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# Chapter 16 - Rings
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# Chapter 16 - Rings
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## Section 16.1 - Rings
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## Section 16.1 - Rings
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**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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**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
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# Chapter 17 - Polynomial Rings
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## Section 17.1 - Polynomial Rings
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## Section 17.1 - Polynomial Rings
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Throughout this chapter, we will assume that $R$ is a commutative ring with identity.
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Throughout this chapter, we will assume that $R$ is a commutative ring with identity.
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$$ ar^2 + br + c = 0 $$
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$$ ar^2 + br + c = 0 $$
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# Section 3.2 - Real & Distinct Roots
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## Section 3.2 - Real & Distinct Roots
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This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/RealRoots.aspx).
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This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/RealRoots.aspx).
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$$ y(t) = c_1 e^{r_1 t} + c_2 e^{r_2 t} $$
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$$ y(t) = c_1 e^{r_1 t} + c_2 e^{r_2 t} $$
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# Section 3.3 - Complex Roots
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## Section 3.3 - Complex Roots
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This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/ComplexRoots.aspx).
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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
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# Section 4 - Laplace Transformations
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## Section 4.1 - Definition
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## Section 4.1 - Definition
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This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/LaplaceDefinition.aspx).
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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
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# Chapter 1 - Mathematics
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## 1.5 - Dyads and Tensors
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## 1.5 - Dyads and Tensors
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**Definition**. A *dyadic* is a representation of two-ish vectors.
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**Definition**. A *dyadic* is a representation of two-ish vectors.
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The curl of an electrostatic or magnetostatic is relatively simple.
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The curl of an electrostatic or magnetostatic is relatively simple.
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$$
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$$
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\begin{align}
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\begin{align}
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\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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\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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V_e(\vb{r}) = \frac{1}{4 \pi \epsilon_0} \int_{V'} \frac{\rho_e(\vb{r'})}{\abs{\vb{r}-\vb{r'}}} \dd{V'}
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V_e(\vb{r}) = \frac{1}{4 \pi \epsilon_0} \int_{V'} \frac{\rho_e(\vb{r'})}{\abs{\vb{r}-\vb{r'}}} \dd{V'}
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$$
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$$
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$$
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$$
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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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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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$$
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$$
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# Breads
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# Breads
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# Pumpkin Bread
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## Pumpkin Bread
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- Preheat oven to $350 \degree$ F.
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- Preheat oven to $350 \degree$ F.
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- 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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- 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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- Mix in apples, add sugar mixture.
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- Mix in apples, add sugar mixture.
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- Wait until apples are softened (approx. 5 minutes).
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- Wait until apples are softened (approx. 5 minutes).
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### Pies
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### Hand Pies
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- Preheat oven to $400 \degree$ F.
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- Preheat oven to $400 \degree$ F.
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- Split pie crust into 4. Place fillin in crust, fold.
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- Split pie crust into 4. Place fillin in crust, fold.
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