Here is the images of the Universe. Everything is the same
PS: “I think that it is not still the time to release all results
around the world”
Friday, May 26, 2017
Wednesday, May 24, 2017
The Change Depends on the Direction of the Motion: Generating All Directions in 3D Space
In physics, science, and engineering, especially fluid dynamics and electromagnetism fields, we usually need to investigate the changes of a function in different directions. In this case, the best way for analyzing and designing is to have different directions all together in our hands, in which we will be able to compare all results to reach the new theorems and the new physical phenomena. The purpose of this article is to make a spreadsheet on an Excel file by using a new method where you will have all unit vectors in 3D space in different directions.
Let me give you an example:
Suppose you have a curve of f (x, y, z) = x^2 + y^2 - 2z^2 and the point P (x, y, z) wants to move from P0 (-1, 3, 2) on this curve toward all directions in amount of 4 unit (∆s = 4 unit). The question is:
What is the maximum change of the function when point P (x, y, z) moves in all directions? In which direction will the maximum change in the function occur?
Solution:
By making an analytic model in Excel, you will be able to
answer the above questions. There is no need for any other software.
Unit Vectors in
2D and 3D
You as well as know, we can easily
get all unit vectors in 2D surface by using bellow formula:
U = i cosθ + j sinθ
On the other hand, we can obtain the direction of a vector in 2D and 3D, by using below formula:
Direction of A = A / │A│
But, how can we prove above formula?
A vector in 3D space can be modelled
as the radius of a sphere in which we will have below function:
w = r (x, y, z) = (x^2 + y^2 + z^2)
^0.5
The gradient vector in any point is:
As you can see, the gradient vector
proves above formula (formula:
Direction of A = A / │A│) where
vector A = xi + yj + zk because we have:
Generating all
directions in 3D space
Consider a particle starts its circular motion
on surface XY and simultaneously has a circular motion on surface XZ
perpendicular to surface XY. In this case, this particle will produce a sphere
where its circular angle on surface XY is “θ” and its circular angle on surface
XZ is “β”.
According to above circular angles
and radius of sphere (r), we can calculate coordination of point P (x, y, z) on
sphere by using below equations:
x = r * cos β * cos θ
y = r * cos β * sin θ
z = r * sin β
Example:
Solve equation: x^2 + y^2 + z^2 = 225
If θ = 25 degree and β = 56 degree
Answer:
We have: r = (225) ^0.5 = 15
x = 15 * cos 56 * cos 25 = 7.602013
y = 15 * cos 56 * sin 25 = 3.544877
z = 15 * sin 56 = 12.43556
(7.602013 ^2) + (3.544877 ^2) +
(12.43556 ^2) = 225
In fact, we have below vector:
V = xi + yj + zk
V = (r * cos β * cos θ)i + (r * cos β * sin θ)j + (r * sin β)k
For r = 1, we have below unit vector in 3D space:
U = (cos β * cos θ)i + (cos β * sin θ)j + (sin β)k
It is clear; the range of changes
for “θ” and “β” is between 0 and 360 degree.
Therefore, for making a spreadsheet
included all directions, we should go below steps:
- Choose ∆θ and ∆β between 0 and 360 degree. For instance, I
considered ∆θ = ∆β = 1
- By using the method stated in article of “Can We Solve a
Nonlinear Equation with Many Variables?” posted on link: http://emfps.blogspot.co.uk/2016/10/can-we-solve-nonlinear-equation-with.html, we should find all combinations of “θ” and “β” for ∆θ and ∆β between 0 and 360 degree. For example, when
I considered ∆θ = ∆β = 1, I will have 130321 combinations on my excel spreadsheet
(361^2 = 130321).
If you choose ∆θ = ∆β = 0.5, you
will have 519841 combinations (directions) on your excel spreadsheet (721^2 =
519841).
Anyway, I think that ∆θ = ∆β = 1 is
enough.
- Using from above equations for r = 1 and each set of
combinations. In this case, you have
130321 rows that it show you all directions which you need to your analysis.
In the next articles, I will show
you how we can utilize this spreadsheet as a template to investigate the changes
of some physical functions.
Sunday, December 4, 2016
The change depends on direction of the motion: Generating Eigenvalues from special matrices (CON)
In the reference with the article of “Can We Solve a Nonlinear Equation With
Many Variables? (Con)” posted on link: : http://emfps.blogspot.co.uk/2016/10/can-we-solve-nonlinear-equation-with_19.html and following to article of "The change depends on direction of the motion: Generating Eigenvalues from special matrices" posted on link: https://emfps.blogspot.com/2016/06/the-change-depends-on-direction-of.html, here is another new theorem as follows:
Theorem: “If and only if there are three points on
a sphere which are members of natural number, they make a matrix 3*3 where the
eigenvalue of this matrix is easily calculated by using below
formulas:
A =
a11
a12 a13
a21
a22 a23
a31
a32 a33
λ1 = a11 - a12
λ2 = a11+ a12 + a13
Example (1):
The points P1 (1, 2, 2) and P2 (2, 1, 2) and P3
(2, 2, 1) are on a sphere with radius of 3 (r =3), these points make matrix A
where we can calculate eigenvalue of this matrix as follows:
A =
1
|
2
|
2
|
2
|
1
|
2
|
2
|
2
|
1
|
λ1 = 1 – 2 = -1
λ2 = 1+ 2 +2 = 5
Example (2):
The points P1 (8, 16, 16) and P2 (16, 8, 16) and
P3 (16, 16, 8) are on a sphere with radius of 24 (r =24), these points make
matrix A where we can calculate eigenvalue of this matrix as follows:
A =
8
|
16
|
16
|
16
|
8
|
16
|
16
|
16
|
8
|
λ1 = 8 – 16 = -8
λ2 = 8+ 16 +16 = 40
Example (3):
The points P1 (4, 8, 8) and P2 (8, 4, 8) and P3
(8, 8, 4) are on a sphere with radius of 12 (r =12), these points make matrix A
where we can calculate eigenvalue of this matrix as follows:
A =
4
|
8
|
8
|
8
|
4
|
8
|
8
|
8
|
4
|
λ1 = 4 – 8 = -4
λ2 = 4+ 8 +8 = 20
Saturday, November 12, 2016
Energy Saving by Using the Solution of a Nonlinear Equation With Many Variables
Following to article of "Can We Solve a Nonlinear Equation with
Many Variables? (Con)" posted on link: http://emfps.blogspot.co.uk/2016/10/can-we-solve-nonlinear-equation-with_19.html, the purpose of this article is to solve
a nonlinear equation with five variables in limited domain and range. I have
started it with a sample of thermal conduction in curtain wall system in
buildings in which this example pursues the energy saving and cost management
in shopfront glass in malls, shopping centers and mini-markets. This is a real
example because the people can directly use from the data and results of this
article to save energy.
Before starting of example, let me
tell you about a problem and a restriction when we are applying
this method to
solve the equations as follows:
1. As you understood, this method
gives us the same domain and range for all variables while we
need to different
domains and ranges to solve the equations in the fields of engineering. For
instance, domains for variables can be between 1 - 100 or 1 – 1000 but
sometimes we have to have real numbers for the variables including: x (0.0025,
0.48) or
y (-23.5, 0.78) or z (4, 563.75) and
so on. I can say to you that we are able to solve easily this problem by using
of some theorems in mathematics and change the domains and ranges of variables to
ideal and desired ones. I have shown you this changes in below example.
2. But, there is a restrictive
factor something like discrete functions instead of continuous functions so
that we have to follow this limitation forever. It means that we should
consider sub-intervals between domains and ranges for our variables. In fact,
the numbers which belong to variables between a domain and rage are discrete
not continuous. Of course, we can decrease and decrease sub-intervals step by
step. The best way is, to apply Arithmetic Progression for sub-intervals.
"Please be informed that we can
use many methods to reach our targets which are new data and results. Therefore,
the most important thing is, to extract new data and results by any method. In
fact, the analysis of these new data and results lead us to reach the gates of new
worlds. The methods are only tools to reach our targets."
A real example of energy saving:
Thermal conduction in shopfront glass
The law of thermal
conduction (Fourier's Law) gives us the great opportunity to calculate the rate of
energy transfer by heat for a slab of infinitesimal thickness (dx) and temperature
difference (dT) in which the rate of│dT/dx│is named the temperature gradient. Simple conductive heat transfer (in
watts) through a uniform body can be calculated from below equation:
P =
k. A.│∆T/ ∆x│ and ∆T =
Th – Tc
and Th >
Tc
Where:
- A is the area of the body (m2)
- ∆x is the body's thickness (m)
- Th is one face of the slab with high temperature (°C)
- Tc
is other face
of the slab with low temperature (°C)
- P is the rate of heat transfer (Watt)
The heat
transfer occurs only if there is a difference in temperature between two parts
of a slab.
For this example,
I used from manual of "CODE OF PRACTICE FOR USE OF GLASS IN
BUILDINGS" an Indian Standard which was adopted by the Bureau of Indian
Standards. You can find this paper posted on below link:
The page 27 of
above Standard shows us the boundaries for maximum area of designed normal
glass in compare with its thickness in shop fronts. (See Table 5.3)
According to Table 5.3, I chose below domains for A (area) and ∆x (thickness):
A = (6, 15) and ∆x = (15, 25)
And so, I consider domains for the temperature of outside and
inside as follows:
Th = (15, 32) and Tc = (-20, 4)
In this case, we will have a maximum of heat transfer equal to
41600 W and a minimum heat transfer equal to 2112 W. (kglass = 0.8 W/m°C)
P max = 41600 W, and Pmin = 2112 W
Therefore, the range for heat transfer is: P = (2112, 41600)
Now, I apply previous method and solve
above equation for five domains and range. The rate
of heat transfer (P) to the Number
of Results has been presented on below graph:
I only apply one sub-interval and get the total sum of results equal
to 195 in which we will have 30 results for P = 12800 W. If you spend more time
and apply many sub-intervals, it is possible, you will find much more results.
Analysis of findings and results
The most important part of this article is, to analyze the results
to reach our target which is the energy saving in the same conditions
accompanied by cost management.
According to the results, here is many outcomes which lead us to handle
the cost management and the energy saving. Let me present only several states
as follows:
1. For instance,
if we compare maximum transfer rate with another results, we will have
below analysis: (Please see below figure)
According to above table, if you decrease the temperature of your shop from 32 °C to 15°C and also increase the thickness of glass from 15 mm to 16.935 mm, then you will save about 40% of your energy consumption. What is your cost management?
Usually shop front glasses have around 8 years guarantee. If you open your shop 12 hour per day, total hours for energy consumption is: Total hours = 8*365*12 = 35040 hour
Then, you have around 40% (0.403846) energy saving which is equal to 16800 W. Therefore, you will be able to save the energy totally around 588772 KWh in the period of 8 years (16800*35040). The cost of electricity power is about 12 cent per KWh. In the result, you will save totally amount of USD70, 640 for 8 years and USD8, 830 per year. If we add future value of each annual saving with an average return rate (Cost of capital) equal to 11% per year, it means that you are saving amount of money around USD 104,718. What is your costs? You should purchase normal glass with thickness 17 mm instead of 15mm. Therefore, you should pay more USD 40 per square meter (Price of normal glass with 15mm thickness is around USD 130 and 17mm is around USD170). Consequently, your additional investment is USD 600. It means that you have still saved the amount of money around USD 104,000.
2. If we compare
some results together, sometimes we can see very exiting outcomes as follows:
According to above results, if we increase the thickness from 15mm
to 25mm, we are saving around 42% energy consumption and if we increase the
thickness from 15mm to 23 mm, we are saving around 35% energy consumption. For
calculation of cost management, you can apply the same method mentioned in Item
(1).
3. In the
reference with below results, if we are able to change the design for area of
7.74 m2 to 6 m2, we have already saved around 23% energy consumption.
As I told you, we can obtain 30 results for P = 12800 W. You can
find these results on below figure for your better analysis and design:
Wednesday, October 19, 2016
Can We Solve a Nonlinear Equation With Many Variables? (Con)
Following to article of "Can We Solve a Nonlinear Equation with Many Variables?" posted on below link:
Let me start second example for an equation with four variables
as follows:
Example (2): Solving of Sphere
Equation
As you know, the sphere equation has
many applications in all fields of engineering and physics. When we talk about
a sphere equation, our discussion can be expanded not only macroscopic systems but also
microscopic particles such as the
quantum model of the Hydrogen atom. Therefore, let me start
by solving of a sphere equation for a limited domain and range as follows:
Consider the sphere equation with
below domain:
If
x^2 + y^2 + z^2 = r^2 x, y,z ϵ N, x, y,z ≤
100
Then the range for the radius of
sphere will be: r ϵ N, r ≤ 173
Now, I apply previous method and get
all results of "x, y, z" for "r" in given range. The number
of
results related to "r" has been presented on below graph:
In this case, total sum of possible
answers is only and only equal to 4935.
Above graph shows us that there is a
maximum number of answers equal to 165 for r = 99. For instance, I have brought
some results on below figure:
As we can see, for r =
15 and r = 150, the number of results are the same equal to 15 while for r = 31, we have 24
answers
We can apply this domain and range
as a template for all macroscopic and microscopic numbers.
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