the universal property: if $M$ is any $R$-module, and $\varphi: A \rightarrow M$ is a map of sets, there eixts a unique $R$-module homomorphism: $\Phi: F(A) \rightarrow M$ such that $\Phi(a) = \varphi(a)$ for all $a \in A$. +**Theorem**. For any set $A$, there is a free $R$-module $F(A)$ on $A$ such that $F(A)$ satisfies the universal property: if $M$ is any $R$-module, and $\varphi: A \rightarrow M$ is a map of sets, there exists a unique $R$-module homomorphism: $\Phi: F(A) \rightarrow M$ such that $\Phi(a) = \varphi(a)$ for all $a \in A$. **Corollary**. If $F_1$ and $F_2$ are free modules on $A$, then there is a unique isomorphism between $F_1$ and $F_2$, which is the identity map on A. **Corollary**. If $F$ is a free $R$-module with basis $A$, then $F \cong F(A)$. -**Definition** For a free module $F$ with basis $A$, if $R$ is commutative, tthen the *rank* of $F$ is the cardinality of $A$. +**Definition** For a free module $F$ with basis $A$, if $R$ is commutative, then the *rank* of $F$ is the cardinality of $A$. ## Section 10.4 - Tensor Products of Modules diff --git a/docs/math/abstract-algebra/DF-12-modules-pids.md b/docs/math/abstract-algebra/DF-12-modules-pids.md index d9ff0dd..f3eec7c 100644 --- a/docs/math/abstract-algebra/DF-12-modules-pids.md +++ b/docs/math/abstract-algebra/DF-12-modules-pids.md @@ -1,4 +1,4 @@ -# Chapter 12 - Modules over Principal Ideal Domains +# Dummit & Foote Chapter 10 Chapter 12 - Modules over Principal Ideal Domains ## Section 12.1 The Basic Theory @@ -8,7 +8,7 @@ $$ M_1 \subseteq M_2 \subseteq \ldots $$ -there exists some $k \in \mathbb{N}$ such thaht given any $n \in \mathbb{N}$ with $n \geq k$, then $M_n = M_k$. +there exists some $k \in \mathbb{N}$ such that given any $n \in \mathbb{N}$ with $n \geq k$, then $M_n = M_k$. **Definition**. A ring $R$ is *Noetherian* if it is Noetherian when viewed as a left $R$-module over itself. @@ -20,7 +20,7 @@ there exists some $k \in \mathbb{N}$ such thaht given any $n \in \mathbb{N}$ wit **Corollary**. If $R$ is a principal ideal domain (PID), then all nonempty set of ideals of $R$ has a maximal element. Additionally, $R$ is as Noetherian ring. -**Proposition**. Let $R$ be an integral doman, and $M$ be a free $R$-module of rank $n < \infty$. Then, given $S$ is subset $M$ with $|S| > n$, the elements of $S$ are $R$-linearly dependent. +**Proposition**. Let $R$ be an integral domain, and $M$ be a free $R$-module of rank $n < \infty$. Then, given $S$ is subset $M$ with $|S| > n$, the elements of $S$ are $R$-linearly dependent. **Definition**. Given $R$ an integral domain and $M$ an $R$-module, @@ -49,5 +49,5 @@ Additionally, if $R$ is a PID, as $\Ann_R(N)$ is an ideal, $\Ann(N) = (n)R$ and **Theorem**. Let $R$ be a principal ideal domain, and $M$ be a free $R$-module of finite rank $m$, and $N$ be a submodule of $M$. Then, 1. $N$ is a free submodule with rank $n \leq m$. -2. There exiss a basis $y_1, y_2, \ldots, y_m$ of $M$ so that $r_1 y_1, r_2 y_2, \ldots, r_m y_n$ is a basis of $N$ for some $r_i \in R$ and $r_1 | r_2 | \ldots | r_n$ +2. There exists a basis $y_1, y_2, \ldots, y_m$ of $M$ so that $r_1 y_1, r_2 y_2, \ldots, r_m y_n$ is a basis of $N$ for some $r_i \in R$ and $r_1 | r_2 | \ldots | r_n$ diff --git a/docs/math/diffeq/1-intro.md b/docs/math/diffeq/1-intro.md index 6c7e36e..af802fd 100644 --- a/docs/math/diffeq/1-intro.md +++ b/docs/math/diffeq/1-intro.md @@ -16,15 +16,15 @@ This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/ \[ a_n(t)y^{(n)}(t) + a_{n-1}(t)+y^{n-1}(t) + \ldots + a_1(t)y'(t) + a_0(t)y(t) = g(t) \] -Note that $a_n(t)$ does not depeond on any derivative of $y$, so the presence of terms such as $e^y$ or $\sqrt{y'}$ signal that the equation is *nonlinear*. +Note that $a_n(t)$ does not depend on any derivative of $y$, so the presence of terms such as $e^y$ or $\sqrt{y'}$ signal that the equation is *nonlinear*. -**Definition**. The *solution(s)* to a differential equation over an inverval $\alpha < t < \beta$ are any funcion(s) $y(t)$ that satisfy the differential equation. +**Definition**. The *solution(s)* to a differential equation over an interval $\alpha < t < \beta$ are any function(s) $y(t)$ that satisfy the differential equation. **Definition**. The *initial conditions* are a condition or set of conditions that constrain the possible solution sets. **Definition**. An *Initial Value Problem* is a differential equation along with the appropriate boundary or initial conditions. -**Definition**. The *integral of validity* for a solution to a differential equation is the largest possible interval containing the initial coniditions for which the solution is valid. +**Definition**. The *integral of validity* for a solution to a differential equation is the largest possible interval containing the initial conditions for which the solution is valid. **Definition**. The *general solution* to a differential equation is the most general form a solution to a differential equation can take without requiring the initial conditions. diff --git a/docs/math/diffeq/2-1st-order.md b/docs/math/diffeq/2-1st-order.md index 9cd6c85..cdb28cd 100644 --- a/docs/math/diffeq/2-1st-order.md +++ b/docs/math/diffeq/2-1st-order.md @@ -24,7 +24,7 @@ $$ \mu(t)\frac{dy}{dt} + \mu'(t)y = \mu(t)g(t) $$ -The left of the preceeding equation is simply the product rule, so we can write $(\mu(t)y(t))' = \mu(t)g(t)$. Take the integral of both sides. +The left of the preceding equation is simply the product rule, so we can write $(\mu(t)y(t))' = \mu(t)g(t)$. Take the integral of both sides. \begin{align} \int (\mu(t)y(t))' dt &= \int \mu(t)g(t) \\ @@ -88,7 +88,7 @@ Let the following differential equation of the following forms be given. \frac{dy}{dx} &= N(y)M(x) \\ \end{align}. -For the sake of simplicty, select the following form: +For the sake of simplicity, select the following form: $$ N(y) \frac{dy}{dx} = M(x) diff --git a/docs/math/diffeq/3-2nd-order.md b/docs/math/diffeq/3-2nd-order.md index f225710..9a5dcd3 100644 --- a/docs/math/diffeq/3-2nd-order.md +++ b/docs/math/diffeq/3-2nd-order.md @@ -14,21 +14,21 @@ $$ ay'' + by' + cy = g(t) $$ This is a second-order differential equation with constant coefficients. -**Definition**. In the event that $g(t) = 0$, we say the equation is *homogenous*. Otherwise, the equation is *nonhomogenous*. +**Definition**. In the event that $g(t) = 0$, we say the equation is *homogenous*. Otherwise, the equation is *nonhomogeneous*. **Definition**. Principal of Superposition. Let $y_1(t)$ and $y_2(t)$ be solutions to a linear, homogenous differential equation. Then, any linear combination of said solutions is also a solution to the differential equation. In other words, with $c_1, c_2 \in \mathbb{R}$, the following is a solution to a differential equation. $$ y(t) = c_1 y_1(t) + c_2 y_2(t) $$ -Given a second-order homogenous differential equation with constant coeffictions, we assume solutions of the following form: +Given a second-order homogenous differential equation with constant coefficients, we assume solutions of the following form: $$ y(t) = e^{rt} $$ -Substituting this equation into the differential equationm, we see the following: +Substituting this equation into the differential equation, we see the following: $$ e^{rt}(ar^2 + br + c) = 0 $$ -Thus, we allow the *charactaristic equation* of the differential equation to be as follows: +Thus, we allow the *characteristic equation* of the differential equation to be as follows: $$ ar^2 + br + c = 0 $$ @@ -36,7 +36,7 @@ $$ ar^2 + br + c = 0 $$ This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/RealRoots.aspx). -When the two roots to the charactaristic equation are discrete roots in the real numbers, we see the following solutions. +When the two roots to the characteristic equation are discrete roots in the real numbers, we see the following solutions. $$ y_1(t) = e^{r_1 t} $$ @@ -50,7 +50,7 @@ $$ y(t) = c_1 e^{r_1 t} + c_2 e^{r_2 t} $$ This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/ComplexRoots.aspx). -Let the solutions to the charactaristic equation be of the following form: +Let the solutions to the characteristic equation be of the following form: $$ r_{1,2} = \lambda \pm \mu i $$ @@ -64,7 +64,7 @@ Recall Euler's Formula: $$ e^{i \theta} = \cos \theta + i \sin \theta $$ -A colliloquy of Euler's formula is the following: +A corollary of Euler's formula is the following: $$ e^{-i \theta} = \cos(-\theta) + i \sin(-\theta) = \cos \theta - i \sin \theta $$ @@ -83,7 +83,7 @@ $$ y(t) = c_1 e^{\lambda t} \cos(\mu t) + c_2 e^{\lambda t} \sin(\mu t) $$ This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/RepeatedRoots.aspx). -Assume the solutions to the charactaristic equations are $r = r_1 = r_2$. Thus, the two equations $y_t(t)$ and $y_2(t)$ are not linearly independent. +Assume the solutions to the characteristic equations are $r = r_1 = r_2$. Thus, the two equations $y_t(t)$ and $y_2(t)$ are not linearly independent. After a *lot* of algebra, we see that @@ -112,7 +112,7 @@ $$ **Definition**. If $W(f, g) \neq 0$, then $f(t)$ and $g(t)$ are said to form a *fundamental set of solutions*, and can be superimposed to form the general solution. -## Section 3.8 - Nonhomogenous Differential Equations +## Section 3.8 - Nonhomogeneous Differential Equations This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/NonhomogeneousDE.aspx). @@ -120,11 +120,11 @@ Assume we have the differential equation as follows: $$ y'' + p(t) y' + q(t) y = g(t) $$ -The equivilent homogenous differential equation is +The equivalent homogenous differential equation is $$ y'' + p(t) y' + q(t) y = 0 $$ -**Theorem**. Assume $Y_1(t)$, $Y_2(t)$ are solutions to the nonhomogenous differential equations. Then, $Y_1(t) - Y_2(t)$ is a solution to the homogenous differential equation. This can be proved by substitution. +**Theorem**. Assume $Y_1(t)$, $Y_2(t)$ are solutions to the nonhomogeneous differential equations. Then, $Y_1(t) - Y_2(t)$ is a solution to the homogenous differential equation. This can be proved by substitution. Thus, with $y_h(t)$ the solution to the homogenous problem, and $y_p(t)$ the solution to this particular problem, we can say that the general form of the solution to this differential equation is @@ -156,7 +156,7 @@ Assume we have the differential equation as follows: $$ y'' + p(t) y' + q(t) y = g(t) $$ -The equivilent homogenous differential equation is +The equivalent homogenous differential equation is $$ y'' + p(t) y' + q(t) y = 0 $$ diff --git a/docs/math/diffeq/4-laplace.md b/docs/math/diffeq/4-laplace.md index 66559af..d5a220f 100644 --- a/docs/math/diffeq/4-laplace.md +++ b/docs/math/diffeq/4-laplace.md @@ -60,7 +60,7 @@ $$ This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/IVPWithLaplace.aspx). -**Theorum**. Given a function $f(t)$ with $C^n$ continuity, then +**Theorem**. Given a function $f(t)$ with $C^n$ continuity, then $$ \mathcal{L} \{ f^{(n)} (t) \} = s^n F(s) - s^{n-1} f(0) - s^{n-2} f'(0) - \ldots - s f^{(n-2)} (0) - f^{(n-1)} (0) @@ -77,6 +77,6 @@ $$ We can take the Laplace transformation of an IVP, solve for $Y(s)$, then take the inverse to find the solution. -## Section 4.6 - Nonconstant Coefficient IVPs +## Section 4.6 - Non-constant Coefficient IVPs This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/IVPWithNonConstantCoefficient.aspx). diff --git a/docs/math/real-analysis/2-reals.md b/docs/math/real-analysis/2-reals.md index ff99b33..e8a491f 100644 --- a/docs/math/real-analysis/2-reals.md +++ b/docs/math/real-analysis/2-reals.md @@ -56,7 +56,7 @@ 2. $|ab| = |a||b|$ 3. $|a + b| \leq |a| + |b|$ -**Corollary**. Givem $a, b \in \mathbb{R}$, then $\abs{\abs{a} - \abs{b}} \leq \abs{a - b}$. +**Corollary**. Given $a, b \in \mathbb{R}$, then $\abs{\abs{a} - \abs{b}} \leq \abs{a - b}$. **Remark**. Every field has at least one absolute value function. @@ -86,7 +86,7 @@ $$ --- -**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 +**Decimals**. Let $x \in \mathbb{R}$ such that $x > 0$. By the archimedean 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 $$ x = b_0 + \frac{b_1}{10} + \frac{b_2}{100} + \ldots + \frac{b_n}{100^n} + \ldots diff --git a/docs/math/real-analysis/3-sequences-series.md b/docs/math/real-analysis/3-sequences-series.md index d0cf29a..a3871e3 100644 --- a/docs/math/real-analysis/3-sequences-series.md +++ b/docs/math/real-analysis/3-sequences-series.md @@ -16,7 +16,7 @@ $$ (x_n) = (a, ar, ar^2, ar^3, \ldots) $$ -**Example**. The *arithmatic sequence*, given base $a \in \mathbb{R}$ and distance $d \in \mathbb{R}$, +**Example**. The *arithmetic sequence*, given base $a \in \mathbb{R}$ and distance $d \in \mathbb{R}$, $$ (x_n) = (a, a + d, a + 2d, a + 3d, \ldots) @@ -52,7 +52,7 @@ $$ 2. $x_n \cdot y_n \rightarrow xy$ 3. If $x_n \neq 0$ for all $n$, then $\frac{1}{x_n} \rightarrow \frac{1}{x}$ -**Theorem**. Suppose $(x_n)$ aand $(y_n)$ are convergent sequences and $N \in \mathbb{N}$. Then, +**Theorem**. Suppose $(x_n)$ and $(y_n)$ are convergent sequences and $N \in \mathbb{N}$. Then, 1. If $x_n \leq y_n$ for all $n \geq N$, then $\lim(x_n) \leq \lim(y_n)$ 2. If $x_n \leq a$ for all $n \geq N$, then $\lim(x_n) \leq a$ @@ -110,7 +110,7 @@ $$ ## Section 3.7 - Series -**Definition**. Let $(x_n)$ be a sequence in $\mathbb{R}$. Then, the *infinite series genearted by $X$* is the sequence $S = (s_n)$ with terms +**Definition**. Let $(x_n)$ be a sequence in $\mathbb{R}$. Then, the *infinite series generated by $X$* is the sequence $S = (s_n)$ with terms $$ s_1 = x_1; \; s_{n+1} = s_n + x_{n+1} diff --git a/docs/math/real-analysis/4-limits.md b/docs/math/real-analysis/4-limits.md index 60ec00e..0285081 100644 --- a/docs/math/real-analysis/4-limits.md +++ b/docs/math/real-analysis/4-limits.md @@ -1,12 +1,12 @@ # Chapter 4 - Limits -## Secrion 4.1 - Limits of Functions +## Section 4.1 - Limits of Functions -**Definition**. Let $A \subseteq \mathbb{R}$. Then, a point $c \in \mathbb{R}$ is a *cluster point* of $A$ if for every $\delta > 0$, the $\delta$-neighborhood of $c$ contains a point $a \in A$ such thhat $a \neq c$. That is, there exists some $a$ such that $0 < |a - c| < \delta$. +**Definition**. Let $A \subseteq \mathbb{R}$. Then, a point $c \in \mathbb{R}$ is a *cluster point* of $A$ if for every $\delta > 0$, the $\delta$-neighborhood of $c$ contains a point $a \in A$ such that $a \neq c$. That is, there exists some $a$ such that $0 < |a - c| < \delta$. **Theorem**. A real number $c$ is a cluster point for a set $A$ if and only if there exists a sequence $(a_n)$ in $A\\ \{c\}$ such that $a_n \rightarrow c$ -**Corollary**. A real number $c$ is a cluster point of a set $A$ if and only if every $\delta$-neighborhood conttains infinitely many points of $A$. +**Corollary**. A real number $c$ is a cluster point of a set $A$ if and only if every $\delta$-neighborhood contains infinitely many points of $A$. **Definition**. The set of every cluster point of $A$ is called the *derived set* of $A$, and denoted $A'$. @@ -18,21 +18,21 @@ --- -**Definition**. Suppose $f: A \rightarrow \mathbb{R}$ is a function with domain $A \subseteq \mathbb{R}$, and let $c \in A$ be a cluster point of $A$. then, a real number $L$ is a *limit of $f$ at $c$* if goven any $\epsilon > 0$, there exists some $\delta > 0$ such that +**Definition**. Suppose $f: A \rightarrow \mathbb{R}$ is a function with domain $A \subseteq \mathbb{R}$, and let $c \in A$ be a cluster point of $A$. then, a real number $L$ is a *limit of $f$ at $c$* if given any $\epsilon > 0$, there exists some $\delta > 0$ such that $$ 0 < |x-c| < \delta \Rightarrow |f(x) - L| < \epsilon $$ -**Therorem**. For a given function and cluster point, there can be at most one limit at said point. +**Theorem**. For a given function and cluster point, there can be at most one limit at said point. **Theorem**. Let $A \subseteq \mathbb{R}$ and $f: A \rightarrow \mathbb{R}$. Then, to show that $lim_{x \rightarrow c} f(x) = L$, it suffices to show that for every sequence $(a_n)$ in $A\\ \{c\}$, the sequence $(f(a_n))$ converges tto $L$. --- -**Definition**. The *extended real numbers* are $\hat{\mathbb{R}} = \mathbb{R} \cup \{ \infty, -\infty \}$ are a totally-ordered set witth supremum and infimum. Note that this set is no longer a field. +**Definition**. The *extended real numbers* are $\hat{\mathbb{R}} = \mathbb{R} \cup \{ \infty, -\infty \}$ are a totally-ordered set with supremum and infimum. Note that this set is no longer a field. -**Definition**. At any point $c$, the limitt of $f$ at $c$ is infinite if given some $\alpha$, there exists some $V_\delta(c)$ such that forr all $x \in V_\epsilon(c)$, then $f(x) \in V_\alpha(\infty)$. +**Definition**. At any point $c$, the limit of $f$ at $c$ is infinite if given some $\alpha$, there exists some $V_\delta(c)$ such that for all $x \in V_\epsilon(c)$, then $f(x) \in V_\alpha(\infty)$. **Definition**. The limit of a function at infinity is defined if for a given $\epsilon$, there exists some $\alpha$ so that there exists some $V_\delta(c)$ such that for all $x \in A$, @@ -78,4 +78,4 @@ $$ f(x) \leq g(x) \leq h(x) \; \text{ for all } x \in A, x \neq c $$ -Then, $\lim_{x \rightarrow c} g(x) = L$. \ No newline at end of file +Then, $\lim_{x \rightarrow c} g(x) = L$. diff --git a/docs/math/real-analysis/5-continuity.md b/docs/math/real-analysis/5-continuity.md index 21f40c7..ee28b1c 100644 --- a/docs/math/real-analysis/5-continuity.md +++ b/docs/math/real-analysis/5-continuity.md @@ -1,4 +1,4 @@ -# Chapter 5 - Continuiy +# Chapter 5 - Continuity ## Section 5.1 - Continuous Functions @@ -8,7 +8,7 @@ $$ |x - a| < \delta \Rightarrow |f(x) - f(a)| < \epsilon $$ -Note that if $a$ is an *isolateed point* of $A$, that is, not a cluster point, then $a$ is automatically continuous. +Note that if $a$ is an *isolated point* of $A$, that is, not a cluster point, then $a$ is automatically continuous. If $a$ is a cluster point of $A$, then this definition collapses to the definition of $\lim_{x \rightarrow a} f(x) = f(a)$. @@ -20,17 +20,17 @@ Note that a function cannot be continuous at a point outside of its domain, even --- -**Definition**. Let $(S, d_S)$ and $(T, d_T)$ be metric spaces. A function $f: S \rightarrow T$ is continuous at a point $a \in S$ if given any $\epsilon > 0$, there exists some $\delta > 0$ such thaat for all $x \in S$, +**Definition**. Let $(S, d_S)$ and $(T, d_T)$ be metric spaces. A function $f: S \rightarrow T$ is continuous at a point $a \in S$ if given any $\epsilon > 0$, there exists some $\delta > 0$ such that for all $x \in S$, $$ d_S(x, a) < \delta \Rightarrow d_T(f(x), f(a)) < \epsilon $$ -**Theorem**. A function $f: S \rightarrow T$ is continuous at a point $a \in A$ if and only if given some neighborhood $V(f(a)) \in B$, there xists some $U(a) \in A$ such that $f(U) \subseteq V$. +**Theorem**. A function $f: S \rightarrow T$ is continuous at a point $a \in A$ if and only if given some neighborhood $V(f(a)) \in B$, there exists some $U(a) \in A$ such that $f(U) \subseteq V$. -## Section 5.2 - Combinations of continuous Functions. +## Section 5.2 - Combinations of continuous Functions -**Theorem**. Let $f, g: A \rightarrow \mathbb{R}$ be continuous at $a \in A$. Then, +**Theorem**. Let $f, g: A \rightarrow \mathbb{R}$ be continuous at $a \in A$. Then, - $f + g$ and $fg$ are continuous at $a$ - If $g(x) \neq 0$ for all $x \in A$, then $\frac{f}{g}$ is continuous at $a$. @@ -51,15 +51,15 @@ $$ **Corollary**. Let $f: A \rightarrow \mathbb{R}$ be a continuous function, with $A$ being a compact subset of metric space $S$. Then, $f(A)$ is closed and bounded. Moreover, there exists a $p, q \in A$ such that $f(p)$ and $f(q)$ are the supremum and infimum of $f(A)$. -**Corollary**. Maximum-Minimum Theorem. If $I = [a, b]$ is a closed and bounded interval and $f: I \rightarrow \mathbb{R}$ is continuous on $I$, then $f$ has an absolute minumum and maximum on $I$. +**Corollary**. Maximum-Minimum Theorem. If $I = [a, b]$ is a closed and bounded interval and $f: I \rightarrow \mathbb{R}$ is continuous on $I$, then $f$ has an absolute minimum and maximum on $I$. --- **Theorem**. Let $S, T$ be metric spaces and $A \subseteq S$. Then, if $f: A \rightarrow T$ is continuous on $A$, and $A$ is a connected subset of $S$, then $f(A)$ is a connected subset of $T$. -**Corollary**. Suppose that $I$ is an interval. Let $f: I \rightarrow \mathbb{R}$ be continuous on $I$. Then, $f(I)$ is an intterval. +**Corollary**. Suppose that $I$ is an interval. Let $f: I \rightarrow \mathbb{R}$ be continuous on $I$. Then, $f(I)$ is an interval. -**Theorem**. (Bolzano's) Invermediate Value Theorem. Suppose $f: [a, b] \rightarrow \mathbb{R}$ is continuous on $[a, b]$ with $a \neq b$. Then, given some $k$ such that $f(a) < k < f(b)$, there exists some $c \in (a, b)$ such that $k = f(c)$. +**Theorem**. (Bolzano's) Intermediate Value Theorem. Suppose $f: [a, b] \rightarrow \mathbb{R}$ is continuous on $[a, b]$ with $a \neq b$. Then, given some $k$ such that $f(a) < k < f(b)$, there exists some $c \in (a, b)$ such that $k = f(c)$. --- @@ -77,4 +77,4 @@ Note that if $f$ is uniformly continuous, it must be continuous on $A$. **Theorem**. Suppose $A \subseteq \mathbb{R}$ and $f: A \rightarrow \mathbb{R}$ is uniformly continuous. Then, if $(x_n)$ is a Cauchy sequence in $A$, $(f(x_n))$ is a Cauchy sequence in $\mathbb{R}$. -**Remark**. Suppose $S, T$ are metric spaces and $f: S \rightarrow T$ is uniformly continuous. Then, if $(x_n)$ is a Cauchy sequence in $S$, $(f(x_n))$ is a Cauchy sequence in $T$. \ No newline at end of file +**Remark**. Suppose $S, T$ are metric spaces and $f: S \rightarrow T$ is uniformly continuous. Then, if $(x_n)$ is a Cauchy sequence in $S$, $(f(x_n))$ is a Cauchy sequence in $T$. diff --git a/docs/physics/electrostatics/2-coulomb.md b/docs/physics/electrostatics/2-coulomb.md index 19c9c48..6d936ba 100644 --- a/docs/physics/electrostatics/2-coulomb.md +++ b/docs/physics/electrostatics/2-coulomb.md @@ -98,11 +98,11 @@ $$ We can verify that $\curl{(\vb{r}-\vb{r'})} = 0$, cancelling the first term. Additionally, $\curl{\frac{1}{\abs{\vb{r}-\vb{r'}}^3}} = -3 \frac{\vb{r}-\vb{r'}}{\abs{\vb{r}-\vb{r'}}^5}$, which when crossed with $\vb{r}-\vb{r'}$, will cancel. Thus, all terms in the curl cancel, so for a static field, the curl is zero. -## Section 2.4 - Eletric and Magnetic Flux Densities +## Section 2.4 - Electric and Magnetic Flux Densities The electric and magnetic flux density vectors are given by $\epsilon_0 \vb{E}$ and $\mu_0 \vb{H}$. -Now, given $S$ is a surfance enclosing $Q_e$ or $Q_m$ total charge, we denotate flux as following: +Now, given $S$ is a surface enclosing $Q_e$ or $Q_m$ total charge, we denote flux as following: $$ \Phi_e = \epsilon_0 \int_S \vb{E} \vdot \vu{n} \dd = Q_e S \text{ or } \Phi_m = \mu_0 \int_S \vb{H} \vdot \vu{n} \dd S = Q_m @@ -131,4 +131,4 @@ $$ **Definition**. This is known as *Gauss' Law*. -With applicable symnetry, the integral factor becomes simply $E(r)*A$, where $A$ is the area of the surface at $r$. +With applicable symmetry, the integral factor becomes simply $E(r)*A$, where $A$ is the area of the surface at $r$. diff --git a/docs/physics/electrostatics/3-electro-magnetic-potentials.md b/docs/physics/electrostatics/3-electro-magnetic-potentials.md index bc5474c..1818561 100644 --- a/docs/physics/electrostatics/3-electro-magnetic-potentials.md +++ b/docs/physics/electrostatics/3-electro-magnetic-potentials.md @@ -14,7 +14,7 @@ $$ \vb{E}(\vb{r}) = \frac{1}{q_e} \vb{F_e}(\vb{r}) = - \frac{1}{q_e} \grad{U_e(\vb{r})} = -\grad{V_e(\vb{r})} $$ -The units of electrostatic potential is Joule/Coublomb, also known as a Volt. Thus, the units of the electric field should be expressed in Volts/meter. Similarly, +The units of electrostatic potential is Joule/Coulomb, also known as a Volt. Thus, the units of the electric field should be expressed in Volts/meter. Similarly, $$ \vb{H}(\vb{r}) = \frac{1}{q_m} \vb{F_m}(\vb{r}) = - \frac{1}{q_m} \grad{U_m(\vb{r})} = -\grad{V_m(\vb{r})} @@ -80,7 +80,7 @@ $$ W_n = \frac{1}{2} \frac{4 \pi \epsilon_0} \sum_{i = 1}^{N} \sum_{j > i}^{N} \frac{Q_{ei}Q_{ej}}{\abs{\vb{r_i}-\vb{r_j}}} $$ -For the sake of symnetry, sum overall charges and divide by 2. +For the sake of symmetry, sum overall charges and divide by 2. $$ W_n = \frac{1}{2} \frac{1}{4 \pi \epsilon_0} \sum_{i = 1}^{N} \sum_{j \neq i}^{N} \frac{Q_{ei}Q_{ej}}{\abs{\vb{r_i}-\vb{r_j}}} @@ -93,7 +93,7 @@ W_n = \frac{1}{2} \sum_{i = 1}^{N}Q_{ei} \sum_{i \neq j}^{N} \frac{1}{4 \pi \eps = \frac{1}{2}\sum_{i = 1}^{N} Q_{ei} V(\vb{r_i}) $$ -We can rewrite this as a Reimann sum and convert to an integral. +We can rewrite this as a Riemann sum and convert to an integral. $$ W_e = \frac{1}{2} \int_V p_e(\vb{r}) V_e(\vb{r}) \dd V ; \quad @@ -129,7 +129,7 @@ $$ We know that $\vb{E}(\vb{r}) = -\div{V_e(\vb{r})}$ and $\vb{H}(\vb{r}) = -\div{V_m(\vb{r})}$ -Combinind this, as well as the first of the Maxwell equations, we see that +Combined this, as well as the first of the Maxwell equations, we see that $$ \div{\vb{E}} = -\div{\grad{V_e}} = - \laplacian{V_e} = \frac{\rho_e}{\epsilon_0} @@ -139,7 +139,7 @@ $$ \div{\vb{H}} = -\div{\grad{V_m}} = - \laplacian{V_m} = \frac{\rho_m}{\mu_0} $$ -The last inequatlity is called the Poisson Equation, or the inhomogenous Laplace equation. +The last inequality is called the Poisson Equation, or the inhomogeneous Laplace equation. To solve this equation, we define a Green function as follows: @@ -147,7 +147,7 @@ $$ \laplacian G(\vb{r}, \vb{r'}) = \delta(\vb{r} - \vb{r'}) $$ -Now, we can construct a potential function in terms of said green function that satisfies the lapalce equation. +Now, we can construct a potential function in terms of said green function that satisfies the Laplace equation. $$ V_e(\vb{r}) = - \int_V G(\vb{r}, \vb{r'}) \frac{\rho_e(\vb{r'})}{\epsilon_0} \dd{V'} @@ -273,4 +273,4 @@ $$ Note that as a quirk of the function, $P_n(1) = 1$ for all $n$. -We can apply these quadrupole and beyond terms to the volate or other equations, however, this becomes very messy. +We can apply these quadrupole and beyond terms to the violate or other equations, however, this becomes very messy. diff --git a/docs/physics/electrostatics/4-conductors.md b/docs/physics/electrostatics/4-conductors.md index 4e1d614..70cdba9 100644 --- a/docs/physics/electrostatics/4-conductors.md +++ b/docs/physics/electrostatics/4-conductors.md @@ -6,23 +6,23 @@ We will focus primarily on electric fields and charges. For the purposes for thi ## Section 4.2 - Electrostatic Properties of a Conductor -In a metal or conductor, there are plentiful charges not bound to a particular atom and are thus free to move throughought the material. +In a metal or conductor, there are plentiful charges not bound to a particular atom and are thus free to move throughout the material. We note that there is no electric fiend inside a conductor, as charges internal to the material would move under the force it generates until they find a configuration that eliminates the field. This may happen, but not in electrostatics. -Additionally, as the field is zero, it follows from Maxwell's equations that there is no charge inside a conductor. However, charge may be present at the surface. For sufficiently symnetric charges, this charge may be calculated. +Additionally, as the field is zero, it follows from Maxwell's equations that there is no charge inside a conductor. However, charge may be present at the surface. For sufficiently symmetric charges, this charge may be calculated. -Consider any two points internal to the conductor. The voltage between said points is defined as $\int_A^B \vb{E} \vdot \dd{\vb{l}}$. Since $\vb{E} = 0$ inside the conductor, the volage difference must be zero. Thus, any two points in or on the surface (TODO: Why on the surface?) of a conductor must be at the same potential. +Consider any two points internal to the conductor. The voltage between said points is defined as $\int_A^B \vb{E} \vdot \dd{\vb{l}}$. Since $\vb{E} = 0$ inside the conductor, the voltage difference must be zero. Thus, any two points in or on the surface (TODO: Why on the surface?) of a conductor must be at the same potential. -The electric field at the surface of a conductor is perpendicular to its surface. Consider some displacement $\dd{\vb{l}}$. Now, $\vb{E} \vdot \dd{\vb{l}} = \vb{E}_s \vdot \dd{\vb{l}}_s + \vb{E}_p \vdot \dd{\vb{l}}_p = \dd{V_s} + \dd{V_p}$, in terms of parallel and perpendicular components. The paralell voltage difference is zero, so the electric field must be zero. +The electric field at the surface of a conductor is perpendicular to its surface. Consider some displacement $\dd{\vb{l}}$. Now, $\vb{E} \vdot \dd{\vb{l}} = \vb{E}_s \vdot \dd{\vb{l}}_s + \vb{E}_p \vdot \dd{\vb{l}}_p = \dd{V_s} + \dd{V_p}$, in terms of parallel and perpendicular components. The parallel voltage difference is zero, so the electric field must be zero. -Consider the surface of a conductor with surface charge density $\sigma_e$. A cyliner with one end inside and one end outside said surface, with its axis normal to said surface, will be a Gaussian "pillbox", which will show that with V being the volume of the pillbox, $\int_V \div{\vb{E}} \dd{V} = \frac{Q_e}{\epsilon_0} = \frac{A\sigma_e}{\epsilon_0}$. Thus, $\sigma_e = \epsilon_0 E$. +Consider the surface of a conductor with surface charge density $\sigma_e$. A cylinder with one end inside and one end outside said surface, with its axis normal to said surface, will be a Gaussian "pillbox", which will show that with V being the volume of the pillbox, $\int_V \div{\vb{E}} \dd{V} = \frac{Q_e}{\epsilon_0} = \frac{A\sigma_e}{\epsilon_0}$. Thus, $\sigma_e = \epsilon_0 E$. ## Section 4.3 - Exercises involving conductors at fixed potentials 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) = \sum_{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 symmetry, $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. @@ -79,4 +79,4 @@ This unit, $\frac{C}{V}$, is known as a Farad. For a sphere, $C = 4 \pi \epsilon ## Section 4.7 - Forces on Charged Conductors in Electric Fields -We know that $\vb{F} = \int \vb{E}_{ext}(\vb{r}) \rho_e(\vb{r}) dV$, where $\vb{E}_{ext}(\vb{r})$ is the external electric field and $\rho_e(\vb{r})$ is the charge density of the object. \ No newline at end of file +We know that $\vb{F} = \int \vb{E}_{ext}(\vb{r}) \rho_e(\vb{r}) dV$, where $\vb{E}_{ext}(\vb{r})$ is the external electric field and $\rho_e(\vb{r})$ is the charge density of the object. diff --git a/docs/physics/electrostatics/5-moving-charges.md b/docs/physics/electrostatics/5-moving-charges.md index 0b8da7f..9c23a85 100644 --- a/docs/physics/electrostatics/5-moving-charges.md +++ b/docs/physics/electrostatics/5-moving-charges.md @@ -1,6 +1,6 @@ # Chapter 5 - Electrodynamics with Moving Charges -## Section 5.1 - Currents in Steady-State Regine +## Section 5.1 - Currents in Steady-State Regime We want to work in a steady-state system. Thus, we restrict ourselves to currents that do not change in time. @@ -24,7 +24,7 @@ If we assume cylindrical coordinates and that $\vb{H}(vb{r}) = H_\varphi(s) \vu{ --- -By Helmholtz Theorem, we know that $\vb{H}(\vb{r}) = \curl{\vb{A}(\vb{r})}$. For a current-carying wire, $\vb{A}(\vb{r}) = \frac{I_e}{4\pi} \int_{\text{wire}} \frac{\dd{\vb{l'}}}{|\vb{r}-\vb{r'}|}$. Applying identities, we see the *Law of Biot and Savart$, where +By Helmholtz Theorem, we know that $\vb{H}(\vb{r}) = \curl{\vb{A}(\vb{r})}$. For a current-carrying wire, $\vb{A}(\vb{r}) = \frac{I_e}{4\pi} \int_{\text{wire}} \frac{\dd{\vb{l'}}}{|\vb{r}-\vb{r'}|}$. Applying identities, we see the *Law of Biot and Savart$, where $$ \vb{H}(\vb{r}) = \int{I_e}{4\pi}\int_{\text{wire}} \frac{-(\vb{r}-\vb{r'}) \cross \dd{\vb{l'}}}{|\vb{r}-\vb{r'}|^3} @@ -60,7 +60,7 @@ $$ Consider an electric charge moving with velocity $\vb{v}$ in a magnetic parallel plate capacitor with charge densities $\plusminus \sigma_m$. That is, $\mu_0 \vb{H} = \sigma_m \vu{z}$. Then, we can apply theorems to see the resulting force. -**Theorem**. *Lorentz Force Law* states that $\vb{F} = q_e \vb{v} \cross \u_0 \vb{H}$ in the presence of a magnetic field. In the presence of both an electic andmagnetic field, $\vb{F} = q_e (\vb{E} + \vb{v} \cross \u_0 \vb{H})$. +**Theorem**. *Lorentz Force Law* states that $\vb{F} = q_e \vb{v} \cross \u_0 \vb{H}$ in the presence of a magnetic field. In the presence of both an electric and magnetic field, $\vb{F} = q_e (\vb{E} + \vb{v} \cross \u_0 \vb{H})$. **Theorem**. *Ampere's Force Law* states that generalizing the previous theorem, we can see that @@ -70,4 +70,4 @@ $$ ## Section 5.4 - Multipole Expansion of a Vector Potential -This is messy. Skipped. \ No newline at end of file +This is messy. Skipped. diff --git a/docs/recipes/cookies.md b/docs/recipes/cookies.md index 43aa3fa..617d6bd 100644 --- a/docs/recipes/cookies.md +++ b/docs/recipes/cookies.md @@ -9,7 +9,7 @@ Original: [Link](https://www.allrecipes.com/recipe/10813/best-chocolate-chip-coo - In a medium bowl, mix together $\frac{1}{2}$ cup butter, $\frac{1}{2}$ cup sugar, $\frac{1}{2}$ cup brown sugar. - To the wet ingredients, add 1 egg and 1 tsp. vanilla. - To the wet ingredients, add a combination of 1 tsp. hot water and $\frac{1}{2}$ tsp. baking soda. -- Mix the wet and dry ingredients. Stir in 1 cup chocolate chips and optionallt $\frac{1}{2}$ cup walnuts. +- Mix the wet and dry ingredients. Stir in 1 cup chocolate chips and optionally $\frac{1}{2}$ cup walnuts. - Bake for 10m at $350 \degree$ F. ## Peanut Butter Cookies @@ -36,7 +36,7 @@ Original: [Link](https://www.allrecipes.com/recipe/10813/best-chocolate-chip-coo ## Snickerdoodle Cookies - Preheat oven to $400 \degree$ F. -- Blend together $\frac{3}{4}$ cup sugar, $\frac{1}{4}$ cup butter (softened), $\frac{1}{4} cup shortening. +- Blend together $\frac{3}{4}$ cup sugar, $\frac{1}{4}$ cup butter (softened), $\frac{1}{4}$ cup shortening. - Add 1 egg, 1 tsp. vanilla - In a separate bowl, combine $1 \frac{1}{4}$ cups flour, 1 tsp. cream of tartar, $\frac{1}{2}$ tsp. baking soda, and $\frac{1}{4}$ tsp. salt - Combine diff --git a/docs/recipes/cupcakes.md b/docs/recipes/cupcakes.md index 2bf21a4..b1d7c12 100644 --- a/docs/recipes/cupcakes.md +++ b/docs/recipes/cupcakes.md @@ -4,9 +4,9 @@ - Preheat oven to $350 \degree$ F - Combine $1 \frac{1}{4}$ cups flour, $\frac{3}{4}$ tsp. baking powder, $\frac{1}{4}$ tsp. baking soda. -- In a separate bowl, combine $\frac{1}{4}$ cup butter, $\frac{3}{4}$ cup sugar, $\frac{1}{4}$ cup vegetable oil, $\frac{1}{4}$ tsp. vaninna +- In a separate bowl, combine $\frac{1}{4}$ cup butter, $\frac{3}{4}$ cup sugar, $\frac{1}{4}$ cup vegetable oil, $\frac{1}{4}$ tsp. vanilla - Add 2 eggs. -- Combine with half of the dry mixture. Add 6 tbsp. milk, 1 lemon of zest and juce. Stir. Add the rest of the dry ingredients. +- Combine with half of the dry mixture. Add 6 tbsp. milk, 1 lemon of zest and juice. Stir. Add the rest of the dry ingredients. - Bake for 15-18 minutes at $350 \degree$ F. ### Frosting diff --git a/docs/recipes/meals.md b/docs/recipes/meals.md index 8e5cce8..289cfee 100644 --- a/docs/recipes/meals.md +++ b/docs/recipes/meals.md @@ -20,4 +20,4 @@ Source: [link](https://www.cooking-therapy.com/banh-bao/). - Knead on floured surface for $\approx 10$ minutes. - Transfer to an oiled bowl, cover, and wait for approx. 1 hour -- Bring water to a simmer, add $1$ tsp. rice wine vinegar. Steam for $\approx 15-17$ minutes. \ No newline at end of file +- Bring water to a simmer, add $1$ tsp. rice wine vinegar. Steam for $\approx 15-17$ minutes. diff --git a/docs/recipes/pies.md b/docs/recipes/pies.md index 559dad1..de287a0 100644 --- a/docs/recipes/pies.md +++ b/docs/recipes/pies.md @@ -4,13 +4,13 @@ - Whisk $1 \frac{1}{4}$ cups of flour with $\frac{1}{4}$ tsp. salt - Cut in $\frac{1}{2}$ cups of cubed butter (chilled), $\frac{1}{4}$ cups cold water. -- Refridgerate +- Refrigerate ## Pie Crust 2 - Mix $\frac{1}{3}$ cup flour, $\frac{1}{3}$ tsp. salt. - Cut in $\frac{1}{2}$ cups of shortening, 3 tbsp. cold water. -- Refridgerate +- Refrigerate ## Pumpkin Pie 1 @@ -46,7 +46,7 @@ From: [Link](https://www.allrecipes.com/recipe/234374/apple-hand-pies/) ### Hand Pies - Preheat oven to $400 \degree$ F. -- Split pie crust into 4. Place fillin in crust, fold. +- Split pie crust into 4. Place filling in crust, fold. - Sprinkle with $\frac{1}{4}$ tsp. white sugar. - Whisk 2 tsp. milk, 1 egg. Brush pastries. - Bake at $400 \degree$ F. for 25-30 minutes. diff --git a/docs/recipes/snacks.md b/docs/recipes/snacks.md index 2d8ea01..b784608 100644 --- a/docs/recipes/snacks.md +++ b/docs/recipes/snacks.md @@ -16,11 +16,11 @@ ## Chocolate Fudge -- Melt 1 bag of chocolate chips on low. Stir in 1 can of sweeteneed condensed milk. Pour into buttered or oiled tray. +- Melt 1 bag of chocolate chips on low. Stir in 1 can of sweetened condensed milk. Pour into buttered or oiled tray. ## Peanut Butter Fudge - Melt $\frac{1}{2}$ cup of butter on medium heat. Stir in a 16oz bag of brown sugar, $\frac{1}{2}$ cup of milk. -- Remove from heat. Stir in $\frac{3}{4}$ cups of peanut butter and 1 tsp banilla. +- Remove from heat. Stir in $\frac{3}{4}$ cups of peanut butter and 1 tsp bvnilla. - Add to $3 \frac{1}{2}$ cups of powdered sugar - Pour into buttered or oiled tray.