Mathematics 2507 Final Exam Wednesday, 7 December 1994

NAME____________________________

Instructions: Use non-erasable ink. Write the answers where indicated and give clear evidence of your reasoning (or points will be taken off). You may attach extra sheets with your work if it is organized enough to be helpful. Graphs should be clearly labeled. Symbolic and graphing calculators are not permitted. On the other hand, an answer like \pi is actually preferable to an answer like 3.141592653589793238462643, so you do not really need a calculator at all.

In this test, an "explicit expression" means an expression that someone who knows calculus could evaluate without reading the statement of the problem or being told where the expression comes from. There should be no undefined terms like f(x,y), etc.

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TOTAL

NAME_____________________

1. (10 points). In the following you are asked to calculate several vectorial derivatives. In some cases the result may not be appropriate, in which case write "N.A." Otherwise, evaluate the derivative and write the answer on the appropriate line.


derivative     grad          div        curl      Laplacian

function or
vector field

x+y+z^2                   ________      ________    ________  ____N.A.___

x i + y j + z^2 k         ________      ________    ________  ____N.A.___



KEY FORMULA OR METHOD (optional for partial credit)____________________

_________________________________________________

2. (10 points) Let F(x,y,z) := x i + (y+z) j+ (z^2+y) k.

a) Is this vector field exact ? ____Y____N. Give the calculation(s) necessary to answer this question here:

b) Evaluate

where C1 is the closed loop given by x(t) = 3 + cos(t), y = 2 - 2 sin(t), z = t2 - 2 pi t,

0 <= t <= 2 pi .

c) Evaluate

where C2 is the non-closed curve given by x(t) = 3 + cos(t), y = 2 - 2 sin(t), z = t^2 - 2\pi t, 0 <= t <= \pi .

KEY FORMULA OR METHOD (optional for partial credit)____________________

_________________________________________________

NAME_____________________

3. (10 points) Let F = x i + y j + 3 x z k and let K be the unit cube K = {(x,y,z): 0 <= x <= 1, 0 <= y<= 1, 0 <= z <= 1}. The outward unit normal at its surface is called n. In this problem you are to evaluate surface integrals of F.n and curl F . n.

a) Let S1 be the entire surface of the cube. Use a theorem to rewrite the following as a different but equal integral:

b) Evaluate the integral of part a):

c) Calculate curl F = ______________________________________

d) Let S2 be the part of the surface of the cube for z > 0. Use a theorem to rewrite the following as a different but equal integral:

NOTE: There are at least two quite distinct correct answers to this.

e) Finally, evaluate the integral:

KEY FORMULA OR METHOD (optional for partial credit)____________________

_________________________________________________

4. (10 points). Consider the surface x(u,v) = u cos(v), y(u,v) = - u sin(v), z(u,v) = v, 0 <= u <= 1, 0 <= v <= pi /3. In case it is useful, note that z = - arctan(y/x). (SEE TABLE)

a) A normal vector to the surface at a particular point is given by N = ________________.

b) The surface area element is: d\sigma = _________________________________

c) The integral for the area in question is: Area = _____________________________

d) The value of the area is Area = ______________________________________

KEY FORMULA OR METHOD (optional for partial credit)____________________

_________________________________________________

NAME_____________________

5. (10 points). The temperature on a sphere of radius 2 centered at the origin is given by T(x,y,z) = x y - y z. Where are the hottest and coldest points and what are their temparatures?

(x,y,z) = _________________ is a ____hottest____/coldest point with T = __________

(x,y,z) = _________________ is a ____hottest____/coldest point with T = __________

(x,y,z) = _________________ is a ____hottest____/coldest point with T = __________

(x,y,z) = _________________ is a ____hottest____/coldest point with T = __________

KEY FORMULA OR METHOD (optional for partial credit)____________________

_________________________________________________

6. (10 points) Let S be the crescent-shaped region in polar coordinates bounded by the curves r = theta2 and r = pi theta /2 for 0 <= \theta <= pi /2. The mass density on S is equal to r.

a) Express the total mass as a double integral where the integral over r is performed first:

b) Express the total mass as a double integral where the integral over \theta is performed first:

c) What is the total mass?

Some possibly useful calculations by Mathematica

In[1]:= Integrate[Sqrt[1+x^2],x]

2
x Sqrt[1 + x ]   ArcSinh[x]
Out[1]= -------------- + ----------
2              2

In[2]:= Integrate[Sqrt[1+1/x^2],x]

2
1 + x             1
Out[2]= x Sqrt[------] - ArcSinh[-]
2              x
x

In[3]:= Simplify[D[ArcTan[y/x],x]]

y
Out[3]= -(-------)
2    2
x  + y

In[4]:= Simplify[D[ArcTan[y/x],y]]

x
Out[4]= (-------)
2    2
x  + y

In[5]:= Simplify[D[ArcCot[y/x],x]]

y
Out[5]= (-------)
2    2
x  + y

In[6]:= Simplify[D[ArcCot[y/x],y]]

x
Out[6]= -(-------)
2    2
x  + y