1 - Compute the gradient vector fields of the following functions: 2- 3- 4- 5- L

Question

1 - Compute the gradient vector fields of the following functions:

2-

3-

4-

5-

Let R be the rectangle with vertices (0,0), (8,0), (0,5), (8,5), and let C be the boundary of R traversed counterclockwise. For the vector field

find

6-

7-

8-

Let be the radial force field . Find the work done by this force along the following two curves, both which go from (0, 0) to (9, 81). (Compare your answers! Compute the line integrals directly from the definition. In the next set, you will do the same integrals using the Fundamental Theorem for Line Integrals.)

A. If is the parabola: , then

B. If is the straight line segment: , then

Explanation / Answer

a)f(x,y)=6x2+8y2

f=12x i +16y j

b)f(x,y)=x6y2

f=6x5y2 i +2x6y j

c)f(x,y)=6x+8y

f=6 i +8 j

d)f(x,y)=6x+8y+6z

f=6 i +8 j +8k

e)f(x,y)=6x2+8y2+6z2

f=12x i +16y j +12z k

2)a)C1

r(t)=<t,t2> ==>r'(t)=<1,2t>

F=<x,y>==>f(r(t))=<t,t2>

integral F. dX

=integral[0 to 9] <t,t2>. <1,2t> dt

=integral[0 to 9] t +2t3 dt

=l[0 to 9] (t2/2) +(2t4/4)

= (92/2) +(94/2)

=3321

b)C2

r(t)=<9t2,81t> ==>r'(t)=<18t,81>

F=<x,y>==>f(r(t))=<9t2,81t>

integral F. dX

=integral[0 to 1] <9t2,81t>. <18t,81> dt

=integral[0 to 1] 162t3+6561t dt

=l[0 to 1] 162t4/4 +6561t2/2

=(162/4) +(6561/2)

=3321

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