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\newcommand{\myCourseNumber}{Math 20E}
\newcommand{\myAssignmentDueDate}{11:00pm Tuesday, April 23, 2024}
\newcommand{\myAssignmentTitle}{Homework Assignment 2}
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\begin{questions}

\question Let $D^*$ be the parallelogram with vertices at \enspace $(-1,3)$, \enspace $(0,0)$, \enspace $(2,-1)$, \enspace and \enspace $(1,2)$. \enspace Let $D$ \enspace be the rectangle \enspace $D = [0,1] \times [0,1]$. \enspace Find a $T$ such that $D$ is the image set of $D^*$ under $T$; that is, $D = T(D^*)$.

\question Let \enspace $T : \mathbb{R}^3 \to \mathbb{R}^3$ \enspace be the spherical coordinate mapping defined by \enspace $(\rho, \phi, \theta) \mapsto (x,y,z)$, \enspace where
\begin{displaymath}
x = \rho \sin(\phi) \cos(\theta), \qquad y = \rho \sin(\phi) \sin(\theta), \qquad z = \rho \cos(\phi) .
\end{displaymath}
Let $D^*$ be the set of points \enspace $(\rho, \phi, \theta)$ \enspace such that \enspace $\rho \in [0,1]$, \enspace $\phi \in [0, \pi]$, \enspace $\theta \in [0, 2 \pi]$.
\begin{parts}
\part Find \enspace $D = T(D^*)$.
\part Is $T$ one-to-one?  If not, can we eliminate a subset $S \subseteq D^*$ so that $T$ is one-to-one on the remainder \enspace $D^* \setminus S = \left\{ (x,y,z) \in D^* \mid (x,y,z) \not\in S \right\}$?
\end{parts}

\question Evaluate $\displaystyle \iint_D x^2 \, dx \, dy$ where $D$ is determined by the two conditions $0 \leq x \leq y$ and $x^2 + y^2 \leq 1$.

\question Evaluate $\displaystyle \iiint_W \sqrt{x^2 + y^2 + z^2} \, e^{-(x^2 + y^2 + z^2)} \, dx \, dy \, dz$, where $W$ is the solid bounded by the two spheres

$x^2 + y^2 + z^2 = a^2$ and $x^2 + y^2 + z^2 = b^2$ with $0 < a < b$.

\question Evaluate $\displaystyle \iint_R (x + y) \, dx \, dy$, where $R$ is the rectangle in the $xy$-plane with vertices at $(0,1), (1,0), (3,4), (4,3)$.

\question Show that the path $\mathbf{c}(t) = \left(\sin(t), \cos(t), e^t\right)$ is a flow line of the vector field \newline $\mathbf{F}(x,y,z) = (y, -x, z)$.

\question Let $\mathbf{F}(x,y,z) = (y z, x z, x y)$.  Find a function $f : \mathbb{R}^3 \to \mathbb{R}$ such that $\mathbf{F} = \nabla f$.

\question Evaluate the path integral $\displaystyle \int_{\mathbf{c}} f(x,y,z) \, ds$ with $f(x,y,z) = x+y+z$ and $\mathbf{c}(t) = (\sin(t), \cos(t), t)$ for $t \in \left[0, 2\pi\right]$.

\question Evaluate $\displaystyle \int_{\mathbf{c}} f \, ds$, where $f(x,y,z) = z$ and $\mathbf{c}(t) = (t \cos(t), t \sin(t), t)$ for $0 \leq t \leq t_0$.

\question Find the average $z$ coordinate on the path $\mathbf{c}(t) = (t \cos(t), t \sin(t), t)$ for $0 \leq t \leq t_0$.

\end{questions}

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