Overview:
As with any subject on Skycab, math and measure is an endeavor in itself. I
will discuss some mathematical theory and try to avoid the arithmetic. The arithmetic
in higher math is tedious. Of course, the arithmetic of higher math is essential
in proofs, modeling, science, engineering, and application, but you can get
a "feel" without cumbersome proofs. I have some real goodies in Calculus!
I will offer an expanded definition of Calculus, beyond credible asserted fundamental
theorems of Calculus. This expanded definition will encompass "higher math",
so that we may explain how "higher math" and Calculus is the math
of science and the math of the space and information age.
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Calculus Theory:
In a nutshell, Calculus can be described in terms of the derivative and the
integral. A derivative often produces a linear equation, or at least a slope-line
equation to decipher an "instantaneous rate of change" along growth
and decay models, or other models of rates of change. The integral can be thought
of as the anti-derivative. I will frequently put "instantaneous rates"
in quotes throughout this page. For all practical purposes, proper Calculus
has derived "true instantaneous rates" to a degree that is only limited
to your data or input. In addition to this, you can substitute "abstract"
variables to achieve "absolute" solutions. Comprehending absolute
solutions in abstraction may be difficult, and I will not delve into too much
detail. So you will see that I continue using relative terms like more absolute,
etc.! This is an important point left for your discovery.
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Derivatives:
A derivative, essentially, is/or produces a result along all real numbers of potentially infinite accuracy,
which simply shows a solution to a secant line of two points on a curve, paralleling
the secant on a definitive point between those two points of the secant, and
on the curve, to infinite accuracies, such that you can achieve an "instantaneous
rate" of change along the curve or a definite point on the curve along
all real numbers. Or is it a tangent line? My memory recalls that the tangent
is a mean solution to the secant, but please let me know if I am off. One example
might be acceleration. As one moves along time, distance increases disproportionately,
or decreases disproportionately (deceleration) producing a data curve. On this
curve, you can find the "instantaneous" acceleration/deceleration
at any particular time and obtain instantaneous velocity.
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Integrals:
In a nutshell, the integral is the anti-derivative, or a return to the function from which a derivative
was obtained. You may not be able to return to the exact function, but this
is OK. You gain enough information to draw valid conclusions and astonishing
conclusions. These concepts can be applied to all kinds of relationships of
change including surfaces, areas, and volumes in dynamic situations. This stuff
can make the brain ache, but I will provide details over time and I will try
to bridge these concepts to space endeavors, especially in the summary section.
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Composite Functions:
This is the ability to combine or calculate sets of data. As an example, composite function g(x)
may represent one set of data - let's say gravity's acceleration and friction's
resistance in the stratosphere. Composite function f(x) may represent gravities
acceleration and friction's resistance in the thermosphere. H(x) may represent
the mesosphere. How much fuel and thrust do you need to overcome these forces
combined? By the way, you can do arithmetic of composite functions.
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Abstraction:
Abstraction relating to math and measure is the ability to understand the conceptual theories relating
to any unit or data set. A "unit" can be defined infinitely. 2 units
plus 2 units equals 4 units. Differentiating along a growth model is the same
if it is bacterial growth, or the inverse of radio active decay. Likewise, integrating
follows the same principle and logic. Maxima and Minima can be evident in radio
waves, or calculating surface areas.
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Sets:
Relationships of sets can also be thought of in terms of proportionalities. If the volume
of sphere is expanding at a certain rate, it's surface area is increasing as
well - but disproportionately! Though the rates of these two data sets are increasing
disproportionately, they are predictable and measurable. They have a relationship!
Dah, this is obvious. Is it? Folks, this stuff is how our world turns. A sphere
is an easy example, but the ability to understand these relationships related
to density, speed, etc., is the stuff of Einstein Relativity. These are the
concepts that allow us to send people to space and bring them home safely. These
are the concepts of scale modeling for Hollywood, architectural engineering,
and more.
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Instantaneous Rates:
Again, if you have data accumulations of growth, decay, exponentiation, acceleration, or other
disproportionate models of data, Calculus aids in determining the behavior of
the data at any instant in time - whatever that time may be (abstraction). This
helps offer predictability as well as other information. Keep in mind another
important concept - natural phenomena often follows easily measured and consistent
behavior, more so in laboratory settings due to isolation and other factors,
but through composites, natural phenomena can be explained to great degrees
of accuracy.
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Growth:
Whether it be a volume, mass, bacteria colony, or how fast you get on the highway, Calculus
can help determine all the changes in timespace. |
Motion:
See Growth and Instantaneous rates.
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Energy:
Energy and matter, which is quantifiable and covers so much (if not everything), and is the basis of physics,
chemistry, and other subjects - that I just do not know where to start. These
points could, and continue to be argued. So many scientists are looking for
unification theories (missing links), to their data and theories. Calculus helps
in this endeavor of unification, and of newly discovered constants, formulas,
and functions. We could yell at each other all day on interpretations of data,
and so we try to present it in an understandable and honest way. We need to
say when we are wrong, or if we are close, or we are debunked, or offer caveats,
etc. This is an art. Data collection, interpretations, and empirical proofs
are essential in the language of science. States of energy can be described
in so many terms of matter from inertia, to potential energy, to kinetics, to
Newtonian Physics, to Particle and Quantum Physics. At this point, I will say
only one thing, and that is that there is a special language to help describe
these ideas - and you will find them in science, math, physics, and chemistry
to name a few, and that there is a special vocabulary with meaningful distinctions
and descriptions.
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Radioactive Decay:
This can be a tricky subject. Once again, we are dealing with so many variables. Carbon Dating for example,
may only be as good as atmospheric and geological data from which you are sampling.
How has life and our planet evolved? These must be considered when attempting
to date artifacts, core samples, fossils, etc. There are other dating methods
available. You have to consider your specimen. Is it unique? How unique? What
does legend say about the area? Are there cave paintings? What is the history
of the area? What do core samples say about the state of the earth 10,000 years
ago? What about 1 million years ago? What is the specimen comprised of? Is it
like anything you have seen? Is it out of place - like a meteorite? These may
be only a few of hundreds of questions that you must ask before dicing, slicing,
and using equipment. Having said this, if you feel you have something special,
there are plenty of dating options available. Comparative or reference data
sources can help narrow down some initial estimates. You could start by asking
an expert. If you must measure decay to hone in on accurate dating, you will
be surprised with the accuracy you can achieve, especially if you can cross-reference
your findings to refine results. Radioactive decay is predictable, and the ability
to measure it is a miracle of science. You use the methods described in Calculus
to interpret the results of your data. Folks, I believe you could turn lead
to gold with this science stuff, if it was not so time consuming and cost prohibitive.
So, you better have a hell of a find if you are going to start doing cross referenced
analyses.
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Maxima and Minima:
Maxima and Minima, I believe is one of the most important functions and fundamentals of calculus.
You may have several billion years worth of data. Your data could be several
gigahertz. Your data may stretch over thousands of pages. You may have so much
data, that the only way to decipher it is using Calculus. In the SETI
section on this website, we further explore data acquisition and interpretation.
One of the best ways to interpret large amounts of data, is to find all the
maxima and minima, and work with that. As a simple example, let's say you have
a radio wave that peaks every billionth of a second. What is the frequency?
1 billion cycles a second? Or is it 1/2 billion cycles from crest to trough,
per second? What about modulations? Is there a pattern in the modulations?
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Inflections:
Folks, this is good stuff so far, ain't it? Inflections offer even more data interpretation possibilities.
Inflections describe unique changes along a curve from a symmetrical increase
to a decrease, or vice versa. Again, this could show periods and patterns worth
noting when interpreting huge amounts of data. In addition to Maxima, Minima,
and Inflections, there are other significant curvatures and lines to consider,
if needed. So as you can see, data interpretation becomes very comprehensive.
This is one reason I will expand the definition of Calculus - not because the
fundamental theorems are not fundamental, but a comprehension of the data being
operated on as well as applications need to be considered, or else the ideas
of the fundamentals are lost in the total construct of what Calculus is and
is capable of doing.
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Infinitesimals:
Folks, I have no idea where I pulled this one from. I believe it comes from Calculus. Let me know
via the contact link above. It sounds cool, and I would be curious what an infinitesimal
is. Perhaps I will look it up some day if I do not get an answer and I will
post it here.
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Single Variable Calculus:
Single Variable Calculus can be quite interesting and tricky, when working beyond the 2 axes Cartesian
Plane. The basic idea of this Calculus is to interpret data as a function of
one variable. When dealing with multiple variables, you use relationship substitutions
to achieve a single variable function. As with so many topics on this page and
on this site, I may add to this in the future, so check back from time to time.
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Data Interpretation:
Like Energy and matter itself, this subject is fundamental. We have discussed it throughout. Data interpretation
in many cases sets political and financial policy. Global Warming policy is
an example of government instituting harmful economic policy against business,
based on sketchy data interpretation. Is business causing global warming, or
is the sun causing global warming? Does the earth go through cooling and warming
cycles regardless of human interference? Is our data trend 10 years, 50 years,
or 100 years? Is weather data from 100 years ago reliable and is there enough
data? Is global warming 1 degree or 2 degrees (if it is true)? Will we warm
1 degree in 100 years, or 2 degrees in 150 years? Can meteorology accurately
predict weather 3 weeks from now? You get my point? It is too easy to run blindly
with an idea whose premise may be false. I believe this is especially risky
when spreading fear that we will destroy our planet for future generations,
as well as shaping policy that is costly to business, the economy, and the consumer
in so many ways.
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Measure:
Measure is a fascinating and extensive subject. How do you measure a photon? How do you measure gravity
or an electric field? How do you measure the mass of an element? What about
temperature? You say for temperature, just use a thermometer. Your thermometer
could alter the sample you are measuring. The same is true if you measure particles
- by measuring a particle you have disturbed its position and velocity. Then
there are standards of measure. The masses of elements, in part, are measured
with respect to carbon-12. One measure of energy (a calorie) is measured against
the heating of 1 gram of water at the phase change of water from liquid to boiling
at 1 standard atmosphere and pressure. The calorie is more commonly represented
by the number of joules (around 4.2 joules). Joules may be defined by another
standard including kinetics and electromotive force. And then you have energy
and momentum conservation laws which allows you to interpret energy broadly
throughout systems. Light and electromagnetism may be standardized by its behavior
in a vacuum with no field influences. And the deviation from this standard in
different media can yield such data as refractive indices and a change in speed
and direction. When measuring infinitesimals and discrete particles, numerical
significance may play a role in deducing individual behavior. Photon scattering
tests are one example of procedures that may be repeated many times to achieve
numerical significance (probability graphing and distribution). Lastly, depending
on what you are measuring, you can convert joules, watts, and horsepower (just
to name a few). This small section just scratches the surface of the standards
of weights and measures.
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Physics:
I will further develop this section in the future. |
Expanded Calculus Definition:
Why have I decided to offer an expanded definition of Calculus? First,
if you have read all of the preceding, you should understand an
expanded definition of Calculus. But if you have not read the preceding,
Couldn't I simply leave the definition to 2 fundamental theorems?
Perhaps a dozen theorems? Exactly! A dozen theorems for starters,
then a few more. Calculus has a rich history of tedious and laborious work.
In college, some courses require that you have completed pre-requisite courses
for any subject. Calculus is no exception. For the sake of the direction and
scope of this website, I will start by simply proclaiming Calculus to be "higher
math". Calculus can help aid in determining so many topics on this website,
that it spans great complexities of numbers and data that have not even concretely
been counted yet, or are infinite. Also, how can you divorce the topics that
Calculus seeks to explain, including "arithmetic", from Calculus itself?
And how do you define arithmetic? Adding and subtracting numbers? Folks, mathematicians
could show you "arithmetic" that would include massive matrices and
arrays, and then a few more before making your mind spin. So, isn't Calculus
in part, the fundamental theorems applied? In order to apply the fundamental
theorems, can you discount the knowledge of what you are applying it to? So
my expanded definition of Calculus is simply higher transcendental math applied.
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Academia:
If you are a part of an institution of finer and higher learning, thank you for visiting!
The quest for understanding of ourselves and our universe is an insatiable thirst.
Once we have achieved the first steps in Maslow's Hierarchy, we can't help but
transcend to the higher levels of understanding and learning. I go against thousands
on global warming, its existence, its intensity, and its causes and solutions.
I go against some of the philosophical and liberal entrenchments that plague
state funded or subsidized institutions, such as some higher learning establishments.
Having said this, institutions of higher learning are a great forum of free
speech, exploration, learning, and discovery. I am requesting this website and
the publications that I have constructed to be considered as a thesis in the
application of an honorary Masters in the sciences - specifically in Physics
or Computer Science. I have acquired enough cumulative credits for a Bachelors,
and enough Fort Knox life experience for a Doctoral. A Masters would assist
me in furthering my goals in technology development, and I certainly have enough
experience beyond university courses. Again, please visit the rest of this website
for information. By the way - none of this website is plagerized! I could spit
it out in a speech before an audience or a debate club! Thanks for your consideration.
P.S. My spell checker has rejected so many words as errors. Perhaps I should contact Websters for submissions of words that should be in the dictionary - assuming the spell checker uses Webster's definitions. Thanks again. |
Summary:
As a quick tantalizer of some possible Calculus ideas regarding space endeavors, I will hint at only
a small fraction of what might be considered for a simple non-orbital launch
through the layers of our atmosphere, and returning the vessel safely to earth.
In a nutshell, we have a few layers of atmosphere, each with unique properties
and challenges. Under a constant condition or base formula, we would deal with
a calculation of the fuel needed to defy gravity (a data curve of timespace),
with a certain payload mass (including variable fuel), and the variable acceleration
(time & velocity) needed to achieve an altitude of space. Folks, these are
so many variables and factors - without ever considering terminal equations
of friction. This is only the start! Remember all those layers of atmosphere?
These layers help create terminal velocities of acceleration and deceleration,
and the force needed - or the resistance, as well as the predictability , and
the change in acceleration due to dwindling fuel supply and mass. Can you say
composite functions (which is just one of hundreds of fundamentals in pre-reqs
to the essentials of Calculus applied). What about cross winds? What about telemetry?
Do you have a little extra fuel for margins of error? How much error? Again,
thanks to some of our ancestors, we have nice curves to follow for all of these
considerations - tried and proven! However, it is difficult to steal a rocket
ship, and purchasing or renting one is expensive, and so we should understand
economics, logistics, planning, engineering, science, and math, as well as the
fundamentals that could make intra-stellar space travel possible. Having said
this, nothing beats the knowledge of how these things are accomplished, even
if you could simply purchase a space vehicle! And the knowledge is applicable
to so many general studies. Having said this, there would certainly be paying
customers for space travel. So the nuts and bolts of space travel are really
left for the science, technology, and engineering pioneers of space travel.
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Pic Group:FOLDER: math
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