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Quiz 9 * Math 335 * Prof. Victor Matveev
1. (16pts) Verify the divergence theorem for vector field F=(0, 0, z) and the volume enclosed between the
surface z x 2 y 2 3 and the z = 1 plane
F dS F dV
W
W
The circular paraboloid and
Math 335-002 * Homework #4 * Due date: February 19
1. Find the domain and range of scalar field in R2, f (x, y) = ln(x y2), and sketch its
level curves. To do this, solve the equation f (x,y)=k=const, and plot these curves for
several values of k
Domain:
Math 335-002 * Spring 2015 * Solution to homework #1
1. For a = (1, -1, 1) and b = (0, 2, 1), find the area of the triangle formed by
these two vectors, and find the projection of a onto the direction of b .
i j k
a 1, 1, 1
1
1
1
1
9 1 4
A a b 1 1 1 3,
Math 335-002
Homework #10 * Spring 2015 * Prof. Victor Matveev
Please show all work in detail to receive full credit. Late homework is not accepted.
1. Use cylindrical coordinates to find the mass and the center of mass of an object with density
r x 2 y
Math 335-002 * Spring 2015
Homework #1
Due date: Thursday, January 29, 2015
Please show all work in detail to receive full credit
1. For a = (1, -1, 1) and b = (0, 2, 1), find the area of the triangle formed by
these two vectors, and find the projection o
Math 335-002 * Homework #8 * Due Monday March 30, 2015
Please show all work in detail to receive full credit. Late homework is not accepted.
1. Let F and G denote two differentiable vector fields in R3. Prove the following product rule by calculating
the
Math 335-002 * Homework #10 * Spring 2015 * Prof. Victor Matveev
1. Use cylindrical coordinates to find the mass and the center of mass of an object with density
r x 2 y 2 enclosed between the z=0 plane and the paraboloid z 1 x 2 y 2 .
From the rotationa
Math 335-002
Homework #9B * Spring 2015 * Prof. Victor Matveev
Please show all work in detail to receive full credit. Late homework is not accepted.
1. Calculate
y dV , where the volume of integration B is bounded in by the planes z=1x2y,
B
z=0, y=0 and
Math 335-002 * Homework #5 * Due date: February 26
1. Sketch this space-curve in R2: c(t) = ( 2 cos(t/2), sin(t/2) ), tR; calculate and plot its
tangent vector at points t= and t=2.
This is an ellipse, since for all parameter values t, x(t) and y(t) satis
Math335
Homework12
Spring2015
1. VerifytheGreensTheoremforthevectorfield F r 1, x
2
,andtheregionenclosed
2
betweenthecurves y x x and y x .Startbysketchingtheregionofintegration.
2
2
3
2. Usedivergencetheoremtofindthefluxofvectorfield F r sin y z
Math 335-002
Homework #9A * Spring 2015 * Prof. Victor Matveev
Please show all work in detail to receive full credit. Late homework is not accepted.
1. Use polar coordinates to integrate
D
ln x 2 y 2
x2 y2
dx dy over a ring domain D defined by
1 x 2 y 2
Math 335-002 * Homework #6 * Due date: March 5, 2015
Please show all work in detail to receive full credit. Late homework is not accepted.
1. Section 3.1:
A function satisfying the Laplaces equation is called harmonic. Which of the
following functions is/
Math 335-002 * Homework #5 * Due date: February 26
Please show all work in detail to receive full credit. Late homework is not accepted.
1. Sketch this space-curve in R2: c(t) = ( 2 cos(t/2), sin(t/2) ), tR; calculate and plot its
tangent vector at points
Math 335-002 * Homework #4 * Due date: February 19
Please show all work in detail to receive full credit. Late homework is not accepted.
1. Find the domain and range of scalar field in R2, f (x, y) = ln(x y2), and sketch its
level curves. To do this, solv
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Math 335-002 * Spring 2015 * Quiz #1
1. Consider the following vector operations. Which of them do/does not make
sense? If the expression is valid, indicate whether the result is a vector or a
scalar (number)
a)
b)
c)
d)
e)
( (a
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Math 335-002 Spring 2015 Quiz #6 Prof. Victor Matveev
1. Consider calculating the mass of a circular cone of base radius R=2 and height H=2
with density r z 2 (see Figure below).
R=2
H=2
a) Set up the integral for calculating this
Math 335-002 * Spring 2015 * Quiz #2
1. Consider the following vector operations. Which of them give zero answer for any
two vectors a and b ? Which do/does not make sense?
a) ( a b ) ( a b ) = | a b |2 not zero unless a = b
b) ( a b ) ( a b )
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Quiz 8 * Math 335 * Prof. Victor Matveev
1. (20pts) Calculate the flux
F dS of the vector field F=(y , y, 0)
2
S
across the curved surface
x y 2 z 2 constrained between the planes x=0 and x=2, with the normal pointing outward. Us
Math 335-002 Spring 2015 Quiz #6 Prof. Victor Matveev
1. Consider calculating the mass of a circular cone of base radius R=2 and height H=2
with density r z 2 (see Figure below).
R=2
Note that the equation of conical
curved side surface is z = r, or,
H=2
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Quiz 8 * Math 335 * Prof. Victor Matveev
1. (16pts) Calculate the flux
F dS of the vector field F=(y , y, 0)
2
S
across the curved surface
x y 2 z 2 constrained between the planes x=0 and x=2, with the normal pointing outward. Us
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Math 335-002 * Spring 2015 * Quiz #5 * Prof. Victor Matveev
1. Suppose F is a C2 (twice differentiable) vector field in R3. Which of the following
expressions are meaningful, and which are nonsense? For those which are
meaningful,
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Math 335-002 * Spring 2015 * Quiz #3
1. Find the domain and range of scalar field f (x, y) = e x y , and sketch its level
curves f =0, f =1, f =2.
2. Use linearization to estimate the value of scalar field in R3, f (r)= ln 2 x y 3
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Math 335-002 * Spring 2015 * Quiz #4
1. Make a rough sketch of these space-curves, along with the tangent vector at the endpoint t=10:
(a) c (t ) e 2 , t , t (0.1, 10)
2
(b) c (t ) ,
t
t , t (0.1, 10)
Hint: both are simple, standa
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Math 335-002 * Spring 2015 * Quiz #2
1. Consider the following vector operations. Which of them give zero answer for
any two vectors a and b ? Which do/does not make sense?
a)
b)
c)
d)
e)
f)
(a b ) (a b )
(a b ) (a b )
b
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Math 335-002 * Spring 2015 * Quiz #5 * Prof. Victor Matveev
1. Suppose F is a C2 (twice differentiable) vector field in R3. Which of the following
expressions are meaningful, and which are nonsense? For those which are
meaningful,
Please sign your name: _
Quiz 9 * Math 335 * Prof. Victor Matveev
1. (16pts) Verify the divergence theorem for vector field F=(0, 0, z) and the volume enclosed between the
surface z x 2 y 2 3 and the z = 1 plane
F dS F dV
W
W
2. (4pts) Without performing
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Quiz 7: Find the line integral of the vector field F = (x2y, 3y1/2, z) along the curve given by
r t
ln t , ln t,
2
1 ln t , t [1, 2] .
Quiz 6 make-up: Consider the volume enclosed in the first octant by the surface x 2 y z 4
a)
Quiz 7: Find the line integral of the vector field F = (x2y, 3y1/2, z) along the curve given
by r t
ln t , ln t,
1 ln t , t [1, 2] .
2
Method 1: reparametrize the curve: r u u 2 , u , 1 u , u [0, ln 2]
1
dr 2u , 1,
2 1 u
du
F r u u 5 ,3 u , 1 u
1
6
F
Math 335-002 * Spring 2015 * Quiz #4
1. Make a rough sketch of these space-curves, along with the tangent vector at the endpoint t=10:
(a) c (t ) e 2 , t , t (0.1, 10)
2
(b) c (t ) , t , t (0.1, 10)
t
dc
0,1
dt
c(10)=(e2, 10)
10
Hint: both are simple,
Math 335-002 * Spring 2015 * Quiz #3
1. Find the domain and range of scalar field f (x, y) = e x y , and sketch its level
curves f =0, f =1, f =2.
Domain: e x y 0 y e x Part of R 2 plane lying below the exponential curve
Range: [0,+)
Level curves: f e x y