$1$E$)$ Let $C$ be a closed curve oriented counterclockwise that is a rectangle in two dimensions with vertices $(0,0),$

$(c+2,0),(c+2,d+1),$ and $(0,d+1)$ in the $xy-$plane. Using the same vector field from above, use

MATLAB to calculate the line integral below with Green's Theorem. Call this result P$1$E$.$

${\displaystyle \underset{C}{\overset{\phantom{}}{}}}$vec$(F)*dvec(r)$

$1$F$)$ Clear any variables you want to reuse from previous exercises by using the clear command on a line in your

script after your answer for IE$.$ Consider the following three$-$dimensional vector field.

$F(x,y,z)=(a+1)y^3,(b+1)z^3,(c+1)x^3$

Plot this vector field in MATLAB over $-3x3,-3y3,$ and $-3z3$ with a step size of $1$ in all

three directions $($figure $2).$ Now, use MATLAB to calculate curl $F$ and call this result PIF$1.$ Next, let $S$ be a

square that lies in the $yz-$plane, centered at the origin, with side length $2(d+1),$ oriented clockwise when viewed

from the positive $x-$axis; let $C$ be its boundary. Using the vector field above, use MATLAB to calculate the line

integral below with Stokes' Theorem. Call this result P$1$F$2.$

${\displaystyle \underset{C}{\overset{\phantom{}}{}}}F*dvec(r)$

$1$G$)$ Clear any variables you want to reuse from previous exercises by using the clear command on a line in your

script after your answer for IF$.$ Consider the following three$-$dimensional vector field.

$F(x,y,z)=sin((b+1)x),cos((c+1)y),z$

Plot this vector field in MATLAB over $-3x3,-3y3,$ and $-3z3$ with a step size of $1$ in all

three directions $($figure $3).$ Next, use MATLAB to calculate div $F$ and call this result P$1$G$1.$ Next, let $B$ be a box,

centered at the origin in three$-$space with side length $2(d+1),$ oriented outward; let $S$ be its boundary surface,

Using the vector field above, use MATLAB to calculate the surface integral below with the Divergence

Theorem. Call this result P$1$G$2.$

${}_{S}F*vec(n)dS$

Once you have these operations written in proper MATLAB syntax, go to your Editor menu and click Run. If

there are values in your Command Window for all operations without error messages, then your code has

successfully run! If you have error messages, read them carefully and try to resolve them, then click Run again.

Part $2$ Exercises

Recall that $a=$ last digit of your student ID$,b=$ second to last digit of your student ID$,c=$ third to last digit of

your student ID$,$ and $d-$ the sum of the last three digits of your student ID$.$ Complete the following operations in

your script, making a new line for each operation.

Consider the two vector fields given below for an electric field, $E(vec(l)),$ and a gravitational field, vec$(F)(vec()).$ For the first

vector field, vec$(r)=($:$x,y$:$),=8\mathrm{.}988{10}^{9}N{m}^2{C}^{-2},$ the Coulomb constant, and $Q=1\mathrm{.}6{10}^{-19}C,$

the charge of a single proton. For the second vector field, vec$()=($:$x,y,z$:$),m=(56+d)kg,$ a human

mass within common range, $M=5\mathrm{.}972{10}^{24}kg,$ the mass of the Earth, and $G=6\mathrm{.}674{10}^{-11}N{m}^{2}k$

$g^{-2},$ Newton's universal gravitational constant.

$2$A$)$ Clear any variables you want to reuse from previous exercises by using the clear command on a line in your

script. Plot the electric field in MATLAB over the intervals $-0\mathrm{.}1x0\mathrm{.}1$ and $-0\mathrm{.}1y0\mathrm{.}1,$ both with

a step size of $0\mathrm{.}02($figure $4).$ Next, calculate the curl and divergence of the electric field. Call these results P$2$AC

and $P2AD,$ respectively.

$2$B$)$ Consider an elementary particle being placed into the electric field generated by the positive point charge of a

proton at the origin. It then travels along an oriented path $C$ defined by the vector below. Calculate the work

done by the electric field on the particle for the first $(d+3)$ seconds using two differ