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{\bf NAME: (print!)}  \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_
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{\bf Section:}  \_\_\_\_
\qquad
 {\bf E-Mail address:}  \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

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MATH 251 (4,6,7 ), Dr. Z. ,  Exam 2, Tue., Nov. 21, 2017, SEC 118
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{\bf FRAME YOUR FINAL ANSWER(S) TO EACH PROBLEM}

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1. \qquad  (out of 10)    
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2.  \qquad  (out of 10)    
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3.   \qquad (out of 10)    
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4.   \qquad (out of 10)    
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5.   \qquad (out of 10)    
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6.    \qquad (out of 10)    
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7.    \qquad (out of 10)    
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8.    \qquad (out of 10)    
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9.    \qquad (out of 10)    
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10.    \qquad (out of 10)    
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{\bf MAKE SURE TO PUT THE TYPE!}

{\bf Types:} Number, Function of $variable$(s), 2D vector of numbers, 3D vector of numbers,
2D vector of functions (aka  2D vector-field), 3D vector of functions (aka 3D vector field), equation of a plane, parametric equation of a line,
equation of a line, equation of a surface, equation of a line, DNE (does not exist), abstract double-integral, abstract triple-integral.


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{\bf 1}. (10 pts.)  

 Find the Jacobian of the transformation from $(u,v,w)$-space to
$(x,y,z)$-space.
$$
x=uv+w \quad, \quad y=uw+v \quad, \quad z=vw+u \quad, \quad
$$
at the point $(u,v,w)=(2,2,2)$.


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The {\bf type} of the answers is:
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{\bf ans.} 
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{\bf 2.} (10 points altogether) 

(i) (3 points) Show that
$$
{\bf F} \, = \, \langle  \,
3\,{x}^{2}yz+yz+\cos \left( x+y+z \right) \, , \, {x}^{3}z+xz+\cos \left( x+y+z \right) 
\, , \, {x}^{3}y+xy+\cos \left( x+y+z \right)  \,
\rangle \quad ,
$$
is a conservative vector field. 
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(ii) (4 point) Find a function $f(x,y,z)$ such that ${\bf F}= \g f$.
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(iii) (3 points) Find the line-integral $\int_C \, {\bf F}.d{\bf r} $ where $C$ is the curve
$$
{\bf r}= \langle \sin t, \cos t +1, \sin 2t \rangle \quad, \quad 0 \leq t \leq \pi \quad .
$$


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The {\bf types} of the answer is: For (ii) \quad \quad \quad \quad \quad \quad \quad \quad \quad 
For (iii)
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{\bf answers} (ii) f(x,y,z)=
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(iii)
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{\bf 3.} (10 points)

 Sketch the region of integration
and change the order of integration.
$$
\int_{1}^{2} \int_0^{e^x+1} F(x,y) \, dy \, dx
$$
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The {\bf type} of the answer is:
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{\bf ans.} 
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{\bf 4.} (10 points)
Use Largange multipliers (no  credit for other methods) to
find the smallest value that $x+y+z$ can be, given that
$xyz=1$.


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The {\bf type} of the answer is:
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{\bf ans.} 
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{\bf 5.} (10 points) Compute the volume integral
$$
\int \, \int \, \int_E 48\,x\,y\,z\, dV \quad
$$
where $E$ is the region in 3D
$$
\{(x,y,z) \, | \, 0 \leq x \leq y \leq z \leq 1 \} \quad .
$$
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The {\bf type} of the answer is:
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{\bf 6.} (10 points)  By converting to polar coordinates, compute
$$
\int _{-3}^{3}\!\int _{-\sqrt {9-{x}^{2}}}^{\sqrt {9-{x}^{2}}}
\,{\frac { \left( {x}^{2}+{y}^{2} \right) ^{2}}{ 243 \pi }}{dy}\,{dx}
$$

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The {\bf type} of the answer is:
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{\bf ans.} 
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{\bf 7.} (10 points) Compute the line integral
$$
\int_C \frac{ 4 \sqrt{3} \, x \, y \, z}{3}  \, ds \quad,
$$
where $C$ is the line-segment joining $(0,0,0)$ and $(1,1,1)$

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The {\bf type} of the answer is:
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{\bf 8.} (10 points)  
 Compute
$$
\int_{0}^{3} \int_{\sqrt{y/3}}^{1}  \, e^{x^3} \, dx \, dy \quad .
$$
(Hint: Not even Dr. Z. can do $\int e^{x^3} \, dx$, so you must be clever,
and first change the order of integration.)


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The {\bf type} of the answer is:
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{\bf ans.} 
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{\bf 9.} (10 points) Compute the volume integral
$$
\int\int\int_E  \frac{5 \, (x^2+y^2+z^2)}{4\pi} \, dV \quad,
$$
where 
$$
E=\{(x,y,z) \, | \, x^2 \, + \, y^2 \, + \, z^2 \leq 1 \} \quad .
$$


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The {\bf type} of the answer(s) is:
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{\bf ans.}
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{\bf 10.} (10 points) Find $\g \cdot {\bf F}$ if
$$
{\bf F} \, = \, \langle \, \sin(xy) \, , \, \sin(yz) \, , \, \sin(xz) \, \rangle \quad .
$$

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The {\bf type} of the answer is:
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{\bf ans.}  
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