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 co
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 . Sinc
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 =
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 reas
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 equa
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 =
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 squa
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
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
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 of
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
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 curv
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 ge
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 reas
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
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
SOLUTIONS TO HW8
Problem 1
a) Conside the surface S = cfw_(x, y, z) : x2 + z2 = 1, 0 y 1, 0 x 1 and the vector
~ y, z) = ( y, 0, 0). If F~ represents the velocity of a fluid, then the amount of fluid
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
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
Homework Set 7, Math1c, Spring 2017
Practical Track
You may use any result from your class notes or the textbook (except exercises or solutions) but you must give a precise reference. Each student mus
Homework 7 Solutions
May 24th, 20017
Question 1
Part a
For the cylinder y2 + z2 = 1, a natural parametrization is y = cos , z = sin . Then, since we have the
additional constraint z = x, we have x = s
Homework Set 8, Math1c, Spring 2017
Practical Track
You may use any result from your class notes or the textbook (except exercises or solutions) but you must give a precise reference. Each student mus
Homework 5 Solutions
May 15, 2017
Problem 1
a) The integral curve of c(t) = ( t13 , et , 1t ) is F (x, y, z) = (3z 4 , y, z 2 ) if c0 (t) = F (c(t).
Indeed, c0 (t) = ( t34 , et , t12 ) and F (c(t) = F
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 -
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 re
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 re
MATH 1C 2013 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 =
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
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
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