Math 6C – Chapter 14 – Take-Home Quiz

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Your work should be neat and easy to read, or it will not be accepted.   

** This test is due on Thursday **

1.     Calculate the line integral,      ,  with   ,   over the elliptical path parametrized by     from t = 0 to t = .

(A) 0                  (B)         (C)             (D)             (E)   

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2.     Calculate the line integral,  ,  with   ,  over the path parametrized by     from t = 0 to t = 1.

(A) –22/15                 (B) 22/15           (C) –15/22            (D) 15/22             (E)  0     

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3.   Calculate   with ,   where  ,   over the path    parametrized by      from t = 0  to  t = 1.

(A) 0             (B) 2               (C) –2             (D) –1              (E)  1 

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4.   Suppose that                  with .        Then

  (A) 0             (B)           (C)         (D)         (E)   

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5.   Calculate    with    ,    over the path  from the  point  to   along the circle .

(A) 0            (B) 2           (C) –2            (D) –1            (E)  1

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6.    Suppose that      with .      Then

   (A)       (B)       (C)       (D)     (E)   

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7.   Calculate   with    ,    over the path  from the point ( 1, 2 )  to ( 3, 1 ) along the line segment connecting these two points.

   (A) 0             (B) 5             (C) –9               (D) 9              (E) –5       

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8.   Calculate   with    , over the path   from the point ( 0, 1 )  to   along the curve  y = cos x.

   (A) 0            (B)           (C)           (D)          (E)           

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9.  If    and   is the boundary of the square with vertices  (1,1), (3,1), (3,3), (1,3),  then       =

    (A)      (B)     (C)       (D)       (E)   

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10.  Suppose    and region R has an area of 2 and has
boundary that is a simple closed curve.   Then    =

   (A) 0             (B) 5               (C) –5             (D) –10              (E)  10       

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11.  Suppose , where 

.   If   is parametrized by   

   with ,   then  

 =

   (A) 0             (B) 1               (C) –1             (D)               (E)        

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12.    If   is the circle               ,  then    =  ?

   (A) 0             (B)               (C)               (D)              (E)           

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  13.  If    and   is the unit circle centered at the origin.  If R is the circular interior of  ,  then  

    =  ?      [ Hint:  this time it’s easier to compute the line integral! ]

   (A) 0             (B)               (C)               (D)              (E)           

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14.  Compute the flux of the field   through the portion of the surface   that is above (or below) the rectangle in the xy-plane with corners at (0,0), (3,0), (3,2), (0,2).

(A) 0            (B) 20             (C) –20             (D) –22              (E) –24

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15.  Compute the divergence of the vector field  .

(A) 0             (B) 1/2              (C) 2/3               (D) 3/4               (E) 4/5

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16.    Compute the outward flux of  through the surface  .

(A) 0             (B) 1/2              (C) 2/3               (D) 3/4               (E) 4/5

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17.   Compute the curl of the vector field .

(A) ( 0, 2y, –2z )          (B) ( 0, –2y, 2z )             (C) ( 2x, 0, –2z )            

                   (D) ( –2x, 2y, 0 )         (E) ( –2x, 0, –2z )      

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18.   Compute the circulation of  around the square curve with 

corners (0,0,1), (2,0,1), (2,2,1), (0,2,1).     

 

   (A) 0            (B) –8             (C) –9             (D) –10              (E) –12    

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19.   Suppose that  with  .   Then

   (A) 0            (B) –8             (C) –9             (D) –10              (E) –12  

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20.  Compute the circulation of the field  featured in #19,  around the square curve with corners (0,0,1), (2,0,1), (2,2,3), (0,2,3).      

    

 (A) 0            (B) –8             (C) –9             (D) –10              (E) –12