MATH 2220 HW8.
Due Wednesday 12 November
(1) Section 6.2 p. 392.
(a) # 29.
(2) Section 7.2 p. 447-451.
(a) # 12.
(b) # 18.
(3) Let C be the solid region bounded by the cone (z 1)2 = x2 + y 2 and the planes
z = 0 and z = 1.
Dene T : R3 R3 by T (x, y, z ) =
MATH 2220 HW8.
Due Wednesday 12 November
(1) Section 6.2 p. 392.
(a) # 29.
For part (a), one way to do this is to change coordinates. The easiest choice is
to take spherical coordinates but modify them slightly. Thus, we take
x = a cos() sin()
y = b sin()
MATH 2220 HW9.
Due Wednesday 19 November
(1) Find
C
F dr where C is the semicircle x2 + y 2 = 1, y 0, oriented anticlockwise,
and F(x, y ) = (y, x).
C may be parametrized as c(t) = (cos t, sin t) with 0 t . So the desired
integral is
( sin t, cos t) ( sin
MATH 2220 HW10.
Due Wednesday 26 November (or earlier)
(1) Let ai , bi , ci be real numbers. Dene a vector eld F : R3 R3 by
F(x, y, z ) = (a1 x + a2 y + a3 z, b1 x + b2 y + b3 z, c1 x + c2 y + c3 z ).
(a) Calculate curl(F).
Recall that curl(F) is
i
j
k
x
MATH 2220 HW11.
Due Friday 5 December
(1) Let S be the surface obtained by rotating the graph of the function
y = |x | + 1
about the xaxis between x = 2 and x = 2.
(a) Let F(x, y, z ) = (0, (x2 2)z, y ). Calculate
curl(F) dS
S
with respect to either orien
MATH 2220 PRELIM 1
You have 1 hour 30 minutes to complete this exam. The exam starts at 7:30pm. Each
question is worth 20 marks. There are 5 questions in total. You are free to use results from
the lectures, but you should clearly state any theorems you u
MATH 2220 PRELIM 1: SOLUTIONS
(1) State whether the following are true or false and justify your answer:
(a) The vectors (5, 7, 1) and (1, 0, 5) are orthogonal.
True, since
(5, 7, 1) (1, 0, 5) = 0.
(b) The angle between the vectors (0, 0, 1) and (5, 7, 1)
MATH 2220 PRELIM (PRACTICE)
You have 1 hour 30 minutes to complete this exam. The exam starts at 7:30pm. Each
question is worth 20 marks. There are 5 questions in total. The exam is printed on both
sides of the paper.
Good luck!
(1) Calculate:
(a)
1
x
sin
MATH 2220 PRELIM 2
You have 1 hour 30 minutes to complete this exam. The exam starts at 7:30pm. Each
question is worth 20 marks. There are 5 questions in total. You are free to use results from
the lectures, but you should clearly state any theorems you u
QUIZ
1. MENTAL
Calculate the following (it is possible to do it without using pen and paper).
(1) The outward ux of the vector eld F(x, y, z ) = (x, 0, 0) through the surface of the
cube [0, 2] [0, 2] [0, 2] with the cube [0, 1] [0, 1] [0, 1] removed.
(2)
MATH 2220 PRELIM (PRACTICE)
You have 1 hour 30 minutes to complete this exam. The exam starts at 7:30pm. Each
question is worth 20 marks. There are 5 questions in total. The exam is printed on both
sides of the paper.
Good luck!
(1) State whether the foll
Math 2220 Exam 2
Thursday, March 29, 2012
Name:
Show all work and explain all answers except as noted.
1. a. If =
x
x2 +y 2 dx
+
y
x2 +y 2 dy ,
calculate d .
b. For each k = 0, 1, 2, 3, describe how k -forms on R3 relate to either vector elds or
scalar fu
Math 2220 Exam 1 review (sample test)
1. a. Find the equation of the tangent plane to the surface dened by
ln(x2 + z 4 1) x2 (y 2) z = 0
at the point (1, 3, 1).
b. Find all values of (a, b) for which the tangent plane to z = x2 y xy 2 x + y + xy at
(x, y
MATH 2220 HW6.
Due Wednesday 15 October
(1) Section 3.5, p. 253-255.
(a) # 8.
The inverse function theorem tells us that we need to check that the matrix
1 + yz xz
xy
y
1+x
0
2
0
1 + 6z
is invertible at (0, 0, 0). Plugging in these values, the matrix beco
MATH 2220 HW4.
Due Wednesday 24 September
(1) Use the derivative to estimate the value of cos(0.02 cos(0.03).
Let f (x, y ) = cos(x cos(y ). Let x0 = (0, 0). Let x = (0.02, 0.03). We estimate
f (x) by using the linear approximation:
f (x0 ) +
f (x0 )(x x0
MATH 2220 HW6.
Due Wednesday 15 October
(1) Section 3.5, p. 253-255.
(a) # 8.
(b) # 12.
(2) Review Exercises, p. 255-259.
(a) # 5.
(b) # 15.
(c) # 22.
(3) Find the maximum and minimum values of the function f (x, y ) = ex + ey on the
line segment in R2 jo
MATH 2220 HW5.
Due Wednesday 8 October
(1) Section 3.3 p. 202.
(a) # 5.
(b) # 6.
(2) p. 222 - 225.
(a) # 6.
(b) # 33.
(c) # 41(b).
(3) Section 3.4 p. 243 - 246.
(a) # 5
(b) # 12.
(c) # 30.
(4) Let f : R2 R be a C function with a critical point at (0, 0).
MATH 2220 HW4.
Due Wednesday 24 September
(1) Use the derivative to estimate the value of cos(0.02 cos(0.03).
(2) Section 2.5 p.159-163
(a) # 11.
(b) # 15.
(c) # 24.
(3) Section 2.6 p.171-173
(a) # 4(b).
(b) # 14(a).
(c) # 15.
(4) The surface of a mountai
MATH 2220 HW3.
Due Wednesday 17 September
(1) Find
lim
(x,y )(0,0) x2
x
+ y2
or show that it does not exist.
(2) Section 2.3, p. 139-141
(a) # 1(d).
(b) # 4(b).
(c) # 5.
(d) # 7(c).
(e) # 9.
2
2
2
(3) The temperature at a point (x, y, z ) R3 is given by T
MATH 2220 SYLLABUS AND COURSE INFORMATION 2008
Note: some of the information in this document has not yet been nalized.
General information
Instructor: Dr. Richard Vale, Malott Hall 583.
rvale@math.cornell.edu
Lecture 1: MWF 0905-0955 in BKL (Baker Labora
MATH 2220 HW1 AND HW2.
Homework 1. Due Wednesday 3 September.
(1) Section 1.3, p. 6165
(a) # 8.
(b) # 24.
(c) # 36.
(2) Find a unit vector parallel to the line of intersection of the planes x 2y + 5z = 2
and 3x y + 5z = 3.
(3) Given the points P = (1, 2,
MATH 2220 FINAL EXAM
You have 2 hours 30 minutes to complete this exam. The exam starts at 7:00pm. Each
question is worth 20 marks. There are 8 questions in total. You are free to use results from
the lectures, but you should clearly state any theorems yo
MATH 2220 FINAL EXAM: SOLUTIONS
You have 2 hours 30 minutes to complete this exam. The exam starts at 7:00pm. Each
question is worth 20 marks. There are 8 questions in total. You are free to use results from
the lectures, but you should clearly state any
MATH 2220 HW1 SOLUTIONS.
Homework 1. Due Wednesday 3 September.
(1) Section 1.3, p. 6165
(a) # 8.
The volume is given by absolute value of the determinant
10
0
3 1
0 3 1 =
2 1
4 2 1
= 1.
So the volume is | 1| = 1.
(b) # 24.
The direction vector of the lin
MATH 2220 HW2 SOLUTIONS.
Homework 2. Due Wednesday 10 September.
(1) Section 2.1 p. 105107
(a) # 2a.
If c < 1, then the level curve is empty. If c = 1 then the level curve is a single
point (0, 0). If c > 1 then the level curve is a circle with midpoint (
MATH 2220 HW3.
Due Wednesday 17 September
(1) Find
lim
(x,y )(0,0) x2
x
+ y2
or show that it does not exist.
The limit does not exist. For example, if we approach (0, 0) along the path x = 0,
the limit is 0. If we approach along the path y = x, the limit
Math 2220 - Prelim #1 Solutions
1. For a xed constant a, let g (t) = (a2 t, at2 , t3 ).
(a) For which values of a is |g (a)|2 = 14?
We have
g (t) = (a2 , 2at, 3t2 ),
so
g (a) = (a2 , 2a2 , 3a2 )
and |g (a)|2 = (a2 )2 + (2a2 )2 + (3a2 )2 = 14a4 . Thus a =
Math 2220 HW 6 Solutions
March 12th, 2014
1. Using double integrals nd the volume of the region enclosed by the coordinate planes and the plane
3x ` 2y ` z 6.
Solution: Our function will be z 6 3x 2y . The region we integrate over in the xy -plane is boun
Math 2220 HW 5 Solutions
March 5th, 2014
1. Let f pu, v q peuv , evu q. For what values of u, v, x and y can f be inverted? Here xpu, v q and y pu, v q
are the rst and second coordinate of f pu, v q. Compute Dpf 1 q where possible.
Solution: Using the inv