Quiz 9
Math 1321 - Accelerated Engineering Calc II
Name:
April 22, 2016
Quiz Score:
/10
Answer each question completely in the area below. Show all work and explain your reasoning.
If the work is at all ambiguous, it is considered incorrect. No phones, calculators, or notes are
allowed. Anyone found violating these rules will be asked to leave immediately. Point values
are in the square to the left of the question. If there are any other issues, please ask the
instructor.
932
2
CHAPTER 13
VECTOR CALCULUS
1. Suppose you’re asked to determine the trajectory that requires the least work for a force field F to
Comparing
equation
with Equation
16, we see that
move a particle from one point to another. If you were
providedthis
with
the additional
information
that F is conservative, how would you respond?
P!A" " K!A" ! P!B" " K!B"
which says that if an object moves from one point A to another poin
Solution: Although many of you answered somethingofalong
the linesforce
of Green’s
Theorem
a conservative
field, then
the sum or
of a
its potential energy
statement about a closed loop, this information was not
provided
in
the
problem.
remains constant. This is called the Law of Conservation of Ener
the vector
fieldof
is Line
calledIntegrals
conservative.
The answer I was hoping for: given that the Fundamental
Theorem
says that the
line integral of a conservative vector field only depends on the endpoints, the actual trajectory
you take does not matter.
13.3 Exercises
2
2. Suppose F = ∇f , a conservative vector field, and C is the curve shown below along with the
1. The figure shows a curve C and a contour map of a function f
9. F!x, y" ! !ln y " 2xy 3 " i " !3 x 2 y 2 "
contours of f .
whose gradient is continuous. Find xC $ f ! dr.
10. F!x, y" ! !x y cos x y " sin x y" i " !x 2
y
C
30
20
40
50
60
11. The figure shows the vector field F!x, y
three curves that start at (1, 2) and end
(a) Explain why xC F ! dr has the same
curves.
(b) What is this common value?
10
y
0
x
3
2. A table of values of a function f with continuous gradient is
given.justify
Find xCyour
$ f ! dr
, where C has parametric equations
Compute the following and
answer:
Z
x ! t2 " 1
y ! t3 " t
0%t%1
F · dr.
y
x
C
0
0
1
2
1
6
4
2
1
0
1
7
5 Fundamental
3 of the
1
2
Solution: This is an immediate consequence
Theorem of Line Integrals,
which says
2
9
2
8
Z
12–18 (a) Find a function f such that F ! ∇
∇f · dr = f (r(b)) − f (r(a)),
C
3–10 Determine whether
or not F is a conservative vector field.
part (a) to evaluate xC F ! dr along the given
2
12. F!x,
i " y 2 j,
y" !isxequal
If it is, find
a function
such thatof
.
F!
$ fcurve.
where a, b are the starting
and
endingf points
the
In this particular case,
this
C is the arc of the parabola y ! 2x 2 fro
to 50 − 10 = 40.
3. F!x, y" ! !2x ! 3y" i " !!3x " 4y ! 8" j
13. F!x, y" ! xy 2 i " x 2 y j,
4. F!x, y" ! e x sin y i " e x cos y j
C: r!t" ! # t " sin 12 & t, t " cos 12 & t $ ,
5. F!x, y" ! e cos y i " e sin y j
x
x
6. F!x, y" ! !2xy " y !2 " i " !x 2 ! 2xy !3 " j,
7. F!x, y" ! ! ye " sin y" i " !e " x cos y" j
x
x
8. F!x, y" ! !3x 2 ! 2y 2 " i " !4 xy " 3" j
y#0
y2
i " 2y arctan x j,
1 " x2
21 / 2
C: r!t" ! t i " 2t j, 0 % t % 1
14. F!x, y" !
15. F!x, y, z" ! yz i " xz j " !x y " 2z" k
C is the line segment from !1, 0, !2" to
Quiz 9
6
Math 1321 - Accelerated Engineering Calc II
April 22, 2016
3. Use Green’s Theorem to evaluate the line integral:
I
xy 2 dx + 2x 2 y dy
where C is the triangle with vertices (0, 0), (2, 2), (2, 4).
C
Solution: Green’s Theorem (ignoring all the technical conditions), says that if F = hP, Qi, then
I
I
ZZ ∂Q ∂P
F · dr =
P dx + Q dy =
−
dA.
∂y
C
C
D ∂x
Here, we see that P = xy 2 and Q = 2x 2 y meaning that Qx = 4xy and Py = 2xy and our
integrand is Qx − Py = 4xy − 2xy = 2xy .
5
4
3
2
1
0.5
1.0
1.5
2.0
2.5
3.0
To know the bounds of integration, we must draw the region. We could set this up as either
type 1 or type 2, but I think type 1 is easier here. Note that x varies between 0 and 2 and then
y is bounded by the two sides of the triangle described by y = 2x and y = x, thus:
Z 2 Z 2x
ZZ ∂Q ∂P
dA =
−
2xy dy dx
∂y
D ∂x
0
x
Z 2
2 y =2x
=
xy y =x dx
0
Z 2
=
3x 3 dx
0
3 4 x=2
=
x
= 12.
4
x=0
Something worth noting: only one person asked about the orientation of the triangle, which
everyone else assumed to be positive. In general, this DOES change the answer!
2/2