MATH 1C 2013 HOMEWORK 5 SOLUTIONS
Problem 1-6pts
10.5.11 Compute the line integral C (dx + dy )/(|x| + |y |), where C is the square with vertices (1, 0),
(0, 1), (1, 0), (0, 1), transferred once in counterclockwise direction.
Solution. Let F (x, y ) = (1/
Solutions, Midterm Exam, Math1c Analytical
Spring 2014
1) (a) Let V = p(U ) and let x V . We will nd an open interval containing
x and contained in V . Since x V , there is y such that (x, y) U . Since
U is open in R2 , there is an open rectangle R = (a,
MATH 1C 2014 HOMEWORK 5 SOLUTIONS
Problem 1-6pts
10.5.11 1st Solution.
Write f = (P, Q), where P (x, y) = 2x + y 2 , Q(x, y) = 2xy + 3y 2 . Then P = 2y = Q , so f is
y
x
conservative. Therefore f d = f d, where is the straight line path (t) = (1 2t, 0)
fr
MATH 1C 2014 HOMEWORK 7 SOLUTIONS
Problem 1-12pts
11.28.5 - 4 pts. Make a sketch of the region S and express the double integral I =
integral in polar coordinates.
S
f (x, y) dx dy as an iterated
S = cfw_(x, y) | x2 y 1, 1 x 1.
Solution. S is the set of p
Homework Set 5, Math1c Analytical, Spring 2014
Due Monday, May 12 at 10:00 am
Do all ve problems. Give details; part of what one must learn - in the analytical track at
least - is precise logical reasoning with no gaps. You may use any result from the cla
MATH 1C 2014 HOMEWORK 6 SOLUTIONS
Problem 1-6pts
11.22.1(e) - 6pts Use Greens theorem to evaluate the line integral
(t) = (2 cos3 t, 2 sin3 t), where t [0, 2].
C
y 2 dx+xdy, when C has the vector equation
Solution. By Greens theorem,
y 2 dx + xdy =
C
R
2
MATH 1C 2013 HOMEWORK 4 SOLUTIONS
Problem 1-12pts
9.13.21 - 6 pts Given n > 1 distinct x1 , , xn and n (not necessarily distinct) numbers y1 , , yn , we want
to nd a and b that minimize the total square error of the linear approximation f (x) = ax + b:
n
Chapter 7
Div, grad, and curl
7.1
The operator
and the gradient:
Recall that the gradient of a dierentiable scalar eld on an open set D in
Rn is given by the formula:
,
,.,
.
(7.1)
x1 x2
xn
It is often convenient to dene formally the dierential operator
Chapter 6
Greens Theorem in the Plane
Recall the following special case of a general fact proved in the previous
chapter. Let C be a piecewise C 1 plane curve, i.e., a curve in R2 dened
by a piecewise C 1 -function
: [a, b] R2
with end points (a) and (b)
Homework Set 8, Math1c Analytical, Spring 2014
Due Monday, June 2 at 10:00 am
Do all ve problems. Give details; part of what one must learn - in the analytical track at
least - is precise logical reasoning with no gaps. You may use any result from the cla
Homework Set 7, Math1c Analytical, Spring 2014
Due Tuesday, May 27 at 10:00 am
NOTE CHANGE IN DATE
Do all four problems. Give details; part of what one must learn - in the analytical track at
least - is precise logical reasoning with no gaps. You may use
Homework Set 6, Math1c Analytical, Spring 2014
Due Monday, May 19 at 10:00 am
Do all four problems. Give details; part of what one must learn - in the analytical track at
least - is precise logical reasoning with no gaps. You may use any result from the c
Homework Set 2, Math1c Analytical, Spring 2014
Due Monday, April 14 at 10:00 am
Do all ve problems. Give details; part of what one must learn - in the analytical track at
least - is precise logical reasoning with no gaps. You may use any result from the c
Chapter 9
The Theorems of Stokes and
Gauss
1
Stokes Theorem
This is a natural generalization of Greens theorem in the plane to parametrized
surfaces in 3-space with boundary the image of a Jordan curve. We say
that is smooth if every point on it admits a
MATH 1C 2014 HOMEWORK 1 SOLUTIONS
Problem 1-12pts
8.3.1(f)-4pts Make a sketch to describe the level set f (x, y, z) = sin(x2 +y 2 +z 2 ) = c, where c = 1, 1/2, 0, (1/2) 2, 1.
Solution. If we want to x f (x, y, z) at a value c, we simply x x2 + y 2 + z 2 a
Chapter 8
Change of Variables,
Parametrizations, Surface Integrals
8.1
The transformation formula
In evaluating any integral, if the integral depends on an auxiliary function of the
variables involved, it is often a good idea to change variables and try t
Chapter 4
Multiple Integrals
4.1
Basic notions
We will rst discuss the question of integrability of bounded functions on closed rectangular
boxes, and then move on to integration over slightly more general regions.
Recall that in one variable calculus, th
Chapter 5
Line Integrals
A basic problem in higher dimensions is the following. Suppose we are given a function
(scalar eld) g on Rn and a bounded curve C in Rn , can one dene a suitable integral of g
over C? We can try the following at rst. Since C is bo
Chapter 2
Dierentiation in higher dimensions
2.1
The Total Derivative
Recall that if f : R R is a 1-variable function, and a R, we say that f is dierentiable
at x = a if and only if the ratio f (a+h)f (a) tends to a nite limit, denoted f (a), as h tends
h
Chapter 3
Tangent spaces, normals and extrema
If S is a surface in 3-space, with a point a S where S looks smooth, i.e., without any
fold or cusp or self-crossing, we can intuitively dene the tangent plane to S at a as follows.
Consider a plane which lies
VECTOR CALCULUS
Notes for Math1c, D. Ramakrishnan
Chapter 1
Subsets of Euclidean space, vector
elds, and continuity
1.1
Introduction
The subject matter of this course is the study of functions
f : Rn Rm
for general n and m (one may summarize the previous
Homework Set 4, Math1c Analytical, Spring 2014
Due Monday, April 28 at 10:00 am
Do all four problems. Give details; part of what one must learn - in the analytical track at
least - is precise logical reasoning with no gaps. You may use any result from the
Homework Set 3, Math1c Analytical, Spring 2014
Due Monday, April 21 at 10:00 am
Do all four problems. Give details; part of what one must learn - in the analytical track at
least - is precise logical reasoning with no gaps. You may use any result from the
MATH 1C 2012 HOMEWORK 6 SOLUTIONS
Problem 1-12pts
11.22.1(e) - 6pts Use Greens theorem to evaluate the line integral
(t) = (2 cos3 t, 2 sin3 t), where t [0, 2 ].
C
y 2 dx + xdy , when C has the vector equation
Solution. By Greens theorem,
y 2 dx + xdy =
C
MATH 1C 2013 HOMEWORK 3 SOLUTIONS
Problem 1-12pts
8.17.1(b) - 6 pts Solution. We get:
F (t) = f (X (t), Y (t)X (t) +
x
f
y (X (t), Y
(t)Y (t),
f
d f
(X (t), Y (t)X (t) +
(X (t), Y (t)Y (t)
x
dt y
d f
f
d f
f
=
(X (t), Y (t) X (t) +
(X (t), Y (t)X (t) +
(X
MATH 1C 2013 HOMEWORK 2 SOLUTIONS
Problem 1-12pts
8.9.9-6pts Compute all rst-order partial derivatives of the function f (x) =
aij = aji .
n
i=1
n
j =1
aij xi xj , where
Solution. By Theorem 8.3or by writing out the limitwe see that f /xi is what we get i
MATH 1C 2013 HOMEWORK 1 SOLUTIONS
Problem 1-12pts
8.3.1(f)-4pts Make a sketch to describe the level set f (x, y, z ) = sin(x2 +y 2 +z 2 ) = c, where c = 1, 1/2, 0, (1/2) 2, 1.
Solution. If we want to x f (x, y, z ) at a value c, we simply x x2 + y 2 + z 2
Solutions, Final Exam, Math1c Analytical
Spring 2013
1) The region R is the interior of the triangle with vertices (0, 0), (1, 0), (1, 1).
3
This is a type I region and since the function f (x, y ) = yex is continuous,
we can apply Fubinis Theorem to get
MA1C 2010 HOMEWORK 3 SOLUTIONS
Problem 1
Let f (x, y ) = |xy |
a. Verify that f /x and f /y are both zero at the origin.
b. Does the surface z = f (x, y ) have a tangent plane at the origin? [Hint: Consider the
section of the surface made by the plane x =
Ma 1c
HOMEWORK ASSIGMENT 2
INSTRUCTOR:
D. RAMAKRISHNAN
278 SLOAN
SPRING 10
4348
ANALYTICAL TRACK
It is advisable to do all the problems. But the only ones which need to be turned
in are those which are underlined and in bold, like 20, and the starred prob