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EE 212/01 Examination No. 1 Spring 2009/10
Kuwait University
Electrical Engineering Department
Name in Arabic : ………………………………………
Student I. D. : …………………………………….…
Signature : …………………………………….…
Problem No. Grade
1
/25
2
/25
3
/25
4
/25
Total
/100
2
Problem 1 (25 points):
a) (10 points): Find a parametric representation of the straight line through (0, 1, 0) and (-2, 3, 1)?
b) (5 points): Find a normal vector of the surface at the point (1, 1, 3)?
c) (5 points): Find the directional derivative of at the point (1, 1, 3) in the
direction of [
]?
d) (5 points): Find the Laplacian for ?
Problem 2:
a) (10 points): Find a curve integral ∫ ( )
for [
] along C the parabola
from A: (0, 0) to B: (2, 4)?
b) (10 points): Find a curve integral ∫ ( )
for [
] along C the line from
A: (0, 0) to B: (2, 4)?
c) (5 points): Show that the above integral is path independent
Problem 3 Given the vector function [
] and the surface S of the box
a) (10 points): Evaluate the surface integral ∬
by using the divergence theorem?
b) (15 points): Evaluate the surface integral ∬
directly?
Problem 4 (25 points): Find the trigonometric Fourier series for the following signal?
f(t)
t -1
2
-2
1
3
EE 212/01 Examination No. 2 Spring 2009/10
Kuwait University
Electrical Engineering Department
Name in Arabic : ………………………………………
Student I. D. : …………………………………….…
Signature : …………………………………….…
Problem No. Grade
1
/25
2
/25
3
/25
4
/25
Total
/100
4
Problem 1:
e) (20 points): Use ( ) ∫ ( )
to find the Fourier transform of
( ) {
?
f) (2.5 points): Sketch the amplitude spectrum?
g) (2.5 points): Sketch the phase spectrum?
Problem 2 (25 points): Find the temperature ( ) in a laterally insulated bar of thermal diffusivity
⁄ and length . The bar has initial temperature ( ) and is kept at
at the ends
Problem 3
a) (10 points): Use the Cauchy-Riemann equations to tell if ( ) ( ) is analytic or
not?
b) (15 points): Find all solutions z of ?
Problem 4 (25 points): Integrate ∮ ̅
counterclockwise around the triangle with vertices z= 0, 2,
2+2i ? ( ̅ is the complex conjugate of z)
5
EE 212/01 Final Examination Spring 2009/10
Kuwait University
Electrical Engineering Department
Name in Arabic : ………………………………………
Student I. D. : …………………………………….…
Signature : …………………………………….…
Problem No. Grade
1
/20
2
/20
3
/20
4
/15
5
/25
Total
/100
6
Problem 1:
h) (5 points): Find a parametric representation of the ellipse curve ( )
?
i) (10 points): Given the function ( ) and the vector function (
) . Show
that ( ) ( ) ?
j) (5 points): Given the function ( ) . Find ( ) ?
Problem 2
a) (10 points): Find the value of so that ∮
∮
for any
closed curve c?
b) (10 points): Calculate the closed curve integral ∮
by Stokes theorem for (
), C
the rectangle with vertices (1, 1, 1), (0, 1, 1), (1, 0, 2), (0, 0, 2)?
Problem 3
c) (10 points): Find the exponential Fourier series of the periodic signal
d) (10 points) Sketch its amplitude spectrum and phase spectrum (for )?
Problem 4 (15 points): Use the method of separating variables to determine solutions of the one-
dimensional wave equation ( )
( )
corresponding to the initial condition
( ) and the boundary conditions ( ) , ( ) , ( )
|
( ).
Problem 5
a) (10 points) use the residue theorem to evaluate the integral ∮
( )( )
counterclockwise around the closed path | | ?
7
b) (10 points) Find the eigenvalues and eigenvectors (in term of ) of the matrix [
] and
then diagonalize the matrix A?
c) (5 points) Use the truth table to tell if the compound proposition (( ) ) is logically
equivalent to the compound proposition ( ( )) ( ) or not?
8
EE212/11 Examination No. 1 Fall 2001/10
Kuwait University
Electrical Engineering Department
Name : ………………………………………
Student I. D. : …………………………………….…
Signature : …………………………………….…
Problem
No.
Grade
1
/ 20
2
/ 20
3
/ 20
4
/20
1
/20
Total
/ 100
9
Problem 1:
a) (5 points) Find two parametric representations of the curve: straight line from (-1, 1) to (1, 1)
with semi-circle from (1, 1) to (-1, 1) , y > 0?
b) (5 points) Find a normal vector of the surface 222 yxz at the point (1, 2, 6)?
c) (5 points) solve part b using different method?
d) (5 points) Find the directional derivative of zyxf 22 at the point (1, 1, 0) in the direction of
(0, 1, 1)?
Problem 2
a) (10 points) Is the curve integral C
dzydyxdxyx 32 path independent? Why
b) (10 points) evaluate the above integral along the curve: straight line from (0, 0, 0) to (1, 1, 1)?
Use the parametric representation in your answer.
Problem 3 (20 points): Use the Green theorem to evaluate
C
xy dyedxe 2 counterclockwise around
the boundary curve C of the region R: the rectangle with vertices (1, 1), (-1, 1), (1, -2), and (-1, -2)?
Evaluate the integral in the xy-plane.
Problem 4 (20 points): Evaluate the surface integral S
dAncurlF )( where
xz
z
F
3
10
2
and the
surface S: y = x, 40 x , 80 z ? Evaluate the integral in the xyz-plane without using the Stokes
theorem.
Problem 5 (20 points): Use the Stokes theorem to evaluate the surface integral S
dAncurlF )( ,
where
xz
z
F
3
10
2
and the surface S: y = x, 40 x , 80 z ? Use the parametric representation in
your answer.
10
EE 212/51 Examination No. 2 Fall 2010/11
Kuwait University
Electrical Engineering Department
Name : ………………………………………
Student I. D. : …………………………………….…
Signature : …………………………………….…
Problem No. Grade
1
/25
2
/25
3
/25
4
/25
Total
/100
11
Problem 1: Find the exponential Fourier series of the following function (simplify your final answer as
much as possible) and sketch its exponential spectra for ?
Hint ∫–
for
Problem 2: The steady state temperatures in the rectangle plate ( ) can be
described by the Laplace equation ( ) ( )
( )
. Let the
temperature ( ) on the upper side and ( ) on the other three sides of the rectangle
plate. Find the steady state temperature at ?
Problem 3: Reduce the differential equation ( )
( )
( ) to
Bessel’s equation by using then find the solution ( ) with a free parameter ?
Problem 4 Use the Cauchy-Riemann equations to show that the complex function ( ) ( ) is analytic then find all solutions of ( ) ?
𝑓(𝑡)
𝑡
12
EE 212/51 Final Examination Fall 2010/11
Kuwait University
Electrical Engineering Department
Name : ………………………………………
Student I. D. : …………………………………….…
Signature : …………………………………….…
Problem No. Grade
1
/20
2
/20
3
/20
4
/20
5
/20
Total
/100
13
Problem 1: Given the vector function [
] and S the tetrahedron surface with vertices (1,0,0),
(0,1,0), and (0,0,1). Use the divergence theorem of Gauss to evaluate the surface integral ∬
.
Problem 2: If the Fourier transform of ( )
( ) √ | |
?
a) (10 points) Use the Parseval’s theorem to find the energy of the signal ( ) ?
b) (10 points) Use the Fourier transform properties to find the Fourier transform of ( )
?
Problem 3: A string of length is fixed at but moves with the displacement
( ) at . The motion of the string can be described by the one-dimensional wave
equation: ( )
( )
where . Find an expression ( ) of the string
motion?
Problem 4: Use the residue theorem to evaluate (counterclockwise) the integral ∮
( )( )
when: (a) (10 points) The closed path C is | | ?
(b) (5 points) The closed path C is | | ?
(c) (5 points) The closed path C is | | ?
Problem 5
a) (11 points) Find the eigenvalues and the eigenvectors of the matrix [
] in term
of . How many independent eigenvectors can you get?
b) (5 points) is the compound proposition ( ) logically equivalent to the compound
proposition ( ) ( )? Why?
14
EE212/01A Examination No. 1 Summer 2011
Kuwait University
Electrical Engineering Department
Name : ………………………………………
Student I. D. : …………………………………….…
Signature : …………………………………….…
Problem
No.
Grade
1
/ 25
2
/ 25
3
/ 25
4
/25
Total
/ 100
15
Problem 1: Given the vector functions
)(
)(
)(
tz
ty
tx
U and
)(10)(5
)(5)(20
)(6
tzty
tzty
tx
V
e) (5 points) Find the dot product VU ?
f) (5 points) From part a: let VUtztytxf ))(),(),(( then find dt
tfd )( ?
g) (5 points) solve part b using the property of dot derivative )( VU ?
h) (5 points) From part a: let VUtztytxf ))(),(),(( then find f ?
i) (5 points) From the above results, verify that Ufdt
tfd
)( ?
Problem 2 Given the curve 1C : the straight line from (0, 0, 0) to (0, 1, 1) and the curve 2C : the
parabola 2yz from (0, 0, 0) to (0, 1, 1)
c) (10 points) Find the parametric representation of the curves 1C and 2C ?
d) (10 points) From part “a” calculate 1
)(C
drrF and 2
)(C
drrF where
zy
F
3
0
0
?
e) (5 points) From part “b” is 2
)()(1 CC
drrFdrrF , why?
Problem 3 (25 points): Show that dzedyyedxey z
C
xx
222 22 is path independent and then
evaluate the integral from (0, 0, 0) to (0, 2, 0) along a curve C?
Problem 4 (25 points): Use the divergence theorem of Gauss to evaluate the surface integral
S
dAnF where
zx
z
F
2
10
2
and S is the surface of the cube 20 x , 20 y , 20 z ?
Evaluate the integral in the xyz-plane (i.e., not in the uv-plane)?
16
EE 212/01A Examination No. 2 Summer 2011
Kuwait University
Electrical Engineering Department
Name : ………………………………………
Student I. D. : …………………………………….…
Signature : …………………………………….…
Problem No. Grade
1
/25
2
/25
3
/25
4
/25
Total
/100
17
Problem 1: Use the properties of the Fourier transform to find the Fourier transform of
( ) ( )?
Problem 2: Convert the ordinary differential equation ( )
to the Legendre
equation by using then find the solution ( )
Problem 3
a) (10 points): Use the Cauchy-Riemann equations to show that ( )
is analytic for all
?
b) (15 points): Find all solutions z of ?
Problem 4: Use the Residue theorem to evaluate ∮
( ) counterclockwise around the
closed path C: circle of radius 2 with center at origin?
18
EE 212/01A Final Examination Summer 2011
Kuwait University
Electrical Engineering Department
Name : ………………………………………
Student I. D. : …………………………………….…
Signature : …………………………………….…
Problem No. Grade
1
/20
2
/20
3
/20
4
/20
5
/20
Total
/100
19
Problem 1:
a) (5 points): Use the Green theorem to evaluate ∮
where C is a
counterclockwise closed curve defined as the square with vertices (1, 1), (-1, 1), (1, -1), (-1,
-1)?
b) (15 points): find the trigonometric Fourier series of ( ) and then sketch its amplitude
spectrum for ?
Problem 2 : Determine the general solution of the wave equation ( )
( )
given
that the following boundary and initial conditions ( ) , ( ) , ( )|
( ) (where α is a constant), ( )
|
? You can use the known results derived in the class and
then derive what you need to answer the problem.
Problem 3
e) (15 points): Find the singular points of the complex function ( )
( ) ?
f) (5 points) use the Residue theorem to evaluate the integral ∮ ( )
counterclockwise
around the closed path : circle of radius 1 with center at
?
Problem 4:
a) (15 points): Find the eigenvalues and the eigenvectors of the matrix [
] in term
of α?
𝑓(𝑡)
𝑡
20
b) (5 points): from your results in part “a”, diagonalize the matrix ?
Problem 5
a) (10 points) Show which rule of inference is used in this argument “If you don’t get a job
you won’t buy a car. If you don’t buy a car, you will use a cap. Therefore, if you
don’t get a job, you will use a cap’’?
b) (10 points) prove the following theorem “Let and be odd integers, then is even’’ using
(a) direct proof (b) proof by contradiction? Hint: every odd integer can be written as
and every even integer can be written as for some other integer .
21
EE212/01 Examination No. 1 Fall 2010
Kuwait University
Electrical Engineering Department
Name : ………………………………………
Student I. D. : …………………………………….…
Signature : …………………………………….…
Problem
No.
Grade
1
/ 50
2
/ 20
3
/ 30
Total
/ 100
22
Problem 1:
j) (10 points) Find the directional derivative of 2zyxf at the point (1, 1, 2) in the
direction of (1, 0, 1) ?
k) (10 points) Find the normal vector of the surface )(2 222 yzx , 20 x , at the point (2, 1,
1)?
l) (10 points) Find the divergence of the vector function ),,( 22 yxyzxyzV ?
m) (10 points) Find the curl of the vector function ),,( 22 yxyzxyzV ?
n) (10 points) Use the theorem 1 to show that the dzzxydyyzdxzC
)16(23 22 is path
independent?
Problem 2 Given the vector function )5,2( 2 yxyxF and the boundary curve C of the triangle
region with vertices (0, 0), (1, 0) and (1, 1).
a) (10 points) Evaluate C
drrF )( counterclockwise?
b) (10 points) Use the Green theorem to evaluate C
drrF )( counterclockwise?
Problem 3 Given the vector function ),,( yxzF and the bounded curve C of the surface S:
42 zyx , 40 x , 20 y and 40 z .
a) (10 points) Find the parametric representation of the surface S and the normal vector?
(20 points) Use the Stokes’s theorem to evaluate C
drF clockwise
23
EE 212/01 Examination No. 2 Fall 2011
Kuwait University
Electrical Engineering Department
Name : ………………………………………
Student I. D. : …………………………………….…
Signature : …………………………………….…
Problem No. Grade
1
/25
2
/25
3
/25
4
/25
Total
/100
24
Problem 1: Find the trigonometric Fourier series of ( )f t (simplify your answer as much as possible)?
Hint 2
1cos( ) sin( ) cos( )
tt at dt at at
a a and
2
1sin( ) cos( ) sin( )
tt at dt at at
a a
Problem 2: The temperatures in a laterally insulated copper bar of length 100 cm is described by the
differential equation ( )
( )
where . If the initial temperature
is and the ends of copper bar are kept at . Find the temperature ( )?
Problem 3: Use Cauchy-Riemann equations to show that if ( ) is analytic or not?
Problem 4 find all solutions of complex equation ( ) ?
𝑓(𝑡)
𝑡
25
EE 212/01 Final Examination Fall 2011
Kuwait University
Electrical Engineering Department
Name : ………………………………………
Student I. D. : …………………………………….…
Signature : …………………………………….…
Problem No. Grade
1
/25
2
/25
3
/25
4
/25
Total
/100
26
Problem 1: a) (10 points) Using Green’s theorem to evaluate ∮
counterclockwise around the closed curve C of region R: rectangular with vertices (1,0), (1,2), (2,0),
(2,2) ?
b) (10 points) Using Fourier transform definition to find the Fourier transform of ( ) ( ) , ? Simplify your answer.
c) (5 points) Show that the compound proposition ( ) is logically equivalent to ?
Problem 2: Reduce the differential equation
to Bessel’s equation
by changing the variable, then find its solution ( )? Use for solving the Bessel’s equation.
Problem 3: Using Residue theorem to evaluate ∮
( ) ( )
counterclockwise around
the circle C: | | ?
Problem 4 Find the eigenvalues and eigenvectors of the matrix [
] ? Show all steps.
