Problem Set #8

Concourse 18.02 Problem Set 8 – Fall 2014
due Thursday, November 20
15.1/4. Illustrate the vector field F ( x, y=
) 2 i + x j by sketching several typical vectors in the field. [You may
also wish to use technology to draw the vector field in an appropriately sized window.]
15.1/5. Illustrate the vector field F ( x, y ) =
( x 2 + y 2 ) 2 ( x i + y j) by sketching several typical vectors in the field.
[You may also wish to use technology to draw the vector field in an appropriately sized window.]
1
15.1/21. Calculate the divergence and curl of the vector field F( x, y, z ) = ( y 2 + z 2 ) i + ( x 2 + z 2 ) j + ( x 2 + y 2 ) k .
15.1/37. Verify the identity ∇ ⋅ r3 = 0 where r = x i + y j + z k and r = r .
r
15.2/9. Evaluate
∫
C
x 2 y dx + xy 3dy where C consists of the line segments from (−1,1) to (2,1) and from (2,1)
to (2,5) .
15.2/13. Evaluate
∫
C
F ⋅ T ds where F( x, y, z ) = y i − x j + z k and C is given parametrically by x = sin t ,
y = cos t , z = 2t , 0 ≤ t ≤ π .
15.2/16. Evaluate
∫
C
xyz ds where C is the straight line segment from (1, −1, 2) to (3, 2,5) .
15.2/19. Find the centroid of a uniform thin wire shaped like the semicircle x 2 + y 2 =,
a2 a > 0 , y ≥ 0 .
15.2/33a. Imagine an infinitely long and uniformly charged wire that coincides with the z-axis. The electric
k ( x i + y j)
force that it exerts on a unit charge at the point ( x, y ) ≠ (0, 0) in the xy-plane is F( x, y ) = 2
.
x + y2
[Can you show why??] Find the work done by F in moving a unit charge along the straight line
segment from (1, 0) to (1,1) .
SN-4B/2. For the following fields F and curves C, evaluate
∫
C
F ⋅ dr without any formal calculation, appealing
instead to the geometry of F and C.
(a) F
= x i + y j ; C is the counterclockwise circle, center at (0, 0) , radius a.
(b) F
= yi− x j;
C is the counterclockwise circle, center at (0, 0) , radius a.
15.3/3. Determine whether the vector field F( x, y ) = (3 x 2 + 2 y 2 ) i + (4 xy + 6 y 2 ) j is conservative. If so, find a
potential function (either by inspection – guess and check – or by using a more formal approach).
15.3/34. Let F( x, y, z ) = yz i + ( xz + y ) j + ( xy + 1) k . Define the function f by f ( x, y=
, z)
∫
C
F ⋅ T ds where C is
the straight line segment from (0, 0, 0) to ( x, y, z ) . Determine f by evaluating this line integral, then

show that ∇f =
F.
15.3/36. Let F = kr r 3 be the inverse-square force field of Example 7 in Section 15.2.
kr k ( x i + y j + z k )
. Show that the work done by F in moving a particle from a point at a
F( x, y, z=
) =
3
r 3 ( x2 + y 2 + z 2 ) 2
1 1
distance r1 from the origin to a point at a distance r2 from the origin is given by=
W k − .
 r1 r2 
1
SN-4C/5. For each of the following, tell for what value of the constants the field will be a gradient field, and for
this value, find the corresponding (mathematical) potential function.
b) =
F e x + y (( x + a ) i + x j)
a) F =( y 2 + 2 x) i + axy j
15.4/4. Apply Green’s Theorem to evaluate the integral
∫
C
( x 2 − y 2 ) dx + xy dy around the positively oriented
curve C that is the boundary of the region bounded by the line y = x and the parabola y = x 2 .
15.4/6. Apply Green’s Theorem to evaluate the integral
∫
C
∫
C
y 2 dx + (2 x − 3 y ) dy counterclockwise around the
9.
circle x 2 + y 2 =
15.4/9. Apply Green’s Theorem to evaluate the integral
y 2 dx + xy dy where C is the positively oriented
1.
ellipse with equation x 2 9 + y 2 4 =
15.4/29. Let R be the plane region with area A enclosed by the positively oriented piecewise-smooth simple
closed curve C (in the xy-plane). Use Green’s Theorem to show that the coordinates of the centroid
( x , y ) of R are
1
x 2 dy

∫
C
2A
x=
y= −
1
y 2 dx

∫
C
2A
1 , by using Green’s theorem. (This curve can be
SN-4D/3. Find the area inside the hypocycloid x 3 + y 3 =
2
2
parameterized by x = cos3 θ , y = sin 3 θ , between suitable limits on θ .)
SN-4E/3. Let =
F x 2 i + xy j . Evaluate
∫
C
F ⋅ n ds if C is given by r (t ) =(t + 1) i + t 2 j , where 0 ≤ t ≤ 1 ; the
positive direction on C is the direction of increasing t.
SN-4E/4. Take C to be the square of side 1 with opposite vertices at (0, 0) and (1,1) , directed clockwise. Let
F
= x i + y j ; find the flux across C.
SN-4F/4. Verify Green’s theorem in the normal form by calculating both sides and showing they are equal if
=
F x 2 i + xy j , and C is the square with opposite vertices at (0, 0) and (1,1) .
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