Maple Lab 4 (Spring 2007 Math 251) 

Section 05-07 

Answer Key 

Exercises 

Note: Use appropriate colors and plot options to make your plots look nice. The appearance of 

          you lab will be factored into your grade. You may add execution groups if you need more. 

          Please delete unused execution groups. 

>

 

 

 

Warning, the assigned names `<,>` and `<|>` now have a global binding
Warning, these protected names have been redefined and unprotected: `*`, `+`, `-`, `.`, D, Vector, diff, int, limit, series
Warning, the name changecoords has been redefined

 

1. Define a vector field  

   and a parametric curve where . 

>

 

(1)

 

>

 

(2)

 

1(a).  Plot the parametric curve. 

>

 

1(b). Compute the line integral  where is the curve parametrized by 

        . Compute the integral by plugging in the parametrization and computing 

        directly. 

>

 

(3)

 

1(c). Compute the line integral  by computing . 

>

 

(4)

 

>

 

(5)

 

>

 

(6)

 

1(d). Check to see if the vector field conservative. 

       If it is, compute  using the fundmental theorem  

       of line integrals. 

>

 

(7)

 

Since is the zero vector field, must be conservative. 

>

 

(8)

 

>

 

(9)

 

>

 

(10)

 

>

 

(11)

 

2. [Section 16.4 #6]  

   Let be the curve which consists of the parabola 

    from to and then the line segment from 

   to.  

   Plot and define the functions: 

   and . 

>

 

(12)

 

>

 

(13)

 

2(a). Compute the line integral  

>

 

(14)

 

>

 

(15)

 

>

 

(16)

 

>

 

(17)

 

>

 

(18)

 

2(b). Use Green's Theorem to turn the line integral from part 2(a) 

       into an equivalent double integral and verify the Green's 

       Theorem holds. 

>

 

(19)

 

>

 

(20)

 

3. Consider the surface where . 

   Plot this surface. 

>

 

3(a). Find formulas for both orientations of this surface using 

       the gradient operator. Decide which orientation points 

       "upward" and which points "downward" and label the 

       orientations accordingly. 

>

 

(21)

 

>

 

(22)

 

>

 

(23)

 

>

 

(24)

 

3(b). This surface can be parametrized using the equations: 

        ,   ,  and   

        where and . 

        Define a such a parametrization: 

>

 

(25)

 

3(c).  Compute both orientations for this surface again, but 

        this time use the parametrization. Again label them as 

        either "upward" or "downward". 

>

 

(26)

 

>

 

(27)

 

>

 

(28)

 

>

 

(29)

 

3(d).  Plot this surface (using 's parametric plot) along  

        with a few sample normal vectors (using your downward orientation). 

>

 

>

 

>

 

>