Work on multicalc ch 0 and 1

Work relevant to #2 and #29.

multiCalc/backgroundReview/crossProducts.tex

@@ -1,19 +1,21 @@
 \subsection{Cross Products}
 \noindent
-A cross product is a way of multiplying two vectors so that the result is a vector. Although the cross product technically only works for 3D vectors, we will first look a a "fake" 2D version to build an intuition.
+A cross product is a way of multiplying two vectors so that the result is a vector.
+Although the cross product technically only works for 3D vectors, we will first look a a "fake" 2D version to build an intuition.
 \begin{equation*}
 	\vec{a}\times\vec{b} = a_1b_1-a_2b_2
 \end{equation*}
-This "fake" 2D cross product gives the area of the parallelogram spanned by $\vec{a}$ and $\vec{b}$.
+This ``fake'' 2D cross product gives the area of the parallelogram spanned by $\vec{a}$ and $\vec{b}$.
 \begin{equation*}
 	\vec{a}\times\vec{b} = \norm{\vec{a}}\norm{\vec{b}}\sin{\theta}	
 \end{equation*}
-where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$.\\
+where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$.
 Another way to think of the magnitude of the cross product, both in 2D and 3D, is as a measure of how perpendicular two vectors are.
 
 \begin{figure}[H]
 	\centering
-	\includegraphics[scale=0.5]{Images/backgroundReview/CrossProduct}
+	\includegraphics[width=0.5\textwidth]{./Images/backgroundReview/CrossProduct.png}
+	\caption{Visualization of the cross product}
 \end{figure}
 
 \noindent
@@ -25,38 +27,39 @@ \subsection{Cross Products}
 \begin{equation*}
 	\norm{\vec{a}\times\vec{b}}=\norm{\vec{a}}\norm{\vec{b}}\sin{\theta}
 \end{equation*}
-where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$.\\
-Each component of $\vec{a}\times\vec{b}$ gives the area of the parallelogram spanned by $\vec{a}$ and $\vec{b}$ in some plane: The x-component of $\vec{a}\times\vec{b}$ gives the area in the yz-plane (x = 0 plane).\\
+where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$.
+Each component of $\vec{a}\times\vec{b}$ gives the area of the parallelogram spanned by $\vec{a}$ and $\vec{b}$ in some plane:
+The $x$-component of $\vec{a}\times\vec{b}$ gives the area in the yz-plane ($x$ = 0 plane).
 $\vec{a}\times\vec{b}$ is perpendicular, also called "normal," to the plane containing $\vec{a}$ and $\vec{b}$. It's direction, is determined by the right hand rule.\\
 
 \noindent
-The cross product table of the standard basis vectors is useful for providing some insight into the properties of the cross product.
+This cross product table of the standard basis vectors is useful for providing some insight into the properties of the cross product.
 \begin{table}[H]
 	\centering
 	\renewcommand{\arraystretch}{1.5}
-	\begin{tabular}{|l|l|l|l|}
+	\begin{tabular}{|c||c|c|c|}
 		\hline
-		$\overrightarrow{\text{row}}\times\overrightarrow{\text{col}}$ & $\hat{i}$  & $\hat{j}$  & $\hat{k}$  \\ \hline
+		$\overrightarrow{\text{row}}\times\overrightarrow{\text{col}}$ & $\hat{i}$  & $\hat{j}$  & $\hat{k}$  \\ \hline\hline
 		$\hat{i}$                                & $0$        & $\hat{k}$  & $-\hat{j}$ \\ \hline
 		$\hat{j}$                                & $-\hat{k}$ & $0$        & $\hat{i}$  \\ \hline
 		$\hat{k}$                                & $\hat{j}$  & $-\hat{i}$ & $0$        \\ \hline
 	\end{tabular}
 \end{table}
 
-\begin{itemize}
-	\item \textbf{NOT} Commutative, but is antisymmetric
-	\begin{equation*}
-		\vec{a}\times\vec{b} = -\left(\vec{b}\times\vec{a}\right)
-	\end{equation*}
-	\item Scalar Associative
-	\begin{equation*}
-		\left(c\cdot\vec{a}\right)\times\vec{b}=\vec{a}\times\left(c\cdot\vec{b}\right)
-	\end{equation*}
-	\item Distributive
-	\begin{equation*}
-		\vec{a}\times\left(\vec{b}\times\vec{c}\right) = \vec{a}\times\vec{b} + \vec{a}\times\vec{c}
-	\end{equation*}
-\end{itemize}
+\begin{enumerate}[label=]
+	\item \textbf{\underline{NOT} Commutative}, but is antisymmetric
+		\begin{equation*}
+			\vec{a}\times\vec{b} = -\left(\vec{b}\times\vec{a}\right)
+		\end{equation*}
+	\item \textbf{Scalar Associative}
+		\begin{equation*}
+			\left(c\cdot\vec{a}\right)\times\vec{b}=\vec{a}\times\left(c\cdot\vec{b}\right)
+		\end{equation*}
+	\item \textbf{Distributive}
+		\begin{equation*}
+			\vec{a}\times\left(\vec{b}\times\vec{c}\right) = \vec{a}\times\vec{b} + \vec{a}\times\vec{c}
+		\end{equation*}
+\end{enumerate}
 
 \noindent
 One can also think of the cross product as the determinant of a matrix.

multiCalc/backgroundReview/dotProducts.tex

@@ -1,32 +1,37 @@
 \subsection{Dot Products}
 \noindent
-A dot product is a way of multiplying two vectors so that the result is a scalar. $\vec{a}\cdot\vec{b} = \norm{\vec{a}}\norm{\vec{b}}\cos{\theta}$ where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$. One way to think of the dot product is as a measure of how much two vectors point in the same direction.\\
-We can also show using the law of cosines that $\vec{a}\cdot\vec{b} = a_1b_1+a_2b_2+...+a_nb_n$.\\
+A dot product is a way of multiplying two vectors so that the result is a scalar.
+$\vec{a}\cdot\vec{b} = \norm{\vec{a}}\norm{\vec{b}}\cos{\theta}$ where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$.
+One way to think of the dot product is as a measure of how much two vectors point in the same direction.
+We can also show using the law of cosines that $\vec{a}\cdot\vec{b} = a_1b_1+a_2b_2+...+a_nb_n$.
 Knowing the lengths of two vectors and their dot product we can calculate the angle between them as
 \begin{equation*}
-	\theta = \arccos{\left(\frac{\vec{a} \cdot \vec{b}}{\norm{\vec{a}} \norm{\vec{b}}}\right)}
+	\theta = \arccos{\left(\frac{\vec{a} \cdot \vec{b}}{\norm{\vec{a}} \norm{\vec{b}}}\right)}.
 \end{equation*}
 
 \begin{figure}[H]
 	\centering
-	\includegraphics[scale=0.33]{Images/backgroundReview/DotProduct}
+	\includegraphics[width=0.25\textwidth]{./Images/backgroundReview/DotProduct.png}
+	\caption{Two vectors and the angle between them}
 \end{figure}
 
 \noindent
 Although similar to scalar multiplication, dot products have some properties that set them apart.
-\begin{itemize}
-	\item Commutative
+\begin{enumerate}[label=]
+	\item \textbf{Commutative}
 		\begin{equation*}
 			\vec{a}\cdot\vec{b} = \vec{b}\cdot\vec{a}	
 		\end{equation*}
 		the same as scalar multiplication.
-	\item Distributive
+	\item \textbf{Distributive}
 		\begin{equation*}
 			\vec{a}\cdot\left(\vec{b}+\vec{c}\right) = \vec{a}\cdot\vec{b}+\vec{a}\cdot\vec{c}
 		\end{equation*}
 		the same as scalar multiplication.
-	\item \textbf{NOT} Associative $\left(\vec{a}\cdot\vec{b}\right)\cdot\vec{c}$ is a nonsense expression. However, like scalar multiplication, dot products are scalar associative.
-	\begin{equation*}
-		\left(c\cdot\vec{a}\right)\cdot\vec{b} = \vec{a}\cdot\left(c\cdot\vec{b}\right)
-	\end{equation*}
-\end{itemize}
+	\item \textbf{\underline{NOT} Associative}
+		$\left(\vec{a}\cdot\vec{b}\right)\cdot\vec{c}$ is a nonsense expression.
+		However, like scalar multiplication, dot products are scalar associative.
+		\begin{equation*}
+			\left(c\cdot\vec{a}\right)\cdot\vec{b} = \vec{a}\cdot\left(c\cdot\vec{b}\right)
+		\end{equation*}
+\end{enumerate}

multiCalc/backgroundReview/vectors.tex

@@ -1,64 +1,76 @@
 \section{Vectors}
 \noindent
-A vector is a quantity with both direction and magnitude. One can think of it as a directed line segment. In multivariable calculus, we mostly will work with vectors in $\mathbb{R}^2$ and $\mathbb{R}^3$, but vectors can exist in other dimensions.\\
+A vector is a quantity with both direction and magnitude.
+One can think of it as a directed line segment.
+In multivariable calculus, we mostly will work with vectors in $\mathbb{R}^2$ and $\mathbb{R}^3$, but vectors can exist in other dimensions.\\
 
 \noindent
-Numerical (scalar) quantities have vector analogues, many of which show up in physics. Speed becomes velocity, distance becomes displacement, and mass becomes weight.\\
+Numerical (scalar) quantities have vector analogues, many of which show up in physics.
+Speed becomes velocity, distance becomes displacement, and mass becomes weight.\\
 
 \noindent
 Say we have a 2D vector, $\vec{v} = \langle v_x, v_y \rangle$.
 
 \begin{figure}[H]
 	\centering
-	\includegraphics[scale=0.5]{Images/backgroundReview/VectorAddition}
+	\includegraphics[scale=0.5]{./Images/backgroundReview/VectorAddition.png}
+	\caption{The $x$ and $y$ components of a vector $v$}
 \end{figure}
 
 \noindent
-Its length, also called magnitude or norm, is notated $\norm{\vec{v}} = \sqrt{v_x^2+v_y^2}$. This pattern of the norm being equal to the square-root of the sum of the squares of the vector's components continues into higher dimensions.\\
+Its length, also called magnitude or norm, is notated $\norm{\vec{v}} = \sqrt{v_x^2+v_y^2}$.
+This pattern of the norm being equal to the square-root of the sum of the squares of the vector's components continues into higher dimensions.\\
 
 \noindent
-The angle a 2D vector forms with the horizontal axis is $\theta = \tan^{-1}{\left(\frac{v_y}{v_x}\right)}$. There is not a useful version of this formula in higher dimensions.\\
+The angle a 2D vector forms with the horizontal axis is $\theta = \tan^{-1}{\left(\frac{v_y}{v_x}\right)}$.
+There is not a useful version of this formula in higher dimensions.
 Using $\theta$ and $\norm{\vec{v}}$, we can see that $v_x = \norm{\vec{v}}\cos{\theta}$ and $v_y = \norm{\vec{v}}\sin{\theta}$.\\
 
 \noindent
-Vectors can be added and subtracted from each other in a way that the result is another vector. We do this numerically by adding the corresponding components of each vector.\\
-For example, if $\vec{a} = \langle 1,3 \rangle$ and $\vec{b} = \langle 4,7 \rangle$, then $\vec{a}+\vec{b} = \langle 1+4, 3+7 \rangle = \langle 5,10 \rangle$.\\
+Vectors can be added and subtracted from each other in a way that the result is another vector.
+We do this numerically by adding the corresponding components of each vector.
+For example, if $\vec{a} = \langle 1,3 \rangle$ and $\vec{b} = \langle 4,7 \rangle$, then $\vec{a}+\vec{b} = \langle 1+4, 3+7 \rangle = \langle 5,10 \rangle$
 and $\vec{b}-\vec{a} = \langle 4-1, 7-3, \rangle = \langle 3,4 \rangle$.
 
 \noindent
 Visually, you can think of $\vec{v}+\vec{w}$ as the vector connecting the tail of $\vec{v}$ with the tip of $\vec{w}$ where the tail of $\vec{v}$ is on the tip of $\vec{w}$.
 
 \begin{figure}[H]
 	\centering
-	\includegraphics[scale=0.33]{Images/backgroundReview/Parallelogram}
+	\includegraphics[scale=0.33]{./Images/backgroundReview/Parallelogram.png}
+	\caption{Visualization of $\vec{v}+\vec{w}$ and $\vec{v} - \vec{w}$}
 \end{figure}
 
 \noindent
 We can also multiply vectors by scalars and get another vector as a result. We do this by multiplying each component of the vector by the scalar. This has the effect of stretching or shrinking the vector and possibly changing the vector's direction if the scalar is negative.
 
 \begin{figure}[H]
 	\centering
-	\includegraphics[scale=0.5]{Images/backgroundReview/ScalarMultiples}
+	\includegraphics[scale=0.5]{./Images/backgroundReview/ScalarMultiples.png}
+	\caption{A vector $\vec{v}$ scaled by different constants}
 \end{figure}
 
 \noindent
-A unit vector is any vector with magnitude 1. Rather than using an arrow like for other vectors, unit vectors are notated with a carat $\left(\wedge \right)$ over top, like $\hat{i}$, which is read as "i hat".\\
-We an transform any vector with non-zero magnitude into a unit vector by dividing the vector by its norm. This normalized vector will point in the same direction as the original vector.\\
+A unit vector is any vector with magnitude 1.
+Rather than using an arrow like for other vectors, unit vectors are notated with a carat $\left(\wedge \right)$ over top, like $\hat{i}$, which is read as ``i hat''.
+We can transform any vector with non-zero magnitude into a unit vector by dividing the vector by its norm.
+This normalized vector will point in the same direction as the original vector.\\
 
 \noindent
-It is common in mathematics for $\hat{i} = \langle 1,0,0 \rangle$ to be the unit vector in the x-direction, $\hat{j} = \langle 0,1,0 \rangle$ to be the unit vector in the y-direction, and $\hat{k} = \langle 0,0,1 \rangle$ to be the unit vector in the z-direction.\\
-Together, $\hat{i}$, $\hat{j}$, and $\hat{k}$ are called the standard basis vectors because all other vectors can be written as linear combination of these.
+It is common in mathematics for $\hat{i} = \langle 1,0,0 \rangle$ to be the unit vector in the x-direction, $\hat{j} = \langle 0,1,0 \rangle$ to be the unit vector in the y-direction, and $\hat{k} = \langle 0,0,1 \rangle$ to be the unit vector in the z-direction.
+Together, $\hat{i}$, $\hat{j}$, and $\hat{k}$ are called the standard basis vectors because all other vectors in $\R^3$ can be written as linear combination of these.
 
 \begin{figure}[H]
 	\centering
-	\includegraphics[scale=0.33]{Images/backgroundReview/UnitVectors}
+	\includegraphics[scale=0.33]{./Images/backgroundReview/UnitVectors.png}
+	\caption{The standard basis vectors $\hat{i}$, $\hat{j}$, and $\hat{k}$}
 \end{figure}
 
 \input{./backgroundReview/dotProducts}
 \input{./backgroundReview/crossProducts}
 
 \noindent
-Now that we have defined the dot product and cross product, we can put the two together as the scalar triple product, which gives the volume of the parallelepiped spanned by $\vec{a}$, $\vec{b}$, and $\vec{c}$.\\
+Now that we have defined the dot product and cross product, we can put the two together as the scalar triple product, which gives the volume of the parallelepiped spanned by $\vec{a}$, $\vec{b}$, and $\vec{c}$.
 \begin{equation*}
 	\vec{a}\cdot\left(\vec{b}\times\vec{c}\right) = \det\begin{bmatrix}
 		a_1 & a_2 & a_2 \\
@@ -69,5 +81,6 @@ \section{Vectors}
 
 \begin{figure}[H]
 	\centering
-	\includegraphics[scale=0.33]{Images/backgroundReview/Parallelipiped}
+	\includegraphics[width=0.5\textwidth]{./Images/backgroundReview/Parallelipiped.png}
+	\caption{Scalar triple product gives volume of parallelepiped spanned by three vectors.}
 \end{figure}

multiCalc/vectorValuedFunctions/CommonVVFs.tex

@@ -8,7 +8,7 @@ \section{Common Vector-Valued Functions}
 \noindent
 For example, if $\vec{r}(t) = \langle \tan{t},6t,\ln{\left(16-t^2\right)} \rangle$,
 \begin{itemize}
-	\item $\tan{t}$ is defined for all real numbers equivalent to $\pm\pi/2$ radians.
+	\item $\tan{t}$ is defined for all real numbers not equivalent to $\pm\pi/2$ radians.
 	\item $6t$ is defined for all real numbers.
 	\item $\ln{\left(16-t^2\right)}$ is defined for $t\in\left(-4,4\right)$.
 \end{itemize}

multiCalc/vectorValuedFunctions/DerivativesOfVVFs.tex

@@ -1,58 +1,64 @@
 \section{Derivatives of VVFs}
 \noindent
-Just like functions from Calc I and II, we can differentiate VVFs. In fact, the limit definitions of the derivative are nearly identical.\\
+Just like functions from Calc I and II, we can differentiate VVFs.
+In fact, the limit definitions of the derivative are nearly identical.
 Let $\vec{r}(t) = \langle x(t), y(t), z(t) \rangle$.
 \begin{align*}
 	\vec{r^\prime}(t) &= \lim_{h\to 0}{\frac{\vec{r}(t+h)-\vec{r}(t)}{h}} \\
-	&= \lim_{h\to 0}{\langle \frac{x(t+h)-x(t)}{h}, \frac{y(t+h)-y(t)}{h}, \frac{z(t+h)-z(t)}{h} \rangle}
+	&= \lim_{h\to 0}{\bigg\langle \frac{x(t+h)-x(t)}{h}, \frac{y(t+h)-y(t)}{h}, \frac{z(t+h)-z(t)}{h} \bigg\rangle}
 \end{align*}
 The limit distributes inside the vector, so
 \begin{equation*}
-	\vec{r^\prime}(t) = \langle x^{\prime}(t), y^{\prime}(t), z^{\prime}(t) \rangle
+	\vec{r^\prime}(t) = \langle x^{\prime}(t), y^{\prime}(t), z^{\prime}(t) \rangle.
 \end{equation*}
 
 \noindent
-Like a position function from Calc I and II, the derivative of a VVF representing position gives a VVF representing velocity, and the 2nd derivative gives a VVF representing acceleration. The magnitude of the velocity VVF, the speed, is commonly notated $v(t)$.\\
+Like a position function from Calc I and II, the derivative of a VVF representing position gives a VVF representing velocity, and the 2nd derivative gives a VVF representing acceleration.
+The magnitude of the velocity VVF, the speed, is commonly notated $v(t)$.\\
 
 \noindent
-There are 5 important properties of the derivatives of VVFs. These properties are similar to single-variable derivatives.\\
-Let $\vec{r}(t)$ and $\vec{s}(t)$ be VVFs, $a(t)$ be a scalar function, and $c$ be a scalar.\\
-\begin{enumerate}
-	\item Linearity
+There are 5 important properties of the derivatives of VVFs.
+These properties are similar to single-variable derivatives.
+Let $\vec{r}(t)$ and $\vec{s}(t)$ be VVFs, $a(t)$ be a scalar function, and $c$ be a scalar.
+\begin{enumerate}[label=]
+	\item \textbf{Linearity}
 	\begin{equation*}
 		\frac{\mathrm{d}}{\mathrm{d}t}c\vec{r}(t) = c\vec{r^\prime}(t)	
 	\end{equation*}
-	\item Product Rule for Scalar Functions
+	\item \textbf{Product Rule for Scalar Functions}
 	\begin{equation*}
 		\frac{\mathrm{d}}{\mathrm{d}t}a(t)\vec{r}(t) = a(t)\vec{r^\prime}(t) + \vec{r}(t)a^{\prime}(t)
 	\end{equation*}
-	\item Dot Product Rule
+	\item \textbf{Dot Product Rul}e
 	\begin{equation*}
 		\frac{\mathrm{d}}{\mathrm{d}t}\vec{s}(t)\cdot\vec{r}(t) = \vec{s}(t)\cdot\vec{r^\prime}(t) + \vec{r}(t)\vec{s^\prime}(t)	
 	\end{equation*}
-	\item Cross Product Rule
+	\item \textbf{Cross Product Rule}
 	\begin{equation*}
 		\frac{\mathrm{d}}{\mathrm{d}t}\vec{s}(t)\times\vec{r}(t) = \vec{s}(t)\times\vec{r^\prime}(t) + \vec{s^\prime}(t)\times\vec{r}(t)	
 	\end{equation*}
-	\item Chain Rule
+	\item \textbf{Chain Rule}
 	\begin{equation*}
 		\frac{\mathrm{d}}{\mathrm{d}t}\vec{r}(a(t)) = \vec{r^\prime}(a(t))a^{\prime}(t)
 	\end{equation*}
 \end{enumerate}
-A quotient rule doesn't make sense because we don't have an operation for dividing two vectors by each other.
+A quotient rule doesn't make sense because we don't have an operation for dividing two vectors by each other.\\
 
 \noindent
-Just like in single variable calculus, we can use the derivative of VVFs to find tangent lines to the curve. Similar to how $f^{\prime}(a)$ represents the slope of $f$ at $a$, $\vec{r^\prime}(a)$ represents the direction of the tangent line at $a$. Remembering the VVF form of a line, the tangent line to $\vec{r}$ at $t$ is 
+Just like in single variable calculus, we can use the derivative of VVFs to find tangent lines to the curve.
+Similar to how $f^{\prime}(a)$ represents the slope of $f$ at $a$, $\vec{r^\prime}(a)$ represents the direction of the tangent line at $a$.
+Remembering the VVF form of a line, the tangent line to $\vec{r}$ at $t$ is 
 \begin{equation*}
-	\vec{l}(t)=\vec{r}(t)+t\vec{r^\prime}(t)	
+	\vec{l}(t)=\vec{r}(t)+t\vec{r^\prime}(t).	
 \end{equation*}
 In fact, tangent lines appear so often, that we have a special unit vector representing the direction of the tangent line.
 \begin{equation*}
 	\hat{T}(t) = \frac{\vec{r^\prime}(t)}{\norm{\vec{r^\prime}(t)}}
 \end{equation*}
-You can remember $\hat{T}$ as the "tangent" vector.
+You can remember $\hat{T}$ as the ``tangent'' vector.
 
 \begin{figure}[H]
 	\centering
 	\includegraphics[scale=0.33]{Images/vectorValuedFunctions/TangentVector}
+	\caption{$\hat{T}$ is $\vec{r^\prime}$ normalized.}
 \end{figure}

multiCalc/vectorValuedFunctions/IntegralsOfVVFs.tex

@@ -1,5 +1,6 @@
 \section{Integrals of VVFs}
-VVFs can also be integrated. The integration operation also distributes inside the vector, for both definite and indefinite integrals.
+VVFs can also be integrated.
+The integration operation also distributes inside the vector, for both definite and indefinite integrals.
 \begin{equation*}
 	\int{\vec{r}(t)\mathrm{d}t} = \left<\int{x(t)\mathrm{d}t},\int{y(t)\mathrm{d}t},\int{z(t)\mathrm{d}t}\right>
 \end{equation*}