Due Wednesday August 4
I prefer exact answers like useful for that problem.
√
2 instead of 1.414. Note that a symbol ´ indicates that graph paper might be
1. Arrange the following flux integrals,
Z
S i
~
· A , with i = 1 , 2 , 3 , 4, in ascending order if and S i are the following surfaces:
~
= −
~i
−
~j
+
~k
• S
1 is a horizontal square of side length 1, oriented upward with one corner at (0 , 0 , 2) and above the first quadrant of the xy -plane.
• S
2 is a horizontal square of side length 1, oriented upward with one corner at (0 , 0 , 3) and above the third quadrant of the xy -plane.
• S
3 is a square of side length 2 in the xz -plane with one corner at the origin, one edge along the positive x -axis, one along the negative z -axis,and oriented in the negative y -direction.
• S
4 is a square of side length 2 with one corner at the origin, one edge along the positive y -axis, one corner at the point (1 , 0 , 1), and oriented upwards.
2. Let S be the cube with side 2, faces parallel to the coordinate planes, and centered at the origin.
(a) Calculate the total flux of the constant vector field ~v = −
~i
+ 2
~j
+
~k out of S by computing the flux through each face seperately.
(b) Calculate the flux out of S for any constant vector field ~v = a~i + b~j + c~k .
3. Explain why if
~ has constant magnitude of 1 on of orientation, then
S and is everywhere normal to S and in the direction
Z
~
· A = Area of S.
S
4. A fluid is flowing along in a cylindrical pipe of radius a running in the ~i direction. The velocity of the fluid at a distance r from the center of the pipe is ~v = u (1 − r 2 /a 2 ) ~i .
(a) What is the velocity of the fluid at the wall of the pipe?
(b) Find the flux through a circular cross-section of the pipe.
5.
´ Compute the flux of the given vector field,
~ through the given surface S .
(a)
(b)
(c)
(d)
(e)
(f)
(g)
~
= z~i + y~j + 2 x~k ; S is the rectangle z = 4 , 0 ≤ x ≤ 2 , 0 ≤ y ≤ 3, oriented in the positive z -direction.
~ = x 2 ~i + ( x + e y ) ~j − ~k ; S is the rectangle y = − 4 , 0 ≤ x ≤ 7 , 0 ≤ z ≤ 3, oriented in the negative y -direction.
~ = ( x − y ) ~i + z~j + 3 x~k ; S is the region in the plane z = x + y above the rectangle 0 ≤ x ≤
2 , 0 ≤ y ≤ 3, oriented upward.
~ = ~r ; S is the surface z = x 2 + y 2 , oriented downward, above the disk x 2 + y 2 ≤ 1.
~
= x~i + y~j ; S is the part of the surface z = 25 − ( x 2
5 centered at the origin.
~
= cos( x 2 + y 2 )
~k
; S is as in part (e).
+ y 2 ) above R where R is the disc of radius
~
= − xz~i − yz~j + z 2
~k
; S is the cone z = p x 2 + y 2 for 0 ≤ z ≤ 6.

6. Compute R
S
~
· A in two ways if possible: directly and using the Divergence Theorem.
(a)
(b)
(c)
~
( ~r ) = ~r and S is the cube enclosing the volume 0 ≤ x ≤ 2, 0 ≤ y ≤ 2, 0 ≤ z ≤ 2.
~ ( x, y, z ) = y~j and S is a vertical cylinder of height 2, with its base a circle of radius 1 on the xy -plane, centered at the origin.
S includes the disks at the top and bottom of the cylinder.
~
( x, y, z ) = − z~i + x~k and S is a square pyramid with base on the xy -plane of side length 1 and height 3.
7. Use the Divergence Theorem to evaluate the flux integral R
S
~
· A , where and S is the sphere of radius 5 centered at the origin, oriented outward.
~
= x 2 ~i + ( y − 2 xy ) ~j + 10 z~k
8. Recall that a function φ ( x, y, z ) is said to be harmonic in a region if div ( grad φ ) = 0 at every point in the region. This equation is also written ∇ 2 a vector operator, ∇ = ∂
∂x
~i + ∂
∂y
~j + ∂
∂z
φ = 0, because div ( grad φ ) = ∇ · ( ∇ φ ). Recall that, as
~k
. Show that
∇ 2 φ ( x, y, z ) =
∂ 2 φ
∂x 2
+
∂ 2 φ
∂y 2
+
∂ 2 φ
.
∂z 2
9. Show that linear functions are harmonic.
10. Use the Divergence Theorem to show that if φ is harmonic in a region W , then R
S
∇ φ · every closed surface S in W such that the volume enclosed by S lies completely within W .
d ~ = 0 for
11. Show that a nonconstant harmonic function φ cannot have a local minimum. (Hint: see your notes).
12. Show that if φ is a harmonic function, then div ( φ grad φ ) = k grad φ k 2 .
13. Use the Divergence Theorem to show that if φ and ψ are harmonic functions in a region W , then
Z
S
φ ∇ ψ · d ~ =
Z
S
ψ ∇ φ · d ~ for every closed surface S in W such that the volume enclosed by S lies completely within W .
14. Compute the given line integrals using Stokes’ Theorem.
(a) R
C
~
· d~r where
~
= − z~i + y~j + x~k and C is a circle of radius 2 around the y -axis at y = 1 with orientation (given by the right hand rule) in the positive y -direction. (i.e., as you look from the origin, down the y -axis at C , the curve is oriented clockwise.)
(b) R
C
~
· d~r where
~
= 1 r
~r and C is the path consisting of straight line segments from (1
(1 , 0 , 0) to (0 , 0 , 1) back to (1 , 0 , 1).
, 0 , 1) to
15. Is there a vector field ~ such that curl ~ = y~i + x~j ? How do you know?
16. Determine whether vector potentials for
~ exist, and if so, find one.
(a)
(b)
~
= 2 x~i + (3 y − z 2 ) ~j + ( x − 5 z )
~k
~
= x 2
~i
+ y 2
~j
+ z 2
~k