1.- Solve: a) find the arc length of the curve parametrized by r(t) = (3cos(t))i

Question

1.- Solve:
a) find the arc length of the curve parametrized by r(t) = (3cos(t))i + (3sin(t))k + (4t)k , 0<t<2pi
b) for F= (yx^3)i + (y^2)j find ?Fdr on the portion of the curve y=x^2 from (0,0) to (1,1).

2.-Let F = -2xz i + y^2 k.
(a) Calculate curlF.
(b) Show that ??R curl Fn dS = 0 for any finite portion R of the unit sphere S given by
x^2 + y^2 + z^2 = 1 with outward facing normal vector n.
(c) Show that ?CFdr = 0 for any simple closed curve C on the unit sphere x^2 + y^2 + z^2 = 1.

3.- Let S be the part of the spherical surface x^2 + y^2 + z^2 = 4, lying in x^2 + y^2 > 1, which is to say outside
the cylinder of radius one with axis the z-axis.
(a) Compute the flux outward through S of the vector field F = y i - x j + z k.
(b) Show that the flux of this vector field through any part of the cylindrical surface is zero.
(c) Using the divergence theorem applied to F, compute the volume of the region between
S and the cylinder.

4.- Let S be the surface formed by the part of the paraboloid z = 1- x^2

Explanation / Answer

Step 1: set values into vector units and find magnitude of [[ r'(t) ]]

r(t) =( 3 cos t ) i + (3 sin t ) k +( 4 t ) k, 0<t<2pi

r(t) = < 3 cos t, 0, 3 sin t + 4 t >   derivatives/

r'(t)= <-3 sin t, 0, 3 cos t + 4>     magnitude of [[ r'(t) ]]/

[[ r'(t) ]] = ( 24 cost + 16)^(1/2)   

Step 2: Now set up intergration   L = intregal (24 cost + 16) ^(1/2) note can you double check the problem

r(t) =( 3 cos t ) i + (3 sin t ) k +( 4 t ) k, 0<t<2pi      is   (3 sin t ) k    j???

I will due whole problem but i need you to post it agian, and i wont skip steps. you dont

have to post it rite away because in the mean time i work on it but i need to if K    is   J??    

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