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\newcommand{\myCourseNumber}{Math 20E}
\newcommand{\myAssignmentDueDate}{11:00pm Tuesday, May 7, 2023}
\newcommand{\myAssignmentTitle}{Homework Assignment 3}
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\begin{questions}
    
\question Evaluate \ $\displaystyle \int_{\mathbf{c}} \mathbf{F} \boldsymbol{\cdot} d\mathbf{s}$, \ where \ $\mathbf{F}(x,y) = (- x \, y, \ x^2)$ \ and $\mathbf{c}$ is the path along the unit circle

$x^2 + y^2 = 1$ beginning at $(1,0)$ and ending at $(0,1)$.

\question Evaluate the line integral \ $\displaystyle \int_{\mathbf{c}} y z \, dx \, + \, x z \, dy \, + \, x y \, dz$, \ where $\mathbf{c}$ consists of the straight-line

segments joining $(1,0,0)$ to $(0,1,0)$ to $(0,0,1)$.

\question Evaluate the line integral \ $\displaystyle \int_C (y^2 + 2 x z) \, dx + (2 x y + z^2) \, dy + (2 y z + x^2) \, dz$, \ where $C$ is an

oriented simple curve from $(1,1,1)$ to $(0,2,3)$.

%%MarsdenTromba 7.2.13
\question Let $\mathbf{c}(t)$ be a path and $\displaystyle \mathbf{T}(t) = \frac{\mathbf{c}'(t)}{\| \mathbf{c}'(t) \|}$ the unit tangent vector.  What is $\displaystyle \int_{\mathbf{c}} \mathbf{T} \boldsymbol{\cdot} d\mathbf{s}$ ?

%%MarsdenTromba 7.2.16
\question Let $S$ be the surface determined by the equation \ $x^3 + 3 x y + z^2 = 2$ \ with \ $z \geq 0$.
\begin{parts}
\part Find a parametrization \ $\Phi : D \subseteq \mathbb{R}^2 \to S \subseteq \mathbb{R}^3$.
\part Find an equation for the tangent plane to $S$ at the point $(1, 1/3, 0)$.
\end{parts}

%%MarsdenTromba 7.2.19
\question The hyperboloid $S$ with equation \ $x^2 + y^2 - z^2 = 25$ \ is parametrized by the mapping
\begin{align*}
\Phi &: [0,2\pi] \times (-\infty, \infty) \longrightarrow \mathbb{R}^3 \\
\Phi(\theta, u) &= \left( \cos(\theta)\cosh(u), \; \sin(\theta)\cosh(u), \; \sinh(u)\right)
\end{align*}
[See Example 5 in Section 7.3 (pg. 381) of your textbook.]
\begin{parts}
\part Find an equation for the plane tangent to the surface $S$ at $(x_0, y_0, 0)$, where $x_0^2 + y_0^2 = 25$.
\part Show that the lines $\boldsymbol{\lambda}_1(t) = (x_0, y_0, 0) + t (-y_0, x_0, 5)$ and $\boldsymbol{\lambda}_2(t) = (x_0, y_0, 0) + t (y_0, -x_0, 5)$ lie in the surface $S$ \ \emph{and} \ in the tangent plane to $S$ at $(x_0, y_0, 0)$.
\end{parts}

%%MarsdenTromba 7.3.23
\question The mapping
\begin{align*}
\Phi &: [0,2\pi] \times [0,2\pi] \longrightarrow \mathbb{R}^3 \\
\Phi(u,v) &= \left(\left(R + r\cos(u)\right) \cos(v), \; \left(R + r\cos(u)\right) \sin(v), \; r\sin(v) \right)
\end{align*}
parametrizes a torus (or doughnut) $T$.
\begin{parts}
\part Show that all points $(x,y,z)$ in the image $T$ satisfy \ $\displaystyle \left( \sqrt{x^2 + y^2} - R \right)^2 + z^2 = r^2$.
\part Show that the image surface $T$ is regular at all points.
\end{parts}


\question Find area of the portion of the unit sphere that is inside the mouth of the cone \ $z \geq \sqrt{x^2 + y^2}$.

\question The cylinder $x^2 + y^2 = x$ divides the unit sphere $S$ into two regions $S_1$ and $S_2$, where $S_1$ is outside the cylinder and $S_2$ is inside the cylinder.

Find the ratio $A(S_1)/A(S_2)$ of the areas of $S_1$ and $S_2$.

\question Find the area of the surface $S$ defined by \ $x + y + z = 1$ \ with \ $x^2 + 3 y^2 \leq 1$.

\end{questions}

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