Formatting Fix

This commit is contained in:
Indigo5684
2025-09-30 13:19:25 -05:00
parent e1146e7b1c
commit a629d475d4
13 changed files with 26 additions and 24 deletions

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# 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:

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# Chapter 17 - Polynomial Rings
## Section 17.1 - Polynomial Rings
Throughout this chapter, we will assume that $R$ is a commutative ring with identity.

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@@ -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).
@@ -46,7 +46,7 @@ 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).

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# 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).

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# Chapter 1 - Mathematics
## 1.5 - Dyads and Tensors
**Definition**. A *dyadic* is a representation of two-ish vectors.

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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' \\

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@@ -62,7 +62,6 @@ $$
V_e(\vb{r}) = \frac{1}{4 \pi \epsilon_0} \int_{V'} \frac{\rho_e(\vb{r'})}{\abs{\vb{r}-\vb{r'}}} \dd{V'}
$$
$$
V_m(\vb{r}) = \frac{1}{4 \pi \mu_0} \int_{V'} \frac{\rho_m(\vb{r'})}{\abs{\vb{r}-\vb{r'}}} \dd{V'}
$$

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# 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.

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@@ -43,7 +43,7 @@ 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.