Download Exam 3 Review Please take this opportunity to become skilled at

Exam 3 Review
Please take this opportunity to become skilled at working with spherical coordinates.
Spherical coordinates are useful in integrating over solids and in parameterizing surfaces,
and so mastery of them is necessary to master the course.
Integrals in Spherical coordinates
1) Use integration in spherical coordinates to find the center of mass of a solid hemisphere
of radius R with center at the origin, of density proportional to the distance to the xy
plane, bounded below by the xy plane. (Volume of a sphere of radius R is (4/3)πR3 . The
answer is (0, 0, 8 ∗ R/15).)
2) Use integration in spherical coordinates to find the center of mass of an orange slice of
radius R of constant density with center at at the origin formed from the planes x = 0 and
the plane which makes a 30◦ angle with the plane y = 0.
3) Use integration in spherical coordinates to find the center of mass of the hemispherical
shell of constant density, if the outer radius is 10 meters and the inner radius is 9 meters,
both spheres centered at the origin, and the hemispheres are the part of the spheres above
the xy plane. (The answer is (0, 0, 4.76m))
Parameterization of curves and surfaces
4) Find a parameterization of the line segment which starts at (1, 2, −3) and ends at
(−1, −4, 5). Remember to say what the domain is.
5) Find a paramterization of the line segement of length 5, which starts at (2, −1, 4) and
with direction vector < 1, 2, 2 >. Remember to say what the domain is.
6) Find a parameterization of the circle of radius 3 centered at the origin. Remember to
say what the domain is.
7) Find a parameterization of the arc of the circle, which starts at (6, 6) passes through
(−6, 6) and ends at (6, −6) which is centered at (0, 0). Remember to say what the domain
is.
8) Find a parameterization of the cylinder of radius 7, whose axis is the z-axis, whose
height is 14, base on the plane z = 3. Remember to say what the domain is.
9A) Find a parameterization of the cone whose tip is at the origin, whose axis is the z-axis,
whose height is 12 and whose radius is 4. Remember to say what the domain is.
9B) Use the linearization of your√
parameterization
to find a parameterization of the tangent
√
plane to the cone at the point ( 2, 2, 6).
10) Find a parameterization of the portion of the sphere of radius 8 centered at (0, 0, 0) in
the first octant. Remember to say what the domain is.
11) Find a parameterization of the portion of the parabola with equation y = 25 − x2
1
which lies above or on the x axis.
12) Find two parameterizations of the portion of the surface with equation z 2 = 16−x2 −y
which lies to the right of or on the xz plane. One parameterization should use rectangular
coordinates, the other cylindrical.
Calculating Line Integrals and Flux Integrals
13A) Calculate
Z
< −y, x, 0 > ·d~r
C
where C is the line segment in problem 5.
13B) Calculate
Z
< y, x > ·d~r
C
where C is the curve segment in problem 7.
14A) Calculate the flux integral of F~ =< x, y, 0 > over the cone of 9A). (Assume the
normal points away from the z-axis.)
14B) Calculate the flux integral of F~ =< x, y, z > over the portion of the sphere of 10).
(Assume the normal points away from the origin.)
Calculating Line Integrals and Flux Integrals by the Fundamental theorem,
Green’s theorem or the Divergence Theorem
15) Decide if the following vector fields are gradient fields–if they are find a scalar potential.
~ =< 3x2 − 3y 2 + sin(x), −6xy + y 2 >
a) V
~ =< 3x2 − 3y 2 , −5xy + y 2 >
b) V
16) Use the fundamental theorem to calculate
Z
< 2x, 2y, 2z > ·d~r
C
where C is the line segment in problem 4.
17) Use Green’s theorem to calculate the flux integral of F~ =< y 4 −x2 +2xy, x3 −y−x−y 2 >
around the circle of radius 4 centered at (0, 0).(Hint: Apply Switch and Flip to F~ , changing
the sign of the first entry after switching entries. Call the new vector field F~⊥ . Then the
line integral of F~⊥ around C is the flux integral of F~ around C. Notice that the divergence
of F~ is the scalar curl of F~⊥ .)
18) Use Green’s theorem to calculate the work done by F~ =< 3x2 − y, 3x + y 2 > on a
particle moving once around the circle of radius 4 centered at (6, 7) in a counterclockwise
direction.
2
19) Use the Divergence theorem to calculate the flux integral of F~ =< y 4 − x + 2xy, x3 −
y − x − y 2 , 2z > on the sphere of radius 4 centered at (6, 7, 8) with outward normal.
Understanding and Applications
20) Show how to do the integral in 13B using the fundamental theorem, instead of direct
calculation.
21) Explain why
Z
< x, y > ·d~r = 0
C
where C is any portion of the circle of radius R centered at the origin.
22A Use Green’s theorem to show that the integral of F~ = (x2K
+y 2 ) < −y, x > around any
fixed simple closed curve which contains the origin in the region D enclosed by the curve,
is the same as the integral of F~ around a circle of radius a centered at the origin, a small
enough so the the circle of radius a also lies inside D.
22B Use the result of 22A to show that the integral around the curve in 22A is 2Kπ.
23 Let
~
G(x,
y, z) =
(x2
G0 M
< −x, −y, −z > .
+ y 2 + z 2 )3/2
What is the flux over the ellipsoid with equation
2(x − 1)2 + 3(y − 2)2 + 4(z − 4)2 = 1?
(Hint: The answer is 0. Now explain why.)
24A Consider the cylinder of radius 2 with the z-axis as axis, 0 ≤ z ≤ 5, oriented using
outward normals. What is the flux of F~ =< −y, x, 0 > over the lateral surface of the
cylinder? How about over the top? (You do not need a parameterization to do this.)
Why?
24B What is the flux of F~ =< 5, 2, −1 > over the top end of the cylinder?
24C What is the flux of F~ =< 3x, 2y, 0 > over the cylinder?
25) The temperature u(x, y, z) over the cylinder with equation x2 + y 2 = 5, 0 ≤ z ≤ 10
is given by u(x, y, z) = 100(x2 + y 2 + z 2 ). What is the heat flux over the cylinder, if the
conductivity is 2?
26) Show that the electric flux over a sphere of radius R centered at (0, 0, 0) is Q/0 if
~ = Q 2 <x,y,z>
E
4π0 (x +y 2 +z 2 )3/2 .
R
~ · ~nds = Q/0 , where Q is the enclosed charge. Suppose
27) Gauss’s Law says that S E
an infinite wire along the z axis has a charge density of ρ Coulombs/m. Suppose we know
3
~ = (k/(x2 + y 2 )l ) < x, y, 0 > for some constants k and
that the electric field has the form E
l. Use Gauss’s law to find k and l. (Hint: it suffices to apply the Gauss law to a cylinder
of length 1m centered on the z-axis, but with a variable radius r. Applying the Gauss law
should give you an answer with a power of r in it. Since the formula is true for all r, the
power must be 0 which allows you to solve for l.) Deriving the formula for the electric field
from Gauss’s law is a common application of the Gauss law in electrostatics.
More Flux Integral Practice
To calculate the flux integral of a vector field over a surface S
1) If the surface is closed (ie. the boundary of a solid) use the divergence theorem.
(If the divergence theorem doesn’t apply because the vector field doesn’t have continuous
partials at points on the solid bounded by S, try 2C).)
2) If the surface is not closed try a sneaky trick.
A) Can you make the surface closed by adding on a piece, where you know the flux
over the piece, and the partials of F~ are continuous on the solid you get?
B) Is the dot product of F~ and a unit normal constant? If so, the flux is the value of
the dot product times the surface area.
C) Can we replace F~ by a simpler vector field which agrees with F~ on the surface we
are integrating over?
3) You can always use the formula.(Find a parameterization, calculate the partials of
the parameterization, take the cross product to find the normal, dot the normal and F~
then integrate.)
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1) (practice final #17) If F~ =< 3x, −2y, 0 >m/min is the velocity of a fluid, find the flux of
the fluid over the hemisphere with equation x2 + y 2 + z 2 = 9, z ≥ 0. Bottom not included,
use orientation with normal vectors pointing away from (0, 0, 0). (If you decide to use the
formula, just set it up.)
2) (practice final #19) Find the flux of the vector field
z
F~ = hx3 , ee sin x, 3zy 2 i
across the surface of the solid that is bounded by the cylinder {x2 + y 2 = 1} and the planes
z = −1 and z = 1(oriented by the outward normal).
Stokes Theorem practice. Stokes theorem will not be on the third test but will
be on the final.
Stokes theorem is useful in calculating line integrals over space curves with caps; it
~ over a surface if G is a curl
is also useful in calculating the integrals of vector fields G
field.Here are some typical problems.
27) p
(Final F’09) Let F~ (x, y, z) = (z − y)~i + (z + x)~j + xyz~k and S be the hemisphere
z = 1 − x2 − y 2 oriented upward.
a) (4 points ) Compute curlF~ .
4
R
~
b) (6 points) Use Stokes Theorem to calculate S curlF~ · dS.
R
28A) Calculate the integral C F~ · d~r where F~ = z + x2 , x + y 2 , z 2 and C is the boundary
of the surface S defined by z = 1 − x − y and x2 + y 2 ≤ 9. The surface S is oriented upward
and C is positively orientated with respect to the orientation of S.
28B) Same problem but F~ = −z + x2 , x + y 2 , z 2
28C) Same problem as 28A)but C is the boundary of the surface defined by z = −x − y
and x2 + y 2 + z 2 ≤ 9
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