From d0bebcf40be3fa09da52d2420a458930e8712ce2 Mon Sep 17 00:00:00 2001 From: Nathan Nguyen <159226326+Indigo5684@users.noreply.github.com> Date: Tue, 8 Oct 2024 21:17:36 -0500 Subject: [PATCH] Formatting Fix --- docs/math/abstract-algebra/16-rings.md | 3 ++- docs/math/abstract-algebra/17-polynomial-rings.md | 3 ++- docs/math/abstract-algebra/18-integral-domains.md | 2 +- docs/math/diffeq/1-intro.md | 2 +- docs/math/diffeq/2-1st-order.md | 2 +- docs/math/diffeq/3-2nd-order.md | 12 ++++++------ docs/math/diffeq/4-laplace.md | 3 ++- docs/physics/electrostatics/1-math.md | 1 + docs/physics/electrostatics/2-coulomb.md | 3 +-- .../electrostatics/3-electro-magnetic-potentials.md | 9 ++++----- docs/physics/electrostatics/4-conductors.md | 2 +- docs/recipes/bread.md | 4 ++-- docs/recipes/pies.md | 4 ++-- 13 files changed, 26 insertions(+), 24 deletions(-) diff --git a/docs/math/abstract-algebra/16-rings.md b/docs/math/abstract-algebra/16-rings.md index c14874a..2e50d19 100644 --- a/docs/math/abstract-algebra/16-rings.md +++ b/docs/math/abstract-algebra/16-rings.md @@ -1,4 +1,5 @@ # Chapter 16 - Rings + ## 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: @@ -139,4 +140,4 @@ $$ Thus, $ab \in P$. By symnetry, assume $a \notin P$. Thus, $b \in P$ by the devinition of a prime ideal, so $b + P = 0 + P$, meaning $R/P$ is an integral domain. -**Theorem**. 16.40: In a commutative ring with identity, every maximal ideal is also a prime ideal. \ No newline at end of file +**Theorem**. 16.40: In a commutative ring with identity, every maximal ideal is also a prime ideal. diff --git a/docs/math/abstract-algebra/17-polynomial-rings.md b/docs/math/abstract-algebra/17-polynomial-rings.md index f0afd84..c9c7c85 100644 --- a/docs/math/abstract-algebra/17-polynomial-rings.md +++ b/docs/math/abstract-algebra/17-polynomial-rings.md @@ -1,4 +1,5 @@ # Chapter 17 - Polynomial Rings + ## Section 17.1 - Polynomial Rings Throughout this chapter, we will assume that $R$ is a commutative ring with identity. @@ -63,4 +64,4 @@ Then, if $p | a_i$ for $0 \leq i < n$, but $p \nmid a_n$ and $p^2 \nmid a_0$, th **Theorem**. If $F$ is a field, then every ideal in $F[x]$ is a principal ideal. -**Theorem**. Let $F$ be a field, and suppose $p(x) \in F[x]$. Then, the ideal $$ is maximal if and only if $p(x)$ is irreducible. \ No newline at end of file +**Theorem**. Let $F$ be a field, and suppose $p(x) \in F[x]$. Then, the ideal $$ is maximal if and only if $p(x)$ is irreducible. diff --git a/docs/math/abstract-algebra/18-integral-domains.md b/docs/math/abstract-algebra/18-integral-domains.md index b16896d..5a292d4 100644 --- a/docs/math/abstract-algebra/18-integral-domains.md +++ b/docs/math/abstract-algebra/18-integral-domains.md @@ -98,4 +98,4 @@ As a direct consequence, we see the following. 1. Given a field $F$, since $F$ is a PID, it is also a UFD. Thus, $F[x]$ is a UFD. 2. The ring of polynomials over integers, $\mathbb{Z}[x]$ is a UFD. -3. Given $D$ is a UFD, $D[x]$ is a UFD. Thus, $D[x_1, x_2]$ is a UFD, and by induction, $D[x_1, \ldots, x_n]$ is a UFD. \ No newline at end of file +3. Given $D$ is a UFD, $D[x]$ is a UFD. Thus, $D[x_1, x_2]$ is a UFD, and by induction, $D[x_1, \ldots, x_n]$ is a UFD. diff --git a/docs/math/diffeq/1-intro.md b/docs/math/diffeq/1-intro.md index 5e8b93d..6c7e36e 100644 --- a/docs/math/diffeq/1-intro.md +++ b/docs/math/diffeq/1-intro.md @@ -36,4 +36,4 @@ Note that $a_n(t)$ does not depeond on any derivative of $y$, so the presence of This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/DirectionFields.aspx). -**Definition**. A directional field is the graph of a $t$ vs. $y(t)$, with vectors drawn at each point with a slope corresponding to $y'(t)$. Notably, each arrow will be pointed right (towards increasing $t$). \ No newline at end of file +**Definition**. A directional field is the graph of a $t$ vs. $y(t)$, with vectors drawn at each point with a slope corresponding to $y'(t)$. Notably, each arrow will be pointed right (towards increasing $t$). diff --git a/docs/math/diffeq/2-1st-order.md b/docs/math/diffeq/2-1st-order.md index c64971b..9cd6c85 100644 --- a/docs/math/diffeq/2-1st-order.md +++ b/docs/math/diffeq/2-1st-order.md @@ -144,4 +144,4 @@ $$ \frac{1}{1-n}v' + p(x)v = q(x) $$ -After solving, be sure to rewrite in terms of $y$. \ No newline at end of file +After solving, be sure to rewrite in terms of $y$. diff --git a/docs/math/diffeq/3-2nd-order.md b/docs/math/diffeq/3-2nd-order.md index 4d895dd..f225710 100644 --- a/docs/math/diffeq/3-2nd-order.md +++ b/docs/math/diffeq/3-2nd-order.md @@ -32,7 +32,7 @@ Thus, we allow the *charactaristic equation* of the differential equation to be $$ ar^2 + br + c = 0 $$ -# Section 3.2 - Real & Distinct Roots +## Section 3.2 - Real & Distinct Roots This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/RealRoots.aspx). @@ -42,11 +42,11 @@ $$ y_1(t) = e^{r_1 t} $$ $$ y_2(t) = e^{r_2 t} $$ -Thus, +Thus, $$ y(t) = c_1 e^{r_1 t} + c_2 e^{r_2 t} $$ -# Section 3.3 - Complex Roots +## Section 3.3 - Complex Roots This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/ComplexRoots.aspx). @@ -120,7 +120,7 @@ Assume we have the differential equation as follows: $$ y'' + p(t) y' + q(t) y = g(t) $$ -The equivilent homogenous differential equation is +The equivilent homogenous differential equation is $$ y'' + p(t) y' + q(t) y = 0 $$ @@ -156,7 +156,7 @@ Assume we have the differential equation as follows: $$ y'' + p(t) y' + q(t) y = g(t) $$ -The equivilent homogenous differential equation is +The equivilent homogenous differential equation is $$ y'' + p(t) y' + q(t) y = 0 $$ @@ -164,4 +164,4 @@ For this method, we must have $y_1(t)$ and $y_2(t)$ known. Through a lot of math $$ y_p = -y_1 \int \frac{y_2(t)g(t)}{W(y_1, y_2)} dt + y_2 \int \frac{y_1(t)g(t)}{W(y_1, y_2)} dt -$$ \ No newline at end of file +$$ diff --git a/docs/math/diffeq/4-laplace.md b/docs/math/diffeq/4-laplace.md index c62be78..66559af 100644 --- a/docs/math/diffeq/4-laplace.md +++ b/docs/math/diffeq/4-laplace.md @@ -1,4 +1,5 @@ # Section 4 - Laplace Transformations + ## Section 4.1 - Definition This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/LaplaceDefinition.aspx). @@ -78,4 +79,4 @@ We can take the Laplace transformation of an IVP, solve for $Y(s)$, then take th ## Section 4.6 - Nonconstant Coefficient IVPs -This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/IVPWithNonConstantCoefficient.aspx). \ No newline at end of file +This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/IVPWithNonConstantCoefficient.aspx). diff --git a/docs/physics/electrostatics/1-math.md b/docs/physics/electrostatics/1-math.md index 48047f6..fffd694 100644 --- a/docs/physics/electrostatics/1-math.md +++ b/docs/physics/electrostatics/1-math.md @@ -1,4 +1,5 @@ # Chapter 1 - Mathematics + ## 1.5 - Dyads and Tensors **Definition**. A *dyadic* is a representation of two-ish vectors. diff --git a/docs/physics/electrostatics/2-coulomb.md b/docs/physics/electrostatics/2-coulomb.md index dee2dd4..19c9c48 100644 --- a/docs/physics/electrostatics/2-coulomb.md +++ b/docs/physics/electrostatics/2-coulomb.md @@ -81,7 +81,6 @@ $$ The curl of an electrostatic or magnetostatic is relatively simple. - $$ \begin{align} \curl{E(\vb{r})} &= \frac{1}{4 \pi \epsilon_0} \int_V \rho_e(\vb{r'}) \curl{(\frac{\vb{r}-\vb{r'}}{\abs{\vb{r}-\vb{r'}}^3})} \dd V' \\ @@ -132,4 +131,4 @@ $$ **Definition**. This is known as *Gauss' Law*. -With applicable symnetry, the integral factor becomes simply $E(r)*A$, where $A$ is the area of the surface at $r$. \ No newline at end of file +With applicable symnetry, the integral factor becomes simply $E(r)*A$, where $A$ is the area of the surface at $r$. diff --git a/docs/physics/electrostatics/3-electro-magnetic-potentials.md b/docs/physics/electrostatics/3-electro-magnetic-potentials.md index fecfa1f..bc5474c 100644 --- a/docs/physics/electrostatics/3-electro-magnetic-potentials.md +++ b/docs/physics/electrostatics/3-electro-magnetic-potentials.md @@ -25,7 +25,7 @@ The units of magnetostatic potential is Joule/Weber, also known as an Ampere. Th With this, we can calculate work. Moving a charge $q$ from $A$ to $B$, we see that $$ -\delta W = \int_A^B \vb{F} \vdot \dd{\vb{l}} = q_e \int_A^B \vb{E} \vdot \dd{\vb{l}} = -q_e \int_A^B \grad{\vb{V}} \vdot \dd{\vb{l}} = -q_e \delta V_e +\delta W = \int_A^B \vb{F} \vdot \dd{\vb{l}} = q_e \int_A^B \vb{E} \vdot \dd{\vb{l}} = -q_e \int_A^B \grad{\vb{V}} \vdot \dd{\vb{l}} = -q_e \delta V_e $$ Strictly speaking, this is a potential difference. To find the absolute potential, assume a point charge $Q$ at the origin, and a charge $q$. We take the work as $q$ moves from $\vb{r'} = \vb{\infty}$ to $\vb{r'} = \vb{r}$. Thus, @@ -62,7 +62,6 @@ $$ V_e(\vb{r}) = \frac{1}{4 \pi \epsilon_0} \int_{V'} \frac{\rho_e(\vb{r'})}{\abs{\vb{r}-\vb{r'}}} \dd{V'} $$ - $$ V_m(\vb{r}) = \frac{1}{4 \pi \mu_0} \int_{V'} \frac{\rho_m(\vb{r'})}{\abs{\vb{r}-\vb{r'}}} \dd{V'} $$ @@ -75,7 +74,7 @@ $$ W_2 = W_{21} = \frac{1}{4 \pi \epsilon_0} \frac{Q_{e1} Q_{e2}}{\abs{\vb{r_2} - \vb{r_1}}} $$ -Superposition applies here. The energy to create $N$ charges is +Superposition applies here. The energy to create $N$ charges is $$ 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}}} @@ -177,7 +176,7 @@ $$ Taking the divergence, we find that $$ -- \laplacian{V(\vb{r})} = G(\vb{r}, \vb{r'}) \frac{Q_e}{\epsilon_0} = \frac{Q_e}{\epsilon_0} \laplacian({\frac{-1}{4\pi \abs{\vb{r} - \vb{r'}}}}) +- \laplacian{V(\vb{r})} = G(\vb{r}, \vb{r'}) \frac{Q_e}{\epsilon_0} = \frac{Q_e}{\epsilon_0} \laplacian({\frac{-1}{4\pi \abs{\vb{r} - \vb{r'}}}}) = \frac{Q_e}{\epsilon_0} \div \frac{\vb{r} - \vb{r'}}{\abs{\vb{r} - \vb{r'}}^3} = \frac{Q_e}{\epsilon_0} \delta(\vb{r} - \vb{r'}) $$ @@ -274,4 +273,4 @@ $$ 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. \ No newline at end of file +We can apply these quadrupole and beyond terms to the volate 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 42b6af8..8b19ed6 100644 --- a/docs/physics/electrostatics/4-conductors.md +++ b/docs/physics/electrostatics/4-conductors.md @@ -16,4 +16,4 @@ Consider any two points internal to the conductor. The voltage between said poin 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. -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, $\sigma_e = \epsilon_0 E$. \ No newline at end of file +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, $\sigma_e = \epsilon_0 E$. diff --git a/docs/recipes/bread.md b/docs/recipes/bread.md index 7827625..d8c9865 100644 --- a/docs/recipes/bread.md +++ b/docs/recipes/bread.md @@ -1,9 +1,9 @@ # Breads -# Pumpkin Bread +## Pumpkin Bread - Preheat oven to $350 \degree$ F. - Combine $1 \frac{2}{3}$ cups flour, $1 \frac{1}{2}$ cups sugar, 1 tsp. baking soda, 1 tsp cinnamon, $\frac{3}{4}$ tsp. salt, $\frac{1}{2}$ tsp. baking powder, $\frac{1}{2}$ tsp. nutmeg, $\frac{1}{4}$ tsp cloves. - In a separate bowl, combine 2 eggs, 1 can of pumpkin, $\frac{1}{2}$ cup canola oil, and $\frac{1}{2}$ cups water. - Combine. Mix in $\frac{1}{2}$ cups of walnuts. -- Add to a greased 9x5 pan. Bake at $350 \degree$ F for 65-80 minutes. \ No newline at end of file +- Add to a greased 9x5 pan. Bake at $350 \degree$ F for 65-80 minutes. diff --git a/docs/recipes/pies.md b/docs/recipes/pies.md index 08f7710..559dad1 100644 --- a/docs/recipes/pies.md +++ b/docs/recipes/pies.md @@ -43,10 +43,10 @@ From: [Link](https://www.allrecipes.com/recipe/234374/apple-hand-pies/) - Mix in apples, add sugar mixture. - Wait until apples are softened (approx. 5 minutes). -### Pies +### Hand Pies - Preheat oven to $400 \degree$ F. - Split pie crust into 4. Place fillin 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. \ No newline at end of file +- Bake at $400 \degree$ F. for 25-30 minutes.