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<h3 style="margin-top:2ex;padding-top:0;">Name-<br>
Math 213, May 2019, Test 3</h3>
<p>All question parts are worth 5 pts, unless otherwise noted. (Total 110).
<ol>




<li>
<img src="0g/T3.tetrahedron.jpg" class="rightalign">
(10 pts) Set up a <i>triple integral</i> to evaluate the volume of the tetrahedron bounded by the coordinate planes and the plane $10x +15y + 6z = 30$, as shown. Then evaluate the
integral. 
</li>

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<li style="page-break-before:always">
Let $\myv v(x,y,z)=xy\uv i+(y^2-2z)\uv j+\sin(yz)\uv k$.

<p>For the tetrahedron, call it $T$, on the previous page...Assume that $\myv v$ represents the flux of some fluid.  Calculate the total flux of fluid through the surfaces of the tetrahedron.  That is, calculate:
$$\oiint_T \myv v\cdot d\myv S,$$
but convert the surface integral to a volume integral, using the divergence theorem.  Carry out any partial derivatives.  Write down the volume integral (with all integration limits) that you'd need to calculate.  But you need not carry out the calculation of the integral!

<li style="margin-top:48ex">Compute the curl of the vector field $\myv v$ from the previous problem.






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<li style="page-break-before:always">
<img src="0g/T3.whichvf.png" style="width: 300px" class="rightalign">
Which of the vector fields below describes the plot at right?
<ol type="a">
<li>$x\hat i+(x-y)\hat j$
<li>$x\hat i+(x-y)\hat j$
<li>$x\hat i+(x-y)\hat j$
<li>$x\hat i+(x-y)\hat j$
</ol>
<br clear="both">





<!---3. Pac Man ------------------>
<li style="margin-top: 5ex">
<img src="0g/T3.pacman.jpg" class="rightalign">
(10 pts) Consider the region $P$ in the shape of Pac-Man.
Find $\iint_P x\,dA$ by converting the integral to <i>polar coordinates</i> then solving the integral.








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<li style="page-break-before:always">
<img src="0g/T3.divfield.jpg" class="rightalign" style="margin-right:-4em">
Let $\myv F$ be the vector field sketched in the diagram.
<ol type="a">
<li>
Where is the curl of $\myv F$ negative / zero / positive?

<li style="margin-top: 8ex">
Where is the divergence of $\myv F$ negative / zero / positive?

<li style="margin-top: 8ex">
Is $\myv F$ a conservative field? Why or why not?
</ol>


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<li style="margin-top: 15ex">
Let $\myv F(x,y)=(3+2xy)\uv i+(x^2-3y^2)\uv j$.
<ol type="a">
<li>
Use an appropriate test to show that $\myv F$ is a conservative vector field.
<li>
Find a potential function for $\myv F$.
<li>
Let $C$ be the curve $\myv r(t)=(\cos t +1)\uv i+(-\sin t)\uv j$ with 
$t=0\to\pi/2$.  Based on your answers to a) and b), indirectly compute 
$\int_C\myv F\cdot d\myv r$.
</ol>
</li>




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<li style="page-break-before:always">
<img src="0g/T3.vfplot.png" class="rightalign" style="margin-right:-4em">
Consider this vector field (at right).<br>
Draw and label curves from (-2,0) to (2,0) such that:
<br><br>
a. $\int_A\myv F\cdot d\myv r \gt 0$<br>
b. $\int_B\myv F\cdot d\myv r \lt 0$<br>
c. $\int_C\myv F\cdot d\myv r \approx  0$<br>


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<li style="margin-top:20ex;">
(10 pts) Let $L$ be the directed line segment from the point (1,4,2) to the point (3,4,1). Given $\myv F=2x\uv i+(y^2+z)\uv j+(x+y)\uv k$, evaluate the line integral
$\int_L \myv F \cdot d\myv r$.  Show all your work.






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<li style="page-break-before:always">
<img src="0g/T3.Jacobian.jpg" class="rightalign">Let $R$ be the region shown in the $xy$ plane, bounded by 4 straight line segments. Using the following change of variables:
$$u=2x-y$$
$$v=4y-x$$
<ol type="a">
<li>Sketch the region in the $uv$ plane that corresponds to $R$.
<li>Find expressions for $x(u,v)$ and $y(u,v)$ in terms of $u$ and $v$ and find the Jacobian of the transformation.
<li>Using the transformation you just found, transform the integral $\\int_R2x\,dA$ to an integral in the $uv$ coordinate system.  Write the integral out in terms of $u$ and $v$, including the limits on the integrals.  But you do not need to evaluate the integral.
</ol>

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<li style="page-break-before:always">
Let $\myv F(x,y)=(\sin x+ xy^2)\uv i+\left(e^y+\frac 12 x^2\right)\uv j$. 
<img src="0g/T3.Green.jpg" class="rightalign" style="margin-right:-2em">
Compute the value of the line integral of $\oint_C \myv F\cdot d\myv r$ where
$C$ is the closed path (at right), by means of...
<ol type="a"><li>Converting the line integral to a double integral using Green's theorem (show the resulting double integral),
<li>Then evaluate the double integral.
</ol>

</ol>


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