__The
Theorem Of General And Universal Equations, For Creating
Generalized Equations,
With The Formulas From Mathematics And Physics ____Created by David
Alderoty __

__The
Theorem Of General And
Universal Equations,__

__For Creating
Generalized Equations, With The __

__Formulas From
Mathematics And Physics__

** **

__Created
by David Alderoty © 2012
David@TechForText.com__

** **

**To
contact the author use the above e-mail address,
or **

**left click on
these words for a website communication form**

** **

**This website is a 5300
word e-book**

** **

** **

*The theorem and
related concepts
presented on this website have practical and theoretical
utility, for all of
the following:*

** **

__For
simplifying
mathematical calculations__

* *

__For
developing
a better understanding of the formulas from physics and
mathematics__

* *

__For
facilitating
creative thinking, which can lead to new theories and
concepts__

* *

__Facilitates
the
understanding of formulas, which can help with the
learning process. __

* *

__*For creating software-based
calculation devices__

** **

** **

__*____With the Theorem of General and
Universal Equations I created
three computer programs that perform multiple
calculations simultaneously.
These devices function online over the Internet, and
they can be accessed from
the following URLs: (NOTE the blue words in this book
are active hyperlinks,
and you can left click on them to open the indicated
webpage.)__

* *

*www.TechForText.com/General-Area-Equation*

** **

** **

*www.TechForText.com/General-Volume-Equation*

** **

** **

__Physics Calculator for
Molecular Speed__

** **

__I devised the
theorem and concept
discussed in this e-book several years ago, and the
following material
represents a simplified an updated version of an e-book I
placed online in
2009. I have recently edited the original e-book and
moved it to the URL
presented below. The original e-book is slightly more
technical, and it
contains some material that is not presented on this
website.__

** **

** **

__The
theorem and related material discussed on this website,
and the older e-book
mentioned above, require a working knowledge in
mathematics and physics. The
primary focus of both of the e‑books is on mathematics and
physics, but
it is important to note that the theorem, and related
techniques can be applied
to just about any discipline that utilizes mathematical
equations. __

** **

__NOTE ON THE TERMINOLOGY
USED IN THIS E-BOOK__

__The
words UNIVERSAL EQUATION, means
in this paper a GENERALIZED
EQUATION that applies to MANY
or ALL calculations in a
specific set or category, such as a formula that
represents the area
calculations of many or all geometric figures. A GENERAL
EQUATION means any equation that applies to two
or MORE calculations in a specific
category,
such as a generalized formula for volume that can be used to
calculate the
volume of cubes and spheres. The word formula(s) and
equation(s) are used as
synonyms in this e-book.__

** **

** **

** **

__Section 1)
Introduction __

**Over
the years,
mathematicians and physicists have tried to find coefficients
that are
universal in nature, which are generally called universal or
fundamental
constants, such as pi, Euler's number, Planck's constant, the
gravitational
constant, and the speed of light. ****These numbers
have theoretical and
practical value in a number of formulas. However, my
paper takes the OPPOSITE
APPROACH, which in NO way contradicts the conventional
concepts of physics and
mathematics. Specifically, the constants I use are
NOT universal, and more
precisely should be called coefficients, but THE EQUATIONS I
DERIVED ARE
UNIVERSAL. Of course, the *universal
equations
provide the same calculated results as the conventional
mathematics and physics
formulas. **

__The above
suggests the question: Why Create
Universal Equations?:__**The answer is universal equations provide a look at the
principles and
concepts of physics and mathematics from a unique
perspective. This is similar
to looking at an object from a different angle, which might
reveal new
information, insights and theories. Universal equations
generally convey
precise principles, which can result in a better understanding
of the
conventional formulas and concepts of physics and mathematics.
The generalized
equations are sometimes useful for deriving new formulas and
they are often
more versatile than conventional formulas. Sometimes,
a universal equation is
useful for carrying out calculations that cannot be done
with conventional
formulas. All of this will become apparent in the
next section, which
presents a simplified derivation of a general area
formula. **

** **

** **

*Section 2) A Simplified
Illustration Deriving A
Universal Area Formula With The **Theorem
Of General
Equations*

__Section 2)
A Simplified Illustration__

__Deriving
A Universal Area Formula__

__With the
____Theorem of
General Equations__

** **

**To
explain the
basic idea of my thesis in a simplified way, I will create a ****universal area formula, and I will
illustrate this step-by-step.
To do this I will start with six conventional area formulas,
with the goal of creating ONE FORMULA
that represents area of two-dimensional
geometric figures, as well as the surface area of
three-dimensional
objects. In this regard, the first step is to
examine the six area
formulas presented below, and then proceed to the next
paragraph, which is
below the formulas. **

** **

__(THE AREA OF A RECTANGLE)__ A = LW The
area = A of a
rectangle, is equal to the length=L
multiplied by width=W
(__The coefficient is 1 and the two lengths
are L and W) __

** **

__(THE AREA OF A TRIANGLE)__** ****
(The area = A of a
triangle, equals base=b
multiplied by height=h,
divided by 2 ( The coefficient is ½ The two
lengths are b and h)**

* *

__(THE AREA OF A SQUARE)__** **** The
area = A of a square, equals the length of the
side = S squared ( The coefficient is 1 and the two lengths
are S and S)**

** **

**(**__THE SURFACE AREA OF A CUBE)__** **** (The
surface area = A of
cube, equals the length of the side = S squared multiplied by 6) ( The coefficient is 6
and the two
lengths are S and S)**

** **

__(____THE AREA OF A CIRCLE)__** **** The
area of a circle area = A,
equals **** multiplied
by the radius squared ( The coefficient is **

** **

__(____THE SURFACE AREA
OF A SPHERE)__** **** ****The
surface area = A of a
sphere, equals 4 times **** ****multiplied
by the square of the radius. ( The coefficient is **

** **

**
If you
examined the six area formulas presented above, you may have
noticed that they
all have two lengths multiplied together. This
essentially involves the length
of two line segments, or distances, multiplied together, which
is multiplied by
a coefficient that relates to a specific geometric figure.
The line
segments that are multiplied together are usually
perpendicular to each other,
with most area formulas. This can easily be seen in one
of the simplest cases,
which is the formula for calculating the area of a rectangle,
which is length
multiplied by width or A = LW.
We
can represent the two lengths of all of the formulas presented
above with L_{g} and W_{g}.
I am representing the **

**
If you
examine the six area formulas once again, you will notice they
all have a
constant or coefficient. For example, the area of a
rectangle has a coefficient
of: 1, for the area of a
circle it is: ****,
and for the surface area of a sphere
it is: ****.
We can represent all of these coefficients
with K_{A}
The resulting concept,
K_{A},
is defined and it is true by definition.
(NOTE:
A logical choice is to use c, for the coefficients, but c is
used in this
e-book, and in many other books on physics, to represent the
speed of light, so
I am using the letter K to prevent confusion. Actually,
another good
alternative is to use an uppercase C to represent the
coefficients.) **

**
With the
above information we can create a general area formula.
That is area involves
two lengths multiplied together, (L _{g}W_{g}**

**
If we want to
actually use this formula to perform calculations, we must
know the value of K**_{A}**
for our geometric figure. For example, if we want to
find the area of a circle
it is obvious that K _{A} = **

** I have
created a software-based calculation device
using the General Area Formula discussed above. This
software functions
online, over the Internet, and you can access it from the
URL presented below:
**

** **

*www.TechForText.com/General-Area-Equation*

** **

*Section 3) The Theorem
of General and Universal
Equations*

__Section 3)
The Theorem of __

__General and
Universal Equations__

** **

**If
you understand
the above example (of how A_{g}=K_{A}L_{g}W_{g}
was derived) you should have no difficulty in understanding
the following.
However, if you did not understand the above, you should
reread the material,
and carry out some calculations with pencil and paper to
facilitate
understanding of the ideas before reading the following: **

** **

**AN
ESSENTIAL NOTE:**
**It is important to keep in mind that the value of the K
terms are determined
by the specific nature of the calculations, with the above t**

** **

__There is another
important concept to keep
in mind that relate to the above theorem, and many of the
techniques presented
in this book__. __Specifically,
the value of the K
term is sometimes partly determined by the units that
are used in a formula.__
For example, if we calculate the area of a square, the
value of K is 1, as
previously explained. However, if we want to use feet
for the measurement, but
we want the calculated result in square inches, the value of
K is not 1.
Actually, with this example K = 144. This
will be clarified with the
following example:

** **

**Let us assume that the
square measures 1.5
feet, then the calculation would be
(144)(1.5)(1.5) = 324 square
inches. Thus, by using the
appropriate value for K it
is possible to create formulas, and software-based
calculation devices, that
calculate using mixed units, such as meters and
centimeters, or even inches and
meters. **

** **

**
Using
some of the concepts described above I created a
calculation device that uses
the same formula, with a number of values for K, to
calculate the speed of gas
molecules, at a specific Kelvin temperature.
Specifically, this
software calculates SIMULTANEOUSLY: Root Mean Square Speed,
**

**Physics Calculator
for Molecular Speed**

** **

**Sometimes, a formula is
needed that
calculates a quantity that does not relate directly to the
variables in the
formula. This idea can be clarified with the following
examples. **

**
If you are selling real estate, you may
want to calculate the selling price of a property, based on
the number of
square feet. This can be calculated directly as
follows. **

** **

**Assuming, the calculations
are based on a
rectangle, the value of K would equal the price per square
foot. Let us assume
that the land measures 50 feet by 100 feet, and it is
selling for $100 per
square foot. With this example the value of
K = $100, and the calculation
is presented below:**

** **

**$100(50)(100) = $500000.**

** **

**A more complex example
similar to the above involves
real estate that is circular in shape, and has a radius of
50 feet, and also
sells for $100 per square foot. In this case the value
of the K = ****,
and the calculation is as follows:**

** **

** **

**Another
example, using the above concept is calculating weight,
based on height,
length, and width. For example, if a foundation
measuring 100 feet, by 100
feet, with a depth of 40 feet must be created to erect a
building, what is the
weight of the material that must be removed. Let us
assume that the average
weight of the material is 100 pounds per cubic foot, and we
want to know how
many tons must be removed to create the foundation.
With this example the
value of K is 0.05 (The value for K is based on 2000
pounds for one ton, which
results (100/2000) = 0.05). The calculations
for the foundation are
as follows: **

**(0.05)(100)(100)(40)=****20,000 tons**

** **

** **

** **

*Section 4) Modifying a
General Equation So It
Will Apply To a Larger Number of Cases*

__Section
4) Modifying a General Equation__

__So It
Will Apply To a Larger Number of Cases__

** **

** **

__es__

__Sometimes
a
general formula can be further modified, so it will apply to
a larger number of
cases:__** To explain precisely what this means I
will return to the
previous example of the general area formula. That is
the formula derived
above A _{g} = K_{A}L_{g}W_{g}**

**Starting
with
three, or more general area formulas: **

**A _{g1}=K_{A1}L_{g1}W_{g1},
**

**Addition:
A _{g1}+A_{g2}+A_{g3} = K_{A1}L_{g1}W_{g1}+**

**and
if A _{g1}+A_{g2}+A_{g3}=
A_{g}_{ }then
**

**A _{g }**

__Section
5) The Potential Utility of__

__Universal
Formulas, Using The __

__General
Area equation as An Example__

** **

**There
are a large
number of formulas in mathematics and physics that can be
generalized, into
universal equations, but this does not necessarily imply that
all of the
resulting equations will have practical or theoretical
utility. However,
usually the generalized equation will provide at the very
least a better
understanding of underlying concepts. For example, with
the general area
formula ( A_{g}=K_{A}L_{g}W_{g}**

**
Usually, when
a general formula is derived its practical or theoretical
value may not be
immediately apparent, and the idea is to attempt some creative
thinking, and
perhaps some experimentation to determine the utility of the
equation. To
illustrate some of these ideas I will return to the general
area equation. **

**
The potential utility of the A_{g}=K_{A}L_{g}W_{g}**

**
However, with
odd shaped objects, including all of the above, there is a
challenge to devise
a method of measuring the area,
and the
values for L_{g} and W_{g}**

The general area equation, A

**
The technique
discussed in the above paragraph, can be carried out with a
circle, instead of
a square or rectangle. In such a case, **_{A }_{ } if
the surface is perfectly flat, without any bumps, holes or
peaks. _{A }** if
the surface is not perfectly flat.
**

**
It is
important to note that the concept of roughness and
smoothness are relative
concepts. For example, a flat piece of glass may look
perfectly smooth, but at
a molecular level it will be rough, and at an atomic level it
will be rougher.
This means that if you calculated the roughness of the glass
(as it appears to
the naked eye,) you would find that **

** **

** **

** **

*Section 6) **Creating
One General
Equation From One or More Other General Equations*

__Section 6)
____Creating One
General Equation__

__From One
or More Other General Equations__

** **

**It
is often
possible to create one general formula from one or more
generalized equations,
which can result in insight as well as new sets of equations.
This can sometimes
be achieved with algebraic manipulation and/or substitution of
terms from one
equation to another. Even an arithmetic operation, such
as multiplication can
sometimes result in a new equation as shown in the following
paragraph.**

__A General
Volume Equation From The General Area Equation A _{g}=K_{A}__

***Note,
as PREVIOUSLY
STATED: it is important to keep in mind that** **the
value of the K terms
are determined by the specific nature of the calculations,
with T**

** **

*Section 7) A General
Equation for Distance*

__Section 7)
A General Equation for Distance__

**The
general
distance equation, (D _{L} = K_{D }D_{n})
is a very
simple formula, that I devised to represent the distance
between two points, as
measured with a straight line segment. D_{L}
represents the linear distance
between two points. D_{n} represents the
nonlinear distance between
the two points, which can involve curved or zigzagged line
segments. This can
be restated in words, as follows: The linear
distance (D_{L})
equal the nonlinear distance (D_{n})
multiplied by a coefficient (K_{D}).
When D_{n} is multiply by the
coefficient K_{D} it equals the linear distance D_{n}
Some
examples will clarify this concept. **

**
Let us assume
the linear distance between TOWN‑A and TOWN‑B is 100 miles,
but the
travel distance between the two towns is 200 miles, because of
the structure of
the roadways. The value of K _{D} can be
calculated as follows: 100 = (200
)(K_{D}) This simplifies to (100/200) = K_{D}
That
is K_{D} = 0.5 with this example. The
value of K_{D}
(0.5) relates to the efficiency of the roads, in relation to
the traveling
distance between TOWNS A and B. This can be converted to
a percentage which is
50%. Improving the roadways between the two towns, would
increase the value of
K_{D}. If the roadways were perfectly
redesigned to maximize the
value of K_{D} the roadways with represent a linear
distance between
the towns, and the K_{D} would be equal to 1.
However, with the general
distance equation, K_{D} can never be greater than
one, assuming the
calculations are based on real-world mathematics.**

**
The general
distance equation, (D _{L} = K_{D }D_{n})
can be further illustrated with a right triangle. This
will become clear if
you examine the right triangle presented below: **

**
With the
right triangle presented above, the linear distance is equal
to the hypotenuse
of the triangle, which can be represented as line segment
AB. In addition, a+b
is equal to the nonlinear distance between points A and B.
However, the most
important idea here is AB= K _{D}(a+b) which is true
but definition.
Keep in mind that AB represents the linear distance in this
case, and the
equation can be rewritten as Distance = K_{D}(a+b),
or The Hypotenuse = K_{D}(a+b).
With
a little algebraic manipulation it is obvious that the value
of K_{D}
is equal to AB/(a+b). This relationship can also be
represented as K_{D} = Distance/(a+b).
**

**If you
examine the triangle, the mathematical relationships presented
above, it
becomes obvious that if angle A increases, the value of K _{D}
also
increases. More precisely stated, as angle A approach 90
degrees, K_{D }approaches
1. Thus, there is an obvious relationship between the concept
of an angle, and
K_{D.}**

**
The value of K _{D
}can easily be calculated, when the general distance
formula involves a
right triangle, with the Pythagorean theorem, (c^{2} =a^{2} +b^{2})
as follows: **

**Distance = Hypotenuse = K _{D}(a+b)=C**

** **

**K _{D}=C/(a+b)**

** **

** **

**Thus by substitution**

** **

* *

* *

* *

* *

*Section 8) Deriving the
General Area and
General Volume Equations with The General Distance
Equation*

__Section 8)
Deriving the General Area and__

__General Volume
Equations with __

__The General
Distance Equation__

**The
general
distance equation, (D _{L} = K_{D }D_{n})
discussed
in the previous section, can be used to derive the general
area equation by
multiplication, as follows: The general area equation, A_{g} = K_{A}L_{g}W_{g}**

** **

**D _{L1}=L_{g}**

** **

**D _{L2} = W_{g}**

** **

**A _{g}**

** **

**A _{g} **

** **

**K _{A}=**

** **

**(D _{L1})(D_{L2}) = L_{g}W_{g}**

** **

**Thus,
by
substitution A _{g} = K_{A}L_{g}W_{g}**

** **

**
The general
distance equation, (D _{L} = K_{D }D_{n})
can also
be used to derive the general volume equation (V_{g}**

** **

**D _{L} = K_{D }D_{n}**

** **

**D _{L3}=H_{g}**

** **

**D _{L1} = L_{g}**

** **

**D _{L2
}=W_{g}**

** **

**V _{g}**

** **

**V _{g}**

** **

**K _{V}=**

** **

** (D _{L1})(D_{L2})(D_{L3}) = K_{v}H_{g}L_{g}W_{g}**

** **

**Thus,
by
substitution V _{g}**

** **

***Note,
as
previously stated: it is important to keep in mind that**
**the value of the
K terms are determined by the specific nature of the
calculations, with T**

** **

** **

** **

*Section 9) The General
Dimension Equation*

__Section 9)
The General Dimension Equation__

** **

**There
are three
generally accepted spatial dimensions, which are height,
length, and width,
which can represent perpendicular line segments of a
geometric form.
The line segments are linear and they form perpendicular
angles with each
other. However, there are theoretical concepts that
define a number of
additional dimensions. **

**
In regard to
the following equation, we can think of a dimension as a
factor, in terms of
multiplication. For example area is equal to two factors
multiplied together,
which is length multiplied by width, multiplied by a
coefficient. Volume
involves three factors multiplied together (height, length and
width),
multiplied by a coefficient. With this idea, we can
create, a general
dimension equation (or multiple dimension formula) with the
general distance
equation, or simply by multiplying a set of factors together,
with a
coefficient represented by K _{g } This results
in the following
equation: **

**
If we assume
that all the factors have the same value, such as with the
dimensions of a
square, circle, cube, or sphere, we can represent the concept
with the equation
listed below. **

**With
the above
equation, the exponent n,
determines the meaning
of G _{d}, when appropriate values for K_{G } are
utilized in
the calculations, as follows:**

** **

**When
n = 1 G _{d} = distance,**

** **

**When
n = 2, G _{d} = the
area of
a square, or the surface area of cube, or sphere**

** **

**When
n = 3, G _{d} = the
volume
of a cube, or sphere**

** **

**Both
of the above
equations represent an infinite set, which may contain
undefined concepts that
have practical or theoretical utility. For example, when
the exponent is 0.25,
0.5, 4, 5,6 what would G _{d} represent? **

** **

** **

** **

** **

*Section 10)**
The General Energy
Equation*

__Section 10)____ The General Energy
Equation__

** **

**The
formula for
kinetic energy is **** .
This formula represents the energy
of a moving object, with a mass m,
and the
velocity v. This
equation can be generalized by
replacing the coefficient with K _{E}.
Thus,
we can represent the general equation as follows:**

** **

**E _{g} = K_{E} mv^{2}**

**The
above equation represents
many formulas that are used to calculate energy, including
Einstein's equation
E=mc ^{2} With Einstein's equation K_{E} = 1.
We can even apply this equation to the energy of
photons, by calculating
the mass equivalent of a photon of a frequency of light.
The mass
equivalent to a photon can be calculated with Einstein's
equation E=mc^{2}
and E=hf (I am using f to represent the frequency of
light, to prevent
confusion with other symbols used in this e-book. h of
course equals Planck's
constant.) hf=mc^{2 } Thus, the following is the
mass equivalent of a
photon m=(hf)/c^{2}
It should be obvious
from the formula that was derived, that the greater the
frequency, the greater
the mass equivalent, and the greater the energy of the photon.
**

** **

** **

*Section 11)**
Deriving A General
Temperature Equation, With The General Energy Formula*

__Section
11)____
Deriving A General Temperature Equation,__

__With The
General Energy Formula__

** **

**The
temperature of
an object relates to the kinetic energy of molecules and atoms
that comprise
the object. When a specific object is heated it gains
energy, as its
temperature rises. Thus, the General Energy formula can
be applied to
temperature of a solid, liquid or gas. In solids, and to
some extent in
liquids, v relates to frequency. Specifically, when
molecules vibrate, even if
they are moving back and forth, and not changing their
position, they are still
moving at a specific speed, which can be represented by v term
in the general
energy equation. In the case of a gas, the molecules are
moving at an average
speed that relates to temperature.**

**Thus,
we
can modify the General Energy formula to create a General
Temperature
Equation as follows E _{g} = K_{E} mv^{2}**

**
It is
relatively easy to algebraically rearrange the General Energy
equation, to
derive a general formula to calculate the speed of moving
particles, as
follows: **

** **

**
With the
equation derived above **** I
created a software-based
calculation device that calculates molecular speed, which
functions online over
the Internet. Left click on the following link to access
this calculation
device.**

** **

**Physics
Calculator for Molecular Speed**

** **

** **

** **

*Section 12) The**
General Equation of
Moving Particles*

__Section
12) The____ General
Equation of Moving Particles__

** **

**The
concept
described in the previous section, involving the
representation of coefficients
for a number of related equations with K, can
also be applied to exponents. To demonstrate this
concept, and its potential
utility I will use the General Energy formula that was
discussed in the
previous sections. E _{g} = K_{E} mv^{2}**

**
The
interesting idea here is the equation
P _{M} = K_{M} mv^{n}
defines an infinite set. It is apparent that
there are three important
concepts in the infinite set defined by this equation, which
are mass,
momentum, and energy. This raises the question are there
other unknown or
undefined concepts in this set, that may have practical or
theoretical
utility? See the following examples: **

** **

**P _{M} = K_{M} mv^{0.25}**

** **

**P _{M} = K_{M} mv^{0.5}**

** **

**P _{M} = K_{M} mv^{e}**

** **

**P _{M} = K_{M} mv^{3}**

** **

**P _{M} = K_{M} mv^{4}**

** **

**P _{M} = K_{M} mv^{5}**

** **

**P _{M} = K_{M} mv^{6}**

** **

**P _{M} = K_{M} mv^{-0.25}**

** **

**P _{M} = K_{M} mv^{-0.5}**

** **

**P _{M} = K_{M} mv^{-e}**

** **

**P _{M} = K_{M} mv^{-3}**

** **

**P _{M} = K_{M} mv^{-4}**

** **

**P _{M} = K_{M} mv^{-5}**

** **

**P _{M} = K_{M} mv^{-6}**

__Below, there is
a hyperlinked table
of contents for this e-book. Left click on any of
the blue words to go to the
section of the e-book you want to read.__

*Section
3) The Theorem of General and Universal
Equations*

*Section
4) Modifying a General Equation So It Will
Apply To a Larger Number of Cases*

*Section
6) Creating One General Equation From One or
More Other General Equations*

*Section
7) A General Equation for Distance*

*Section
8) Deriving the General Area and General
Volume Equations with The General Distance Equation*

*Section
9) The General Dimension Equation*

*Section
10) The General Energy Equation*

*Section
11) Deriving A General Temperature Equation,
With The General Energy Formula*

*Section
12) The General Equation of Moving Particles*