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HISTORY 405

Fall 2011
Vectors
Since many of the quantities we will be dealing with
are vector in nature (have direction and
magnitude), we need to have a good understanding
of the math and notation defining vectors.
In most general terms, a vector in any 3dimensional
coordina
University of New Hampshire  Main Campus  DUPLICATE
Early American History
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Fall 2011
ECE602
Practice Final Exam A
1. Write a fourterm Maclaruin series for f ( x)
2
To aid in your
(1 3x)3
calculations, fill in the table below.
n
1
2
f(n)(x)
f(x) _
3
2. In this problem, you are to determine the error in the series given below,
when the ar
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Early American History
HISTORY 405

Fall 2011
ECE602
Practice Exam 3C
1. A uniform, spherical and timevarying charge distribution exists in the region
0 < R < 4 meters. If the volume charge density in that region is given by:
v ( R, t )
0.02 sin(500t )
Coul/m3
2
R
Calculate the current as a functio
University of New Hampshire  Main Campus  DUPLICATE
Early American History
HISTORY 405

Fall 2011
ECE602
Practice Exam 3C Solutions
1. A spherical and timevarying charge distribution exists in the region 0 < R < 4
meters. If the volume charge density in that region is given by:
v ( R, t )
0.02 sin(500t )
Coul/m3
2
R
Calculate the current as a functi
University of New Hampshire  Main Campus  DUPLICATE
Early American History
HISTORY 405

Fall 2011
EE602
Practice Exam #3B
1. Calculate the rate of fluid passing through the portion of the plane
4x+2y+2z=8 that lies in the first quadrant for the fluid velocity given below:
V xax ay az
meters/second
In the xy plane dl1 dxax dya y and 2 y 4 x 8 dy 2dx
University of New Hampshire  Main Campus  DUPLICATE
Early American History
HISTORY 405

Fall 2011
ECE602
Practice Exam #3B
1. Calculate the rate of fluid passing through the portion of the plane
4x+2y+2z=8 that lies in the first quadrant for the fluid velocity given below:
V xax ay az
meters/second
2. Find the total charge enclosed by a sphere of ra
University of New Hampshire  Main Campus  DUPLICATE
Early American History
HISTORY 405

Fall 2011
EE602
Practice Exam #3A
1. Calculate the work required to move completely around the contour formed by
2
2
2
the intersection of the sphere x +y +(z3) =25 and the plane y=4 if force is
2
2
given by: F z ax 2 x a y xzaz
Newtons
2. Find the rate at which f
University of New Hampshire  Main Campus  DUPLICATE
Early American History
HISTORY 405

Fall 2011
EE602
Practice Exam #2B
1. The end of a cannon (i.e., its muzzle) is located at the point (200, 300, 150) meters. If
the cannon is fired at t = 0, provide the answers below for the trajectory information given:
(initial cannonball speed = 450 m/s; = 65 a
University of New Hampshire  Main Campus  DUPLICATE
Early American History
HISTORY 405

Fall 2011
EE602
Practice Exam #3
1. Calculate the work required to move completely around the contour formed by
2
2
2
the intersection of the sphere x +y +(z3) =25 and the plane y=4 if force is
2
2
given by: F z ax 2 x a y xzaz
Newtons
Using Stokes dS rdrd a y
F
University of New Hampshire  Main Campus  DUPLICATE
Early American History
HISTORY 405

Fall 2011
EE602
Practice Exam #2A
1. One plane is defined by the equation ax + 6y +z = 5, where a is a constant. Another
plane is defined by the two vectors below:
A= 3ax  2ay +az
B = 2ax +2a y +3az
a) For what value(s) of the constant a, if any, are the two pla
University of New Hampshire  Main Campus  DUPLICATE
Early American History
HISTORY 405

Fall 2011
ECE602
Practice Exam #1B
1. In this problem, you are to calculate an answer for the definite integral below
using a 4term Maclaurin series.
1
I x 2e3 x dx
0
a) Write a 4term Maclaurin series for the integrand
Integrand _
b) Write a 4term Maclaurin seri
University of New Hampshire  Main Campus  DUPLICATE
Early American History
HISTORY 405

Fall 2011
EE602
Practice Exam #1
1. (40%) A point in Cartesian coordinates is defined by x = 1, y = 3, and z = 4. Define
that same point in (specify angles in degrees):
a) cylindrical coordinates: r = _; = _; z = _
b) spherical coordinates: R = _; = _; = _
c) def
University of New Hampshire  Main Campus  DUPLICATE
Early American History
HISTORY 405

Fall 2011
Laplacian
The Laplacian of a scalar function at a particular point
indicates how the value of the function at that point
compares with its value in surrounding regions.
63
64
If T(x, y,z) is temperature in the region,
50
61
65
positive at P, since the sur
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Early American History
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Fall 2011
Series Example 2
Show that our Maclaurin expansion for ex works for
the complex number x=j/2. We know the solution
to this from Eulers Identity:
j
e j cos + j sin e 2 = cos + j sin = j = 190
2
2
Now, calculate the answer using a 6term Maclaurin Series
e
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Fall 2011
Comments on Inequalities
Multiplying both sides of an inequality by 1 changes the
direction of the inequality. For example:
5 >1
5 < 1
When working with absolute values, we need to check the case
where the argument is positive and negative. For example:
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Early American History
HISTORY 405

Fall 2011
ECE602
Final Exam B
1. In this problem, you are to approximate the value of the integral below by
putting the integrand into series form, and then integrating that series on a
termbyterm basis.
2
5x
f(x) = e dx
x
1
a. Write a fourterm Maclaurin series
University of New Hampshire  Main Campus  DUPLICATE
Early American History
HISTORY 405

Fall 2011
ECE602
Practice Final A Exam
1. Write a fourterm Maclaruin series for f ( x)
2
To aid in your
(1 3x)3
calculations, fill in the table below.
n
f(n)(x)
1
2
3
18
216
3240
1 3x
4
f(x) 2+18x+108x2+540x3
1 3x
5
1 3x
6
2. In this problem, you are to determ
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Early American History
HISTORY 405

Fall 2011
Curl F
The Curl is a measure of how a vector field rotates (circulates).
y
If F represents fluid velocity, then F
results in a vector defining the paddlewheel axis orientation that will result in
maximum rotation. Obeys the right
x
hand rule: F points int
University of New Hampshire  Main Campus  DUPLICATE
Early American History
HISTORY 405

Fall 2011
Cartesian Coordinate System
z
ax , a y , and az are unit vectors pointing in the
increasing x, y, and z directions, respectively
az
ax
y
ay
x
Any vector can be represented by:
R = f x ax + f y a y + f z az where f x , f y , and f z are scalar values or
University of New Hampshire  Main Campus  DUPLICATE
Early American History
HISTORY 405

Fall 2011
Stokes Theorem Example #3
Applying Stokes Theorem to an Open Contour
Calculate the work required to move a particle over the semicircle
shown if F = x 2 ax + 5 xa y + 3zaz Newtons
y
Contour (does not include xaxis)
2
2
d = rd a = 2d a
x
First solve wit
University of New Hampshire  Main Campus  DUPLICATE
Early American History
HISTORY 405

Fall 2011
Static Electric Fields
Static electric fields are created by charges that are not varying with time
Coulombs Law: Describes the force exerted on 2 small charges,
separated by a distance R
F1 =
Q1Q2
a21
4 R 2
a21
R
Q2
Q1
= permittivity (farads/meter)
F1 =
University of New Hampshire  Main Campus  DUPLICATE
Early American History
HISTORY 405

Fall 2011
Finding the Vector Differential Surface Element
In many cases, like our previous example, ds is easy to find. In general,
ds d 1 d 2 where d 1 and d 2 are any differential length elements
tangent to the surface to be integrated over. ds is normal to the s
University of New Hampshire  Main Campus  DUPLICATE
Early American History
HISTORY 405

Fall 2011
Potential Due to a Volume Charge Distribution
VAbs ( R ) =
v dv '
1
4 R R '
z
Example: Set up the integral to find V
along the axis of a cylindrical volume
charge distribution as shown
Observer
(0,0,z)
RR'
(Volts)
dv is located at (r, z)
dv
R =z
y
( z z
University of New Hampshire  Main Campus  DUPLICATE
Early American History
HISTORY 405

Fall 2011
Starting Point for Review: Vectors
Im assuming youre good at:
Algebra & functions
f(x) f(x,y) f(x,y,z)
Calculus concepts
Definition of derivative
Integration in 1, 2 and 3 dimensions
Coordinate Systems
Cartesian, Cylindrical and Spherical
Complex
University of New Hampshire  Main Campus  DUPLICATE
Early American History
HISTORY 405

Fall 2011
ECE602
Practice Final C
1. A fourterm Maclaurin series for the function: f ( z )
3
is given by:
(1 4 z ) 2
f ( z ) 3 24 z 144 z 2 768z 3
a) What is the exact value of f(z) when z=0.01j0.05 (give answer in polar
form)?
f(0.01j0.05) = _
b) What is the v
University of New Hampshire  Main Campus  DUPLICATE
Early American History
HISTORY 405

Fall 2011
Electric Potential
The electric potential between 2 points (the potential difference)
difference)
is defined as the work required to move a oneCoulomb charge
onefrom one point to the other.
Example: find the potential difference from the bottom plate to
University of New Hampshire  Main Campus  DUPLICATE
Early American History
HISTORY 405

Fall 2011
Gauss Law Example
Determine the electric field caused by a spherical cloud of
uniform charge density V (Coul/m3) that exists for 0 R b
Region with
charge
distribution
Assumption: by symmetry
E = ER aR
Spherical
Gaussian
Surface
E
Must analyze in 2 regions
University of New Hampshire  Main Campus  DUPLICATE
Early American History
HISTORY 405

Fall 2011
ECE602
Practice Final C Solutions
1. A fourterm Maclaurin series for the function: f ( z )
3
is given by:
(1 4 z ) 2
f ( z ) 3 24 z 144 z 2 768z 3
a) What is the exact value of f(z) when z=0.01j0.05 (give answer in polar
form)?
f(0.01j0.05) = 2.8602 
University of New Hampshire  Main Campus  DUPLICATE
Early American History
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Fall 2011
Topics for Final Exam
Infinite Series Taylor, Maclaurin & Binomial
Absolute Convergence
LHopitals Rule
Inequalities
Real and Complex Variables
Solving Integrals Using Series
Linear Algebra
Putting equations into matrix form
Matrix multiplication
University of New Hampshire  Main Campus  DUPLICATE
Early American History
HISTORY 405

Fall 2011
ECE602
Final Exam B
1. In this problem, you are to approximate the value of the integral below by
putting the integrand into series form, and then integrating that series on a
termbyterm basis.
2
5x
f(x) = e dx
x
1
a. Write a fourterm Maclaurin series