Math 13 Worksheet #14: Curl and divergence
(1) Find the curl and divergence of F (x, y, z) =< ex sin y, ey sin z, ez sin x >.
(2) Determine if the vector eld F (x, y, z) =< 1, sin z, y cos z > is conservative. If it is, nd
a function f such that f = F .
(
Math 13 Worksheet #19: Stokes Theorem
(1) Verify Stokes Theorem for the vector eld F (x, y, z) =< y, x, ez > on the surface
dened by S = cfw_(x, y, z) : z = 1 x2 y 2 , x2 + y 2 1, with outward unit normal
vector.
(2) Use Stokes Theorem to evaluate to eval
(3) Evaluate ff curlF - ndS, where S is the cap of the unit sphere that lies bciow the
S
:Lyplane and inside the cylinder :52 +y2 = l with outwards-pointing normal vector and
9
where F :1: 1,2 x< 1 22,3;2213my2 >_
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Math 13: Written Homework # 3
Due April 16 at 5pm
Please make sure your homework is stapled, if necessary before handing it in. Do not use
paper clips or any variation of folding techniques to connect papers.
Solutions should be justied in a rigorous way.
Math 13: Written Homework # 5
Due April 30 at 5pm
Please make sure your homework is stapled, if necessary before handing it in. Do not use
paper clips or any variation of folding techniques to connect papers.
Solutions should be justied in a rigorous way.
Math 13, Spring 2014 Homework Solutions Week 8
(1) (Chapter 16.7, Problem #4) Suppose that f (x, y, z) = g( x2 + y 2 + z 2 ),
where g is a function of one variable such that g(2) = 5. Evaluate
f (x, y, z) dS, where S is the sphere x2 + y 2 + z 2 = 4.
S
So
Math 13: Written Homework # 9
Due May 28 at 5pm
Please make sure your homework is stapled, if necessary before handing it in. Do not use
paper clips or any variation of folding techniques to connect papers.
Solutions should be justied in a rigorous way. I
Math 13: Written Homework # 2
Due April 9 at 5pm
Please make sure your homework is stapled, if necessary before handing it in. Do not use
paper clips or any variation of folding techniques to connect papers.
Solutions be justied in a rigorous way. If you
Math 13: Written Homework # 1
Due April 2 at 5pm
Please make sure your homework is stapled, if necessary before handing it in. Do not use
paper clips or any variation of folding techniques to connect papers.
Solutions be justied in a rigorous way. If you
Worksheet #16
(1) Let a = 2i + 3j and b = 2i 3j and c = 5j. Find the following: (a) 2a 4b (b) a b
(c) |a|c a
(2) Find the cosine of the angle between a and b and make a sketch.
(a) a =< 1, 2 > b =< 6, 0 > (b) a =< 4, 7 > b =< 8, 10 >
(3) Write the vector
Math 13 Worksheet #13: Greens Theorem
(1) Use Greens Theorem to evaluate C F dr, where F =< y, 2x > and C is the boundary
of the region bounded by the x-axis and the curve y = 1x2 , transversed in the clockwise
direction.
(2) Evaluate C F dr where F =< 1
Math 13 Worksheet #10: Vector elds and work
(1) Draw the vector eld F (x, y) =< y, x y >.
(2) Is the vector eld F (x, y) =< 2xy, x2 + 1 > conservative? If so, nd the corresponding
potential function f .
(3) Find the gradient vector eld of f (x, y, z) = x
Math 13 Worksheet #11: Fundamental Thm for Line Integrals
For the following problems, use the Fundamental Thm for line integrals, if applicable, to evaluate
C F dr. Otherwise show that the vector eld is not conservative.
(1) F (x, y, z) =< z, 1, x > with
Worksheet #19
(1) Evaluate the limit.
lim
t2
sin(t)
t2 2t
i + t + 4j +
k
t2
ln (t 1)
Solution:
t2 2t
sin(t)
i + t + 4j +
k
t2
ln (t 1)
lim
t2
= 2i +
6j + k
(2) Sketch the curve r(t) =< t2 , t, 1 >. Use arrows to indicate the direction in which t
increases
Math 13 Worksheet #7: Vectors, dot product, cross product, and planes
(1) Set up the equation to nd the angle between the vectors P Q and P R with P (3, 1, 2),
Q(8, 2, 4), and R(1, 2, 3).
(2) Compute P Q P R. Geometrically what is the result?
(3) Find the
Worksheet #23
(1) Find the equation of the tangent plane to the surface z = 2e3y cos(2x) at (/3, 0, 1).
(2) Find all points on the surface z = x2 2xy y 2 8x + 4y, where the tangent plane is
horizontal.
(3) Use the total dierential dz to approximate the ch
Worksheet #18
(1) Find a parametric equation for the line through (1, 2, 3) and (4, 5, 6).
(2) Write both the parametric equations and the symmetric equations for the line through
the point (1, 1, 1) parallel to the vector < 10, 100, 1000 >.
(3) Show that
Worksheet #25
(1) Find the equation of the tangent plane and the normal line to the surface x+y +z = exyz
at (0, 0, 1).
(2) Find the directional derivative of f (x, y) = ex sin y at P (0, /4) in the direction of
a =< 1, 3 >.
(3) Find a unit vector in the
Worksheet #15
(1) For A(4, 4, 1) and B(4, 1, 4), nd a vector a with representation given by the directed
line segment AB . Draw AB and the equivalent representation starting at the orgin.
(2) Find a + b, 2a + 3b, |a|, and |a b| where a = 2i 4j + 4k, and b
Math 13: Written Homework # 7
Due May 14 at 5pm
Please make sure your homework is stapled, if necessary before handing it in. Do not use
paper clips or any variation of folding techniques to connect papers.
Solutions should be justied in a rigorous way. I
Math 13 Worksheet #8: Change of variables and the Jacobian
1
(1) Use the transformation x = 4 (u + v), y = 1 (v 3u) to evaluate the integral
4
(4x + 8y)dA,
R
where R is the parallelogram with vertices (1, 3), (1, 3), (3, 1), and (1, 5).
1
2
sin(9x2 + 4y 2
Worksheet #17
(1) Let a = 3i + 2j 2k, b = i + 2j 4k, and c = 7i + 3j 4k
ab
Solution:
i j k
a b = 3 2 2 =< 4, 10, 4 >
1 2 4
a (b + c)
Solution:
i j k
3 2 2
6 5 8
a (b + c) = a < 6, 5, 8 >=
=< 6, 36, 27 >
a (b + c)
Solution:
a (b + c) = a < 6, 5, 8 >= 8
Math 13 Worksheet #18: Divergence Thm
(1) Verify the conclusion of the Divergence Theorem for the vector eld F (x, y, z) =<
x2 , y 2 , z 2 > with the region R the unit ball centered at the origin.
(2) Evaluate the integral S F (x, y, z) ndS for F (x, y, z
Worksheet #17
(1) Let a = 3i + 2j 2k, b = i + 2j 4k, and c = 7i + 3j 4k
ab
a (b + c)
a (b + c)
(2) Find |u v| and determine whether u v is directed into the page or out of the page.
|v| = 5
= /4
|u| = 4
(3) Let P (1, 3, 1), Q(0, 5, 2), and R(4, 3, 1).
Math 13 Worksheet #12: Line integrals
(1) Give the vector eld that is being integrated.
xy 2 dx + (xy z)dy + cos ydz
C
(2) Compute C F dr where F (x, y, z) =< yz, x, z 2 > with C the straight line segment
from the origin to (1, 0, 4).
(3) Compute C F dr w