# Chapter 1 : Mathematical Preliminaries¶

## Example 1.1,Page Number 45¶

In [26]:
import numpy as np

#variable Declaration
A = np.array([1,2,3]) # A is a vector

#calculations
l=np.linalg.norm(A)   # magnitude or length of vector A
a=A/l                 # direction of vector A

#results
print "magnitude of vector:",round(l,2)
print "direction of vector",np.around(a,3)

magnitude of vector: 3.74
direction of vector [ 0.267  0.535  0.802]


## Example 1.2,Page Number 45¶

In [2]:
import numpy as np

#variable Declaration
A=np.array([2,5,6])  # vector A
B=np.array([1,-3,6]) # vector B

#calculations
Sum = A+B  # summation of two vectors
Sub = A-B  # subtraction of two vectors

#results
print "summation of two vectors:",Sum
print "subtraction of two vectors:",Sub

summation of two vectors: [ 3  2 12]
subtraction of two vectors: [1 8 0]


## Example 1.3,Page Number 45¶

In [3]:
import numpy as np

#Variable Declaration

A = np.array([1,1,2]) # vector A
B = np.array([2,1,1]) # vector B

#Calculations
k = np.dot(A,B) # dot product of vector A and B

#Results

print "dot product of vector A and B:",k

dot product of vector A and B: 5


## Example 1.4,Page Number 45¶

In [4]:
import numpy as np

#Variable Declaration

A = np.array([2,1,2]) # vector A
B = np.array([1,2,1]) # vector B

#Calculations
Cross = np.cross(A,B) # dot product of vector A and B

#Results

print "cross product of vector A and B:",Cross

cross product of vector A and B: [-3  0  3]


## Example 1.5,Page Number 46¶

In [5]:
import numpy as np

#Variable Declaration

A = np.array([1,3,4]) # vector A
B = np.array([1,0,2]) # vector B

#Calculations
k = np.dot(A,B) # dot product of vector A and B

#Results

print "dot product of vector A and B:",k

dot product of vector A and B: 9


## Example 1.9,Page Number 46¶

In [6]:
from __future__ import division
import math

#variable declaration
p = [1,2,3] # coordinates of point p
x = 1       # x coordinate of P
y = 2       # y coordinate of P
z = 3       # z coordinate of P

#Calculations
rho = math.sqrt(x**2+y**2) #radius of cylinder in m
phi = (math.atan(y/x))*(180/math.pi) # azimuthal angle in degrees
z = 3 # in m

#results
print "radius of cylinder is:",round(rho,2),"m"
print "azimuthal angle is:",round(phi,2),"degrees"
print "z coordinate is:",z,"m"

radius of cylinder is: 2.24 m
azimuthal angle is: 63.43 degrees
z coordinate is: 3 m


## Example 1.10,Page Number 47¶

In [27]:
from __future__ import division
from math import cos,sin,pi,atan
import numpy as np

#Variable Declaration

A = np.array([4,2,1])  # vector A
A_x = 4  # x coordinate of P
A_y = 2  # y coordinate of P
A_z = 1  # z coordinate of P

#calculations
phi = atan(A_y/A_x)  # azimuthal in radians
A_rho = (A_x*cos((phi)))+(A_y*sin((phi)))   # x coordinate of cylinder
A_phi = (-A_x*sin(phi))+(A_y*cos(phi))      # y coordinate of cylinder
A_z = 1  # z coordinate of cylinder
A = [A_rho,A_phi,A_z]  # cylindrical coordinates if vector A

#Result
print "cylindrical coordinates of vector A:",np.around(A,3)

cylindrical coordinates of vector A: [ 4.472  0.     1.   ]


## Example 1.12,Page Number 47¶

In [22]:
from __future__ import division
from math import sqrt,acos,atan
import numpy as np

#Variable Declaration
P = np.array([1,2,3])  # coordinates of point P in cartezian system
x = 1 # x coordinate of point P in cartezian system
y = 2 # y coordinate of point P in cartezian system
z = 3 # z coordinate of point P in cartezian system

#calculations
r = sqrt(x**2+y**2+z**2)  # radius of sphere in m
theta = acos(z/r)  # angle of elevation in degrees
phi = atan(x/y)  # azimuthal angle in degrees

#results
print "radius of sphere is:",round(r,3),"m"
print "angle of elevation is:",round(theta,3),"radians"
print "azimuthal angle is:",round(phi,3),"radians"

# note : answer in the book is incomplete they find only one coordinate but there are three

radius of sphere is: 3.742 m
angle of elevation is: 0.641 radians
azimuthal angle is: 0.464 radians


## Example 1.17,Page Number 48¶

In [9]:
import numpy as np

#variable declaration
A_p=22  # power gain

#calulation
A_p_dB=10*(np.log10(A_p)) # power gain in dB

#result
print "power gain is:",round(A_p_dB,3),"dB"

power gain is: 13.424 dB


## Example 1.18,Page Number 48¶

In [10]:
import numpy as np

#variable declaration
A_v=95  # voltage gain

#calculation
A_v_dB=20*(np.log10(A_v)) # voltage gain in dB

#result
print "voltage gain is:",round(A_v_dB,3),"dB"

voltage gain is: 39.554 dB


## Example 1.19,Page Number 48¶

In [11]:
import numpy as np
from math import sqrt

#variable declaration
A_p = 16  # power gain

#calculations
A_p_Np = np.log(sqrt(A_p)) # power gain in Np

#results
print "power gain is:",round(A_p_Np,3),"Np"

power gain is: 1.386 Np


## Example 1.20,Page Number 49¶

In [12]:
import numpy as np

#variable declaration
A_i = 34  # current gain

#calculations
A_i_Np = np.log(A_i) # current gain in Nepers

#result
print "power gain is:",round(A_i_Np,3),"Np"

power gain is: 3.526 Np


## Example 1.21,Page Number 49¶

In [13]:
from __future__ import division
import cmath
from math import sqrt,pi

#variable declaration
A=2+4j  # complex number A

#calculations
magnitude = abs(A)  # magnitude of complex number A
phi = cmath.phase(A)*(180/pi)  # phase of complex number A in degrees

#results
print "magnitude of complex number A is:",round(magnitude,3)
print "phase of complex number A is:",round(phi,3),"degrees"

magnitude of complex number A is: 4.472
phase of complex number A is: 63.435 degrees


## Example 1.22,Page Number 49¶

In [14]:
from __future__ import division
from math import pi
import cmath

#variable declaration
A = 1+3j  # complex no. A

#calculations
c = A.conjugate()  # conjugate of complex no. A
magnitude = abs(A)  # magnitude of complex number A
phi = cmath.phase(A)*(180/pi)  # phase of complex number A in degrees

#results
print "magnitude of complex number A is:",round(magnitude,3)
print "phase of complex number A in degrees:",round(phi,3)
print "conjugate of complex no. A:",c

magnitude of complex number A is: 3.162
phase of complex number A in degrees: 71.565
conjugate of complex no. A: (1-3j)


## Example 1.23,Page Number 49¶

In [24]:
from __future__ import division
from math import cos,sin,radians
import numpy as np

#variable declaration
rho = 5   # magnitude of the complex number A
phi = 45  # phase of a complex number A in Degrees

#calculations
x = rho*cos(radians(phi))  # real part of complex number A
y = rho*sin(radians(phi))  # imaginary part of complex number A
A = complex(x,y)  # complex number A

#results
print "real part of complex number A:",round(x,3)
print "imaginary part of complex number A:",round(y,3)
print "complex number A:",np.around(A,3)

real part of complex number A: 3.536
imaginary part of complex number A: 3.536
complex number A: (3.536+3.536j)


## Example 1.24,Page Number 49¶

In [16]:
#Variable Declaration

A_1 = 2+3j # complex number A_1
A_2 = 4+5j # complex number A_2

#calculation
A = A_1 + A_2

#Result
print "sum of complex numbers A_1 and A_2 is:",A

sum of complex numbers A_1 and A_2 is: (6+8j)


## Example 1.25,Page Number 49¶

In [17]:
#Variable Declaration

A_1 = 6j # complex number A_1
A_2 = 1-2j # complex number A_2

#calculation
A = A_1 - A_2

#Result
print "Difference of complex numbers A_1 and A_2 is:",A

Difference of complex numbers A_1 and A_2 is: (-1+8j)


## Example 1.26,Page Number 49¶

In [18]:
#Variable Declaration

A = 0.4 + 5j # complex number A
B = 2+3j     # complex number B

#calculation
P = A*B

#Result
print "Product of complex numbers A and B is:",P

Product of complex numbers A and B is: (-14.2+11.2j)


## Example 1.27,Page Number 50¶

In [25]:
import numpy as np

#Variable Declaration

A = 10+6j # complex number A
B = 2-3j  # complex number B

#calculation
D = A/B

#Result
print "Division of complex numbers A and B is:",np.around(D,3)

Division of complex numbers A and B is: (0.154+3.231j)


## Example 1.28,Page Number 50¶

In [6]:
from sympy import Symbol,solve

#variable Declaration

x = Symbol('x')
p = (x)**2 + 2*x + 4

#calculations
Roots = solve(p,x)

#result
print "The roots of the given quadratic equation are:",Roots

for i in range(len(Roots)):
print "Root %i = %s" % (i + 1, str(Roots[i].n(5)))

The roots of the given quadratic equation are: [-1 - sqrt(3)*I, -1 + sqrt(3)*I]
Root 1 = -1.0 - 1.732*I
Root 2 = -1.0 + 1.732*I


## Example 1.29,Page Number 50¶

In [1]:
import numpy as np

#variable Declaration
a = np.array([[1,2,3],[4,5,6],[7,8,9]]) # Determinant
i = 2 # Third row of the determinant
j = 1 # second column of the determinant

#Calculations

b = np.delete(np.delete(a, i, axis=0), j, axis=1) # minor of 8
m = np.linalg.det(b) # determinant of minor of 8

i = 2 # Third row of the determinant
j = 2 # Third column of the determinant
c = np.delete(np.delete(a, i, axis=0), j, axis=1) # minor of 9
n = np.linalg.det(c) # determinant of minor of 9

#Result
print "The minor of 8 is:",b
print "the determinant of minor of 8 is:",m
print "The minor of 8 is:",c
print "the determinant of minor of 9 is:",n

The minor of 8 is: [[1 3]
[4 6]]
the determinant of minor of 8 is: -6.0
The minor of 8 is: [[1 2]
[4 5]]
the determinant of minor of 9 is: -3.0


## Example 1.30,Page Number 50-51¶

In [2]:
import numpy as np

#variable declaration and Calculations
a = np.array([[1,3,2],[6,1,5],[7,9,8]]) # Determinant
i = 1 # second row of the determinant
j = 0 # first column of the determinant
b = np.delete(np.delete(a, i, axis=0), j, axis=1) # minor of 6
b_c = (-1)**(i+j) * b #cofactor of 6

i = 1 # second row of the determinant
j = 2 # Third column of the determinant
c = np.delete(np.delete(a, i, axis=0), j, axis=1) # minor of 5
c_c = (-1)**(i+j) * c #cofactor of 5

#Result
print "The cofactor of 6 is:",b_c
print "The cofactor of 5 is:",c_c

## note : the answer of cofactor of 5 is wrong in the book.The sign should be negative.

The cofactor of 6 is: [[-3 -2]
[-9 -8]]
The cofactor of 5 is: [[-1 -3]
[-7 -9]]


## Example 1.31,Page Number 51¶

In [21]:
from math import factorial

f1 = factorial(4)   # factorial of 4
f2 = factorial(6)   # factorial of 6
print "factorial of 4 is:",f1
print "factorial of 6 is:",f2

factorial of 4 is: 24
factorial of 6 is: 720