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hw1.tex
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\documentclass{homework}
\title{Homework 1}
\author{Kevin Evans}
\studentid{11571810}
\date{September 2, 2020}
\setclass{Physics}{341}
\usepackage{amssymb}
\usepackage{mathtools}
\usepackage{amsthm}
\usepackage{amsmath}
\usepackage{slashed}
\usepackage{relsize}
\usepackage{threeparttable}
\usepackage{float}
\usepackage{booktabs}
\usepackage{boldline}
\usepackage{changepage}
\usepackage{physics}
\usepackage[inter-unit-product =\cdot]{siunitx}
\usepackage{setspace}
\usepackage[makeroom]{cancel}
%\usepackage{pgfplots}
\usepackage{enumitem}
\usepackage{times}
\begin{document}
\maketitle
\begin{enumerate}
\item For the LHS, if we begin by evaluating $\bvec{B} \cross \bvec{C}$, \begin{align*}
\bvec{A} \cross \left(\bvec{B} \cross \bvec{C} \right) & = \bvec{A} \cross \abs{
\begin{matrix}
\uvec{x} & \uvec{y} & \uvec{z} \\
B_x & B_y & B_z \\
C_x & C_y & C_z
\end{matrix}
} \\
& = \bvec{A} \cross \left[
\left(B_y C_z - B_z C_y\right) \uvec{x}
+ \left( B_z C_x - B_x C_z \right) \uvec{y}
+ \left( B_x C_y - B_y C_x \right) \uvec{z}
\right] \\
& = \abs{
\begin{matrix}
\uvec{x} & \uvec{y} & \uvec{z} \\
A_x & A_y & A_z \\
\left(B_y C_z - B_z C_y\right)
& \left( B_z C_x - B_x C_z \right)
& \left( B_x C_y - B_y C_x \right)
\end{matrix}
} \\
& = \left(A_y B_x C_y - A_y B_y C_x - A_z B_z C_x + A_z B_x C_z\right) \uvec{x} \\
& \quad + \left( A_z B_y C_z - A_z B_z C_y - A_x B_x C_y + A_x B_y C_x \right) \uvec{y} \\
& \quad + \left( A_x B_z C_x - A_x B_x C_z - A_y B_y C_z + A_y B_z C_y \right) \uvec{z} \\
& = \left( A_y C_y + A_z C_z \right) B_x \uvec{x}
+ \left( A_z C_z + A_x C_x \right) B_y \uvec{y}
+ \left( A_x C_x + A_y C_y \right) B_z \uvec{z} \\
& \quad - \left(A_y B_y + A_z B_z\right) C_x \uvec{x}
- \left( A_x B_x + A_z B_z\right) C_y \uvec{y}
- \left( A_x B_x + A_y B_y \right) C_z \uvec{z}
\end{align*}
Then for the RHS, if we expand the dot products and scale the components of $\bvec{B}$ and $\bvec{C}$, \begin{align*}
\bvec{B} \left(\bvec{A} \cdot \bvec{C}\right) - \bvec{C} \left(\bvec{A} \cdot \bvec{B}\right) & =
\left(A_x C_x + A_y C_y + A_z C_z\right) B_x \uvec{x} + \left(\cdots\right) B_y \uvec{y} + \left(\cdots\right) B_z \uvec{z} \\
& \quad - \left(A_x B_x + A_y B_y + A_z B_z\right) C_x \uvec{x} - \left(\cdots\right) C_y \uvec{y} - \left(\cdots\right) C_z \uvec{z}
\intertext{Removing the terms that subtract out (all the $A_i B_i C_i \uvec{e}_i$), we are left with the LHS result,}
& = \left( A_y C_y + A_z C_z \right) B_x \uvec{x}
+ \left( A_z C_z + A_x C_x \right) B_y \uvec{y}
+ \left( A_x C_x + A_y C_y \right) B_z \uvec{z} \\
& \quad - \left(A_y B_y + A_z B_z\right) C_x \uvec{x}
- \left( A_x B_x + A_z B_z\right) C_y \uvec{y}
- \left( A_x B_x + A_y B_y \right) C_z \uvec{z}
\end{align*}
\qed
\item \begin{enumerate}
\item $\grad{f(x, y, z)} = 2x \uvec{x} + 3y^2 \uvec{y} + 4z^3 \uvec{z}$
\item $\grad{f(x, y, z)} = 2xy^3z^4 \uvec{x} + 3x^2 y^2 z^4 \uvec{y} + 4x^2y^3 z^3 \uvec{z}$
\item $\grad{f(x, y, z)} = e^x \sin(y) \ln(z) \uvec{x} + e^x \cos(y) \ln(z) \uvec{y} + \frac{e^x \sin(y)}{z} \uvec{z} $
\end{enumerate}
\item \begin{enumerate}
\item $\div{\bvec{v}_a} = 2x + 3y^2 + 4z^3$
\item $\div{\bvec{v}_b} = x + y + z$
\item $\div{\bvec{v}_c} = 2yz - 3y$
\end{enumerate}
\pagebreak
\item \begin{enumerate}
\item $\begin{aligned}[t]
\curl[x^2 \uvec{x} + y^3 \uvec{y} + z^4 \uvec{z}] & = \abs{
\begin{matrix}
\uvec{x} & \uvec{y} & \uvec{z} \\
\pdv{x} & \pdv{y} & \pdv{z} \\
x^2 & y^3 & z^4
\end{matrix}
} \\
& = 0
\end{aligned}$
\vspace{1em}
\item $\begin{aligned}[t]
\curl[xy \uvec{x} + yz \uvec{y} + zx \uvec{z}] & = \abs{
\begin{matrix}
\uvec{x} & \uvec{y} & \uvec{z} \\
\pdv{x} & \pdv{y} & \pdv{z} \\
xy & yz & zx
\end{matrix}
} \\
& = \left(-y\right) \uvec{x} -z \uvec{y} - x\uvec{z}
\end{aligned}$
\vspace{1em}
\item $\begin{aligned}[t]
\curl[2z \uvec{x} + y^2z \uvec{y} - 3yz \uvec{z}] & = \abs{
\begin{matrix}
\uvec{x} & \uvec{y} & \uvec{z} \\
\pdv{x} & \pdv{y} & \pdv{z} \\
2z & y^2 z & -3yz
\end{matrix}
} \\
& = \left(-3z - y^2\right) \uvec{x} + 2 \uvec{y}
\end{aligned}$
\end{enumerate}
\vspace{1em}
\item \begin{enumerate}
\item $\laplacian[x^2 + y^3 + z^4] = 2 + 6y + 12z^2$
\item $\begin{aligned}[t]
\laplacian[x^2 y^3 z^4] & = \div[2xy^3 z^4 \uvec{x} + 3x^2y^2z^3 \uvec{y} + 4x^2y^3z^3 \uvec{z}] \\
& = 2y^3z^4 + 6x^2yz^3 + 12x^2y^3z^2
\end{aligned}$
\item $\begin{aligned}[t]
\laplacian[e^x \sin(y) \ln(z)] & = \div[ e^x \sin(y) \ln(z) \uvec{x} + e^x \cos(y) \ln(z) \uvec{y} + \frac{e^x \sin(y)}{z} \uvec{z}] \\
& = e^x \sin(y) \ln(z) - e^x \sin(y) \ln(z) - \frac{e^x \sin(y)}{z^2} \\
& = -\frac{e^x \sin(y)}{z^2}
\end{aligned}$
\item $\laplacian[xy \uvec{x} + yz \uvec{y} + zx \uvec{z}] = 0$ \qquad (as they're all first order)
\end{enumerate}
\end{enumerate}
\end{document}