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\newcommand{\myCourseNumber}{Math 20E}
\newcommand{\myAssignmentDueDate}{Tuesday, November 7, 2023}
\newcommand{\myAssignmentTitle}{Homework Assignment 4}
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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 $(0,2,3)$.

\question Let $S$ be the surface parametrized by
\[ x = \cos(u) \sin(v) \quad y = \sin(u) \sin(v) \quad z = \cos(v) \] for $u \in [0,2\pi]$ and $v \in [0,\pi]$.
\begin{parts}
\part Find an expression for a unit vector normal to $S$ at the image of a point \newline $(u,v) \in [0,2\pi] \times [0,\pi]$.
\part Identify the surface $S$.
\end{parts}

\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}

\question The image $S$ of the parametrization
\begin{align*}
\Phi &: [-\pi,\pi] \times [0, \pi] \to S \subseteq \mathbb{R}^3 \\
\Phi(u,v) &= (a \cos(u)\sin(v), \ b \sin(u)\sin(v), \ c \cos(v))
\end{align*}
is an ellipsoid.
\begin{parts}
\part Find an equation for the ellipsoid $S$ by evaluating the expression \ $\displaystyle \left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 + \left(\frac{z}{c}\right)^2$.
\part Show that $\Phi$ is not regular when $v = 0$ or $v = \pi$.
\part Show that the image surface $S$ is regular at all points of $S$.
\end{parts}

\question Find the area of the unit sphere $S$ parametrized by
\begin{align*}
\Phi &: [0,2 \pi] \times [0, \pi] \to S \subseteq \mathbb{R}^3 \\
\Phi(\theta,\phi) &= (\cos(\theta)\sin(\phi), \ \sin(\theta)\sin(\phi), \ \cos(\phi))
\end{align*}

\question Find the 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}

\end{document}
