Additional Supplements notes

Useful Properties Supplemental notes for http://ocw.mit.edu/courses/mathematics/18-950-differential-geometry-fall-2008/lecture-notes/ch1_revised.pdf

Relation of two vectors given one vector having magnitude length 1

Relation of two vectors where one is of magnitude length 1

Supplement to http://ocw.mit.edu/courses/mathematics/18-950-differential-geometry-fall-2008/lecture-notes/ch1_revised.pdf

A personal self philosophy

Inspired at the moment to the previous post.  Generally I tend to remain topically silent much on the matters of a self philosophy or anything relating to spiritual matters.  I would have to admit at times that I am far from perfect from a given self philosophy on the matters of personal choice.  Whether pertaining to things such as caffeine or nicotine, or other matters, and I could certainly work at times perhaps much to improve myself in terms of thinking and mindset.  I'd caution I don't consider myself a great spiritual adviser, and my training s with respect to theological matters are limited.  Although I've found myself interested at times in reading, or in other words, I would be simply a professed lay reader.

But given this, I found myself curious to the matter of the spiritual and any corresponding self philosophy.  I'd offer at least I believe many may have some spiritual belief or self philosophy in life as to the ways they 'get by' in life.

Here are some things to be considered:

1.  Self contradiction.   Have you ever said that you would commit yourself to living in one way, and then having found yourself doing just the opposite?  It seems there is a mindfulness with respect to the commitments made in one's life.  When I were younger, I might have been inclined to in my thinking more 'austere' commitments whether impractically working towards a set of goals that I've found little patience in pursuing in the long haul.  Learning patience it seems to me is a wisdom gained in life as is commitment.

2.  Learning to love the days that you have.  How often I could recall being so immersed in the idiosyncrasies of a day, or having been a day dreamer thinking I would exist anywhere but in the world that were here and now.  No doubt day dreaming could be the genesis of much inspiration and creative force, but it seems if you hadn't done much to change your environment and dwelling elsewhere, is it always constructive?  From my limited perspective it would be one thing if another had little choice, its another thing when you are empowered to make the changes that are afforded to you.

3.  Too much television or media.  When is enough, enough?  If you find yourself drawn into heated discussions constantly where you believed so much in the causes that were being framed for you to believe, were you living for others or yourself?  If you have the responsibility for others in speaking the truth of what you know with certainty with respect to cause and condition, there isn't neglect for this, but where often I wonder we learn the most of others could be found in our personal contact and experience with others, some of which could be neglected in media and televised presentations.  If you find your blood pressure rising watching another segment of what were supposedly an issue, the personal question of pertinence maybe in order.  How am I directly effected by such an issue?  How carefully does one judge a given world?  Is it upon the reliance of so many others in saying?  Is it formed solely by the consensus of selective channels and presentations, or is it based upon the direct contact and testimony of individuals that could attest their condition personally?  Maybe if having been angry too much at things that were going on hundreds if not thousands of miles away in another world, and having only precluded contact and little personal experience and wisdom, there is wisdom found in better judging one's immediate personal circumstance and condition?  Your spiritual wisdom and guidance is not a television?  Certainly if you find yourself being guided to the condition of loving less and being more hateful towards others, could one say that one were the better in the spiritual sense and lived a healthy life for it?  Honestly I find myself, personally attracted to the sobriety and objective interests of sciences, arts, philosophies, religion, sociology and what not.  I tend to be attracted more these days to objectivity rather then opinion pieces...granted this is an opinion piece in its own right at the moment. I am not so interested in being angry at what others do or don't believe, in as much as being attracted and shaping the social environment in which I interact.  I have to admit I am selective.  I am biased on this point. I've found as of recently.  I turn off the laptop and read, or make quieter space both literally and mentally in mind.  I selectively limit my television viewing.  I've refrained from playing video games which have drawn attention from much else in life.  I actually feel better mentally.  My mind seems a bit clearer.  I have decent focus and concentration.  I sleep more with a natural circadian rhythm these days.

4.  Constructive time use.  If you're work in life were a guidance here, I've found avoiding dwelling on things which go beyond my control and time.  It seems to me organization and structure in life helps a lot.  Whether in sense of creative expression, art, or as I've read from others, hopefully, one paradoxically lives neither in a present life but in thought of some vicarious idea of what life should be later down the road which may never come to be.  If worry is there to serve change, then change, so that worry comes to pass, but if it serves no purpose legitimately in one's life as expressed in a warranted condition, then perhaps analysis of worry should be made?  Is it legitimate, is it right to the conditions of one's life?  Or is it an excuse for dwelling in thought?  It seems if one were to harbor worry so much to the detriment of personal health, and nothing of eventual change is made of a given environment, something must change with respect to worry?  Secondly I'd mention on the part of cognitive behavioral patterns, if your life is structured around the idea of dwelling negatively in some manner, the conditioning of patterns in thought are self reinforcing?  In other words, it should seem the mind is conditioned to structure some routine in daily life whether deliberated more thoughtfully or not.  If one finds oneself into positive mindful habits with that required so much less momentum and effort on our part, maybe it is easier to engage in such pattern of behavior without so much effort and thought in successive applications of retraining the mind, and finding the wisdom of an order and discpline.  Personally I am not into rigorous physical training regimens, or self punishments, nor seeking of disciplines that could be so austere as to be dropped with respect to the everyday practical conditions that one faces.  Locomotively, maybe walking could be preferable to running, or at least over the years, I've learned to slow my pace down, and neither produce so much lactic acid.  I've read so much to the balance and baseline of energy levels.  I've had my up and down days in the past, but my energy seems more balanced.  Eating adequately seems important, as does being well rested.  I am not so much into being over stimulated like I were when I were a kid.  This isn't to say that at times I weren't perfect in this sense of being or seeking at times.

5.  Admittedly a self pro crastinator at times.  Maybe this isn't so good all the time, but then I found a philosophy professors statement on this issue compelling.  Think of the positive accomplishments that you have made, and dwell less on what hasn't been accomplished.  It seems the antipathy of structure and order at times, but at least I'd offer if there were an analog in nature, nature may not always serve to be the most efficient or 'productive' in the engineered sense.  If we learn to find ways of managing ourselves in the sense of finding structure and productivity in the balanced sense, we are also neither machines that struggle minds and bodies to the artificial structures that we might have created for our place and sense in life.  Learning to find relaxation and rest are important.  Admittedly 5 seems in contradiction to 1, and that it seems is part of the mystery of human nature.  We aren't entirely logically driven, nor without sense of emotion and something at times irrational, but at least it seems there is something of a happy medium here?

6.  Peers and friendship.  Wisdom comes in chosing friends wisely.  Many probably are already well aware of this.  If someone has something of constructive criticism to offer, and this is neither meant simply to serve for some purpose of personal dislike, or any opinion that could likely never change could be reflective of the environment one chooses to live in.  Those that dwell in the experience of so much fixations in life to the detriment of relationships and all else, providing little time for others are probably not great friends.  There could be much added to this list.  Enough said.

7.  Dating and relationships.  Not much of an expert here sorry.

8.  Violence.  I see this from the perspective of human evolution.  Those that suffer in violence in the long term sense probably don't fare well, and this is probably for good reason.  When violence cropped up so long ago, likely it seems it were neither driven in the artificial sense to wars extending decades, centuries or extended lengths of time.  Likely when relating to internal conflicts, maybe these were likely to end as quickly having begun.  Certainly if the human mind were resilient to violence and conflict in the short term sense, evolution would be less of aid to those subjected in the long term sense to violence in the social organizational sense.  While some could claim it is part of human nature, civilizations has at times made something unnatural at times by violence...through the power of technology and industries themselves which aren't exactly natural I believe to human physiology and psychology.  We believe we adapt to the weapons that we have fashioned, but where clearly have we adapted?  What inhibition to violence should exist as it might have once existed in more natural orders with respect to the wisdom and strength of elders in a given community to correct any of violent behavior (a natural equalizer) lost with respect to the kid that could pull the trigger of a handgun without so physical reprecussions that might have once existed?  The modern weapon is too easy a faciliatator of violence.  Much of the natural world is dictated by the reprecussion of violence in animal behavior that neither exists given the artificial structures of the modern human world.  I believe this is the great imbalance of our western civilization.  A bruised eye, cuts that can heal, and someone having retributive fear of engaging in such violence are much different then tools that potentially facilitate violence to obscene levels without so much fear.  This aside we are intelligent beings, if it should seem that violence is unnecessary with respect to resolving differences, it is that our societies should have so much to say with respect to law, ethics, and our spiritual beliefs.  What remains of the natural order of human nature pertaining to violence, when peoples could be mobilized by mechanized transport, armed, sold to artificial propagandas promoting violence.  How natural is this relative to the ancient human colonies once having existed and limited so much on foot in communicating through natural alliances of neighbors to promote some level of violence that should pale by comparison.  Is the actions of human nature so unnatural in propensity to violence, or is its modulation profoundly induced by the artificial structures and technologies of civilization which again has promoted this to obscene levels unparalleled in evolutionary history.  The tools of communication, logistic transports, weapons, all reflect the great imbalance of civilization with respect to human nature itself when employed in such a way.  If we are taught to listen through selective channels to hate without question, taught to kill or be killed, to be armed or die, then what choices do we perceive otherwise in life?  If we see only the normative condtion and being in others around us and we fear questioning, how are we to change otherwise?   Thus referring to another post on the subject matter, why violence (and political violence here in America ranks the highest in most past wartime situation excepting the Korean war) should occurs likely has come from the propaganda and the artifical constructions of a given society itself in promoting this in the systemic sense, has it not?  Does this reflect some other imbalance of a given civilization.  I am not here to say that human nature is necessarily so different then it ever has been with respect to some basic condition, but I would say that are social environments themselves could be conducive to the promotions of artificial levels of violence that have little to do with human nature.  Our spiritual imbalances occur when we are not right in the mind to see through what we are told to believe.

At times it seems much more courage could exist for a person to speak to enemy with the intent of non violence, so much better in my opinion for a non violent man or woman of courage if such virtue should exist.  


And some of the very basics of a personal philosophy...

Make a self environment of peace.  Make a loving environment for self and family.  Those closest to you, whom love, trust, and respect you are part of your home.  Avoid dwelling in hate, fear, and worry.  Be willing to learn and see beyond immediate differences.  

The relation the cosine angle between vectors and their inner product

This shows the plane triangle with vectors x, y, and y-x where the coordinate axis are defined in terms of the plane of vector x, or in three dimensions, the vector x on the relative plane of such coordinate axis can be thought as having x₁ = relative x component, x₂ = relative y component. Noticing that the direction y₃ a relative z component is perpendicular to component plane of x₁ and x₂.

First, I'll address why the inner product must be zero for two perpendicular vectors.

For two given vectors x = (x₁,x₂, …, x_n) and y = (y₁, y₂, …, y_n), it should be noted that two vectors are perpendicular if

This results from the Pythagorean formula where the magnitude (distances) of each side length of the vectors on the left side of the equation squared and summed equal the magnitude (distance) of of the vector x-y on the right side of the equation.  One would note the relation of vectors above with application to the Pythagorean formula, while the figure above demonstrates relations here in three dimensions, this application extends to n dimensions likewise.

Expanding the right side of the equation (1) we have

we'd note that expansion of the left side of equation (1) leads to cancellations of the terms

  

Thus we are left with the requirement for the given equality that the central group of terms in equation (2) are equal to zero, or the “dot product” (or inner product) of vectors x and y are orthogonal if

Now getting to the relation of the cosine angle between two vectors x and y.

We'll demonstrate the relation of the inner product using the figure above which is shown in two dimensions. Note this could be extended to n dimensions but for now we'll focus on two dimensions in deriving the relation between the cosine angle of two given vectors and an inner product.

From the figure above one can note the following relations,

  

Now using the relation θ = β - α and using the known trigonometric sum/difference identity and substituting the relations above we have

This can be generalized to any vectors of n dimensions such that

Sources compiled from Linear Algebra and Applications, Strang, 1988.

Prime natural as evidenced in biological life

Watching 'Through the Wormhole with Morgan Freeman' which discussed alien communication through prime numbers which discussed the possibility through genomics.  Interestingly enough I would suggest the gestation cycle of the Magicicada which is found to be in prime number cycles 13 and 17 years.  It were theorized natural selection produced such variation on the basis of survival advantages relative to predators whose gestation maximums were non prime variants.  Interestingly enough while this sort of gestation cycle I would guess is extremely rare, it does give natural advantages in terms of evolution.

Building simple mathematical diagrams with free software on the quick

Some methods in constructing formulas and diagrams on the quick here.   There may be some decent software out there for free which does much the same, or otherwise pay for software that can construct the same.  Aside from free math plotters which abound on the internet, and especially free software for python (if you program in this language) like scipy, matlibplot, gnuplot, and the like.  If you were wanting to set up diagrams really fast, you could use programs like Trimble Sketchup (used to be Google sketchup) which I've found to be the quickest and easiest in setting up plot points and setting up perpendicular angles on arbitrary axis.  Generally with graph construction, I make sure the camera is set to a top view, and then plot any number of given points.  Sketchup is nice in providing a fill for a given set of inter connected points, like lines, squares, arcs, and so forth which makes for much more graphical aid constructions in math modeling in some respects, albeit for more complex curvatures and shapes, you'd probably resort to other methods, or learn, if the interface is still available, using Ruby interface which is sort of python like but different. Once having constructed your graph and having appropriated your view perspective to the appropriate level from such perspective you can obtain a 2d rendering by simply hitting the print screen.  If you are running on windows, keep in mind you'd have to open up a graphics program (like Microsoft Paint which is provided for free) and then click the paste button in Windows 7 to paste the bit map photo into your program from Window's clipboard, or alternately if you are running SketchUp from wine on a linux distro, the print screen method (in Linux Mint and Ubuntu) are the same, the output print screen goes to your home folders Pictures folder generally speaking.  Here, generally you can do some simple photo cropping and the graph is ready for import into any desired text editor or alternate drawing program.  In this case if I want to add formulation into the graph, I've used the LibreOffice Drawing application.  This is nice since you can provide graphical formulation overlays to a given imported photo, and customization in terms of the typesetting of formulation text relative the photo (or in this case the graph) can be situated generally as desired although I haven't figured out how to rotate formula object text overlays in this case.  Once the given graphics are as desired you can either export to a desired imaging format, or as I've done I simply snapshot the perspective view again, and crop the photo in a given free image editing software (Gimp in Linux, or Shotwell does the trick).  Keep in mind if you hadn't like the standard gray imaging fill textures native to drawn mesh objects in SketchUp you could import any desired texture for filling the surface if you hadn't like the graphical diagrams object textures.  Just need to add this in Sketchup before rendering your snapshot, and then follow the steps as desired for adding formula objects to the given text.  Generally speaking this works well for simple mathematical diagrams especially when dealing with those requiring models designed with orthogonal (perpendicular) lines and simple curvatures.  

Exploration of the Bézier curve

Special case of the Bézier curve is as follows.

As pre requisite recommend reading about the Bézier curve and De_Casteljau's algorithm which provides geometric interpretation to curve construction here.

Where a curvature in a local coordinate system has no inflection points and can be described by one local maximum where this is defined in terms of a local y position where a relative coordinate system defined by the plane between points P₀ and P₃), and given two points P₁ and P₂ such that the relative positions y_1r and y_2r on such plane are equidistant from the tangent of the line P₀ and P₃. A local y_r maximum is given by

where and θ_P0,P3 represents the angle of the tangent between the segment (P₀,P₃)

Proof:

We can rotate and translate the given coordinate axis such that such that the P₁ represents a given origin 0,0 on a relative coordinate system, and furthermore we can ensure that rotation of the coordinate plane also matches the tangent line of line P₀, P₃, so that y₀, and y₃ are zero while x₀ and x₃ are not equal. By assumptions above

y₁ and y₂ are equal and remain so under coordinate changes so that are Bézier equation becomes for the y relative in the parametric form  

where y_r(t) is the parametric form of the Bézier equation in our rotated and translated coordinate plane. Keep in mind that the parametric form on the rotated plane preserve the magnitude and the relative positions of P₀, P₁, P₂, and P₃.

taking the derivative with respect to t we have

as long as x_r(t) is not simultaneously 0 at t, the derivative dy_r/ dx_r takes the form dy_r / dt / dx_r / dt

setting this equal to zero means that we are concerned with the zero in the numerator,

so we solve for y_r '(t) = 0

In this case a maximum slope on the relative coordinate system is found at

t = 1/2

Substituting t = 1/2 into the equation y_r (t) means

or

we need rotate our coordinate P₁ to relative coordinate along side providing a translation of such coordinates as this provides a sense of scale in the rotated coordinate system on the relative coordinate system relative to original coordinate positions, or this is to say, expressing the local y in terms of control point P₁.

A translation can be defined with an affine transformation.

In order to express P₀ as a point of origin on our given transformed coordinate system.

or in the case of our coordinate P₀ this is written  

while a rotation in coordinates is defined by

  

Thus we can see that P₀ is zero under the series of transforms with

  

where this resultant vector (0,0,1) time the rotation matrix is also zero.

We can see that P₃ is zero on the transformed y relative under the series of transforms with the affine translation.

rt*tr*P₃  

for the affine transform

and for rotation

Here examining the y relative component of the transformed P₃ position we should notice that y relative coordinate component is

geometrically one can see this from the following graph showing the rotation axis and original axis

The graph will demonstrates points are already affine translated

Here where θ in the graph equals the given rotation angle defined above we can see the y' components cancel when subtracted which results from the rotation matrix transform and the point P₃ - P₀ lay on the zero axis of the new rotated coordinate system.

Similarly under transformation y_1r can be expressed in terms of P₁ as

Thus for any coordinate point we have a transform which is

  

  

substituting eq. (2) into (1) we have

  

We can check that the t = 1/2 doesn't simultaneously equal to zero here, for x_r'(t).

If we wanted to find the original positions x(t), y(t) for t = 1/2 we can either re transform coordinates

using the inverse of the transform matrix above, or we could compute the Bézier curve for t = 1/2

which should remain relatively invariant in so far as the arrangement of points overall for the given

curvature. In other words, rotation and translation only changes the position of all points equally not the relative arrangement of points for such curvature. Why is this?

The Bézier curve geometrically computes on the basis of the relative arrangement of points in determining the shape of the curve, the relative arrangement of control points control the shape of the curvature, but this remains invariant (unchanged) under rotation and translation.

In other words, if the relative angles remain the same between control points alongside preservation of magnitude, these will determine the points on the curve which should be relatively the same irrespective of choice in coordinate plane. Geometrically speaking the points of the polygon formed initially by the control points are the same relatively speaking in so far as distance magnitudes, and the t parameterization on the relative coordinate systems trace the same relative distance when constructing a solution on the basis of relative spatial subdivisions. In the case of the cubic, a triangle is formed irrespective axis at point t₀ on any axis consisting of the same side lengths, and this in turn leads to a line subdivision of the same magnitude line in length irrespective of axis at t₀ which determines the point on the Bézier curve. The points themselves are variant under rotation and translation but not the shape of the curve.

It would be worth noting that in the original coordinate system the relative maximum computed t = 1/2 may not the maximum of the curve, or a given slope = 0 at such point, but the relative coordinate