Spherical and Cylindrical Coordinates
Eric Auld
January 14, 2016
Bellwork
I
Sketch the region of integration of the following integral, and
then reverse the order of integration to get an equivalent
integral:
Z y =1 Z x=y
dx dy
y =0
I
x=y
Suppose we have
Math 32B: Calculus of several variables, Winter 2017
Instructor: Damir Yeliussizov
Textbook
J. Rogawski and C. Adams, Multivariable Calculus, 3rd ed.
General information and outline of the course can be found h
ere.
Homework
There will be weekly homework
Math 32A 2013 Midterm 1
Problem 1 [20 Points): Let E = (a . b) denote a non-zero 2D vector.
Part A (10 points): Find a vector (unique up to a sign) which is orthogonal to E and has the same
length as 271 . [Show a little work]
Part B (10 Points): Find the
History 1A
Introduction to Western Civilization:
Ancient Civilizations to ca. A.D. 843
Fall 2015
Dr. David Phillips
Teaching Assistant: Elle Harvell
Office: Bunche 2113
Email: eharvell@ucla.edu
Office Hours: Wednesday 1-2 pm
Friday 11-12 pm
Sections 1A Mo
1. (5 points) Determine if the lines given by the vector parametrizations
r1(t) = (5,—16,19)+t(1,—3,4) and r2(t) = (5,—1,—11) +t(—2, 1,2)
intersect and, if so, ﬁnd the point of intersection.
I-I- Ema (deﬁed: ad: r‘C-h) = 9,672) . than we have.
a 5* +. we
MATH 32A
PRACTICE FINAL EXAMINATION
Monday, June 6th 2016
Please show your work. You will receive little or no credit for a correct answer to a
problem which is not accompanied by sufficient explanations. If you have a question about
any particular proble
Math 32A, Lecture 1
Multivariable Calculus
Midterm 1
Instructions: You have 50 minutes to complete the exam. There are ﬁve problem's, worth
a total of ﬁfty points. You may not use any books, notes, or calculators. Show all your work;
partial credit will b
Math 32A
Spring 2016
Sample for Exam 1
1/27/16
Time Limit: 50 Minutes
Name (Print):
Student ID Number:
Teaching Assistant
This exam contains 9 pages (including this cover page) and 8 problems. Check to see if any pages
are missing before you turn in your
MATH 32A
FIRST MIDTERM EXAMINATION
Practice Midterm
Please show your work. You will receive little or no credit for a correct answer to a
problem which is not accompanied by sufficient explanations. If you have a question about
any particular problem, ple
P RACTICE M IDTERM 1
PLEASE NOTE:
The midterm will be in
room 2160E Dickson,
NOT the usual classroom.
Instructions.
Please show your work. You will receive little or no credit for an answer
not accompanied by appropriate explanations, even if the answer i
Building Meeting 11/9/15
DONATIONS
1. Cirque du Soleil KURIOS: Cabinet of Curiosities
Wednesday Jan. 13 8-10 PM. 100 residents total (50 Hitch Suites/50 Rieber Terrace).
$200 from Rieber Terrace (150 for bus, 50 for tickets). Each resident would pay $35
MATH 32B
PRACTICE FIRST MIDTERM EXAMINATION
Please show your work. You will receive little or no credit for a correct answer to a
problem which is not accompanied by sufficient explanations. If you have a question about
any particular problem, please rais
Problem Set 2
Due Friday, January 29
Challenge Problems
CP5.
A 4-kg block rests on top of a 5-kg block, which rests on a frictionless table. The coefficient
of friction between the two blocks is such that the blocks start to slip when the horizontal
force
Name:
Last
First
Student ID #
Section:
Math 32A Sections 2 & 4 Fall 2015
Score
Problem Set # 9
Problem (1) A function f (x, y) is given by
f (x, y) = Arctan(2x + y)
could be expressed as a function h(s, t) where
x(s, t) = s2 t and y(s, t) = st2 .
Find an
Math 32A Quiz 5B Solution
Problem 1. Let r(t) be given by r(t) = t, et , tet . Find the decomposition of a(t) = r (t)
into tangential and normal components at t = 0.
Solution: First we compute that r (t) = 1, et , (1 + t)et and r (t) = 0, et , (2 + t)et .
Math 32A Quiz 5A Solution
Problem 1. Prove that aN =
av
v
. (Hint: start by decomposing a as aT T + aN N.)
Solution: Writing a = (aT T + aN N), we have
av
(aT T + aN N) v
=
v
v
aT T v + aN N v
=
v
aN N v
=
v
Note the last equality follows from the fact th
Math 32A Quiz 4B Solution
Name:
SID:
Problem 1. Find the length of the cycloid
r(t) = t sin(t), 1 cos(t)
on the interval 0 t 2. Recall the identity
sin2 (t/2) =
1 cos(t)
.
2
Solution: We compute
r (t) = 1 cos(t), sin(t)
r (t) =
(1 cos(t)2 + sin2 (t)
=
1 2
Math 32A Quiz 5A Solution
Problem 1. Prove that aN =
av
v
.
Solution: From Section 14.5 we have the formulas
a 2 aT
av
.
aT =
v
aN =
2
Because of the dot product formula |u v| = u v cos() (where is the angle between
the vectors), we have
aN =
a 2 aT 2
=
=
Math 32A Quiz 3B Solution
1
1
s
,
ds.
2 1 + s2
1+s
1
1
s
,
ds =
2 1 + s2
1+s
Problem 1. Evaluate
0
Solution:
We have
0
1
0
1
ds,
1 + s2
1
0
s
ds . Recall that
1 + s2
s
1
ds = arctan(s) + C. To evaluate
ds, we make the substitution u = 1 + s2
2
1+s
1 + s2
Math 32A Quiz 4A
Name:
SID:
Problem 1. Find the value of t in [0, 2] such that the speed of the cycloid
r(t) = ht sin(t), 1 cos(t)i
is at a maximum.
Solution:
We compute
r0 (t) = h1 cos(t), sin(t)i
q
0
kr (t)k = (1 cos(t)2 + sin2 (t)
q
= 1 2 cos(t) + cos2
Math 32A Quiz 2B Solution
Problem 1. Find parametric equations for the line through P0 = (3, 1, 1) perpendicular to
the plane 3x + 5y 7z = 29.
Solution: A normal vector to the plane is given by n = 3, 5, 7 , so our desired line goes
through (3, 1, 1) in t
Math 32A Quiz 3A Solution
Problem 1. Determine whether the space curves given by r 1 (t) = t, t2 , t + 1 and r 2 (s) =
s, s, s 1 intersect, and if they do, determine where.
s, s, s 1 to solve for s and t. Setting
Solution: Set r 1 (t) = t, t2 , t + 1 = r
Math 32A Quiz 2A Solution
Problem 1. Find the equation of the plane that contains the lines r 1 (t) = ht, 2t, 3ti and
r 2 (t) = h3t, t, 8ti.
Solution: Note that r 1 (0) = h0, 0, 0i = r 2 (0), so (0, 0, 0) lies on both lines. We then have
that v 1 = r 1 (1
Math 32A Quiz 1B Solution
1
2
2 along v = 0 .
Problem 1. Find the projection of u =
0
1
You must fully simplify your answer to receive full credit.
Solution: We use the formula
projv (u) =
uv
vv
2
4/9
2
0 =
0 .
v=
9
1
2/9
1