Line integrals

$\newenvironment {prompt}{}{} \newcommand {\ungraded }[0]{} \newcommand {\todo }[0]{} \newcommand {\oiint }[0]{{\large \bigcirc }\kern -1.56em\iint } \newcommand {\mooculus }[0]{\textsf {\textbf {MOOC}\textnormal {\textsf {ULUS}}}} \newcommand {\npnoround }[0]{\nprounddigits {-1}} \newcommand {\npnoroundexp }[0]{\nproundexpdigits {-1}} \newcommand {\npunitcommand }[1]{\ensuremath {\mathrm {#1}}} \newcommand {\RR }[0]{\mathbb R} \newcommand {\R }[0]{\mathbb R} \newcommand {\N }[0]{\mathbb N} \newcommand {\Z }[0]{\mathbb Z} \newcommand {\sagemath }[0]{\textsf {SageMath}} \newcommand {\d }[0]{\,d} \newcommand {\l }[0]{\ell } \newcommand {\ddx }[0]{\frac {d}{\d x}} \newcommand {\zeroOverZero }[0]{\ensuremath {\boldsymbol {\tfrac {0}{0}}}} \newcommand {\inftyOverInfty }[0]{\ensuremath {\boldsymbol {\tfrac {\infty }{\infty }}}} \newcommand {\zeroOverInfty }[0]{\ensuremath {\boldsymbol {\tfrac {0}{\infty }}}} \newcommand {\zeroTimesInfty }[0]{\ensuremath {\small \boldsymbol {0\cdot \infty }}} \newcommand {\inftyMinusInfty }[0]{\ensuremath {\small \boldsymbol {\infty -\infty }}} \newcommand {\oneToInfty }[0]{\ensuremath {\boldsymbol {1^\infty }}} \newcommand {\zeroToZero }[0]{\ensuremath {\boldsymbol {0^0}}} \newcommand {\inftyToZero }[0]{\ensuremath {\boldsymbol {\infty ^0}}} \newcommand {\numOverZero }[0]{\ensuremath {\boldsymbol {\tfrac {\#}{0}}}} \newcommand {\dfn }[0]{\textbf } \newcommand {\unit }[0]{\mathop {}\!\mathrm } \newcommand {\eval }[1]{\bigg [ #1 \bigg ]} \newcommand {\seq }[1]{\left ( #1 \right )} \newcommand {\epsilon }[0]{\varepsilon } \newcommand {\phi }[0]{\varphi } \newcommand {\iff }[0]{\Leftrightarrow } \DeclareMathOperator {\arccot }{arccot} \DeclareMathOperator {\arcsec }{arcsec} \DeclareMathOperator {\arccsc }{arccsc} \DeclareMathOperator {\si }{Si} \DeclareMathOperator {\scal }{scal} \DeclareMathOperator {\sign }{sign} \newcommand {\arrowvec }[1]{{\overset {\rightharpoonup }{#1}}} \newcommand {\vec }[1]{{\overset {\boldsymbol {\rightharpoonup }}{\mathbf {#1}}}\hspace {0in}} \newcommand {\point }[1]{\left (#1\right )} \newcommand {\pt }[1]{\mathbf {#1}} \newcommand {\Lim }[2]{\lim _{\point {#1} \to \point {#2}}} \DeclareMathOperator {\proj }{\mathbf {proj}} \newcommand {\veci }[0]{{\boldsymbol {\hat {\imath }}}} \newcommand {\vecj }[0]{{\boldsymbol {\hat {\jmath }}}} \newcommand {\veck }[0]{{\boldsymbol {\hat {k}}}} \newcommand {\vecl }[0]{\vec {\boldsymbol {\l }}} \newcommand {\uvec }[1]{\mathbf {\hat {#1}}} \newcommand {\utan }[0]{\mathbf {\hat {t}}} \newcommand {\unormal }[0]{\mathbf {\hat {n}}} \newcommand {\ubinormal }[0]{\mathbf {\hat {b}}} \newcommand {\dotp }[0]{\bullet } \newcommand {\cross }[0]{\boldsymbol \times } \newcommand {\grad }[0]{\boldsymbol \nabla } \newcommand {\divergence }[0]{\grad \dotp } \newcommand {\curl }[0]{\grad \cross } \newcommand {\lto }[0]{\mathop {\longrightarrow \,}\limits } \newcommand {\bar }[0]{\overline } \newcommand {\surfaceColor }[0]{violet} \newcommand {\surfaceColorTwo }[0]{redyellow} \newcommand {\sliceColor }[0]{greenyellow} \newcommand {\vector }[1]{\left \langle #1\right \rangle } \newcommand {\sectionOutcomes }[0]{} \newcommand {\HyperFirstAtBeginDocument }[0]{\AtBeginDocument }$

We work some examples of line integrals and vector fields.

Work in groups of 3–4, writing your answers on a separate sheet of paper.

Carefully sketch the following vector fields:

(a): $\vec {F}(x,y) = \vector {1,1}$
(b): $\vec {G}(x,y) = \vector {x,y}$
(c): $\vec {H}(x,y) = \vector {y,-x}$
(d): $\vec {I}(x,y) = \vector {x,-y}$

Let $C$ be the circle of radius $1$ , centered at the origin, drawn once in a counterclockwise direction. Using the fields above, compute:

(a): $\int _C \vec {F}\dotp \d \vec {p}$
(b): $\int _C \vec {G}\dotp \d \vec {p}$
(c): $\int _C \vec {H}\dotp \d \vec {p}$
(d): $\int _C \vec {I}\dotp \d \vec {p}$

Let $T$ be the triangle with vertices at $(-2,-1)$ , $(2,-1)$ , and $(0,2)$ drawn once in a counterclockwise direction. Using the fields above, compute:

(a): $\int _T \vec {F}\dotp \d \vec {p}$
(b): $\int _T \vec {G}\dotp \d \vec {p}$
(c): $\int _T \vec {H}\dotp \d \vec {p}$
(d): $\int _T \vec {I}\dotp \d \vec {p}$

Let $A$ be the path connecting $(-2,-1)$ to $(2,-1)$ to $(0,2)$ . Note, this is not a closed path. Using the fields above, compute:

(a): $\int _A \vec {F}\dotp \d \vec {p}$
(b): $\int _A \vec {G}\dotp \d \vec {p}$
(c): $\int _A \vec {H}\dotp \d \vec {p}$
(d): $\int _A \vec {I}\dotp \d \vec {p}$

The work $W$ done by a force $\vec {F}$ on an object moving along a curve $C$ , parameterized by $\vec {p}(t)$ is given by: $W = \int _C \vec {F}\dotp \d \vec {p}$

A raindrop of mass $0.001\unit {kg}$ slides from the top of a flat windshield to the bottom. If the windshield is $\frac {1}{2} \unit {m}$ high and $\frac {1}{2}\unit {m}$ deep, how much work is done by gravity? (Assume that the acceleration due to gravity is $10 \unit {m}/\unit {s}^2$ .)

A skateboarder of mass $70\unit {kg}$ rolls from the left side of a circular half-pipe of radius $3\unit {m}$ to the right side. How much work is done by gravity?