MonDie
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The calculus may be overkill, but I still want to get a grasp of this math since I already started and since it will help me comprehend the book.
Anyway, the odd thing about the brackets is how they have an interval like the integral has an interval. I have now seen two examples that are clearer than the example in Schaum's. These examples clearly show that the content in the brackets is the derivative of the function in the integral across the equal sign. It is puzzling why the brackets also have an interval after them. For the integral, the interval is static but the variable in the function is dynamic. The integral essential says to take the average output of that function for the given interval—once x is given a value—and then multiply that by the width of the interval. In contrast, the derivative says to determine the slope of the line at a particular value of x—once x is given a value. This bracketed content appears to be some sort of inbetween, between the integral and the derivative, even though I do not know what it would want me to do other than solve for y at x as if it were an ordinary function f(x).0 -
If they are designed for exam cramming, then that might be the problem. Anyway, I posted the squared brackets in the opening post.
I get confused at the inductor's equations for power and for the amount of energy stored in its magnetic field. The energy in its magnetic field is confusingly expressed in watts (w subscript L) rather than joules. Furthermore, I am confused by the change of interval from t1-t2 to i1-i2 followed by the change to a bracketed [i-squared-1 minus i-squared-2].
[math]p = vi = L \frac{di}{dt} i = \frac{d}{dt}[ \frac{1}{2} Li^2][/math]
[math]w_L = \int_t^t p dt = \int_i^i Li di = \frac{1}{2} L [i^2_2 - i^2_1][/math]
The book then concludes that "the energy stored in the magnetic field of an inductance is WL = (1/2)Li^2."
Obviously v is converted into the form of inductance times the time derivative of current. Then di/dt is multiplied by dt to get di, and this somehow changes the integral's interval (or the x-axis of the integral's function) from units of time to units of current. I am not sure exactly why this happens even though I see it happening a lot in the chapter. I presume that the function is now being plotted against the backdrop of changing current rather than changing time, but that is all I can say. Lastly, I have no clue what to make of the empty time derivative d/dt, nor the bracketed squares of i.
I do not think it matters that the original content has brackets that are taller than the fraction inside the brackets.
Anyway, it looks as though you found the derivative of x^2/2, which is x, and then set it equal to the integral of its derivative, as if the antiderivative of a derivative is equal to the original, undifferentiated equation. Is that right?
Regardless that is only the first step, whereas I am still baffled at how the second step mysteriously removes the variable from the equation.
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I did not use a particular source to understand how the integral is the antiderivative. I guess I just had to get my mind on math once again. Distance traveled versus speed is more intuitive than work versus power. For motion along a straight path, speed (s) is the time derivative of distance traveled (d), which we might express as s = dd/dt, because speed is the rate at which position is changing. Why should this imply that you can calculate distance traveled from the integral of speed? Assuming that the integral of speed has time spent traveling as its x-axis, then the integral of speed for an interval of time is the average speed for that interval multiplied by the length of that interval, or the average speed multiplied by the time spent traveling, which will give you distance traveled. Thus s = dd/dt and d = integral(s dt) as long as our x-axis remains as time, thus the dt. Note that you can put an undefined k into the integral since the starting position does not matter if we are trying to calculate total distance traveled rather than position traveled to.
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It would seem that part of what I want is a good explanation of the fundamental theorem(s) of calculus. I found something on berkeley dot edu, but it is actually a proof and not intuitive at all. https://math.berkeley.edu/~ogus/Math_1A/lectures/fundamental.pdf
I am confused by page two by the primes inside the functions: f(x') instead of f'(x).
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I decided to actually do a search for NIH articles via the Google search engine. Looking at research from 2000 onward, I found 3 pro and 4 con. 2 of the 3 pro-findings are free-to-read, but unfortunately we cannot read about the null findings unless we pay up. The second one appears to come from a Polish journal. That is Polish.
PRO
Evidence that the lunar cycle influences human sleep. (Current Biology, 2013)
https://www.ncbi.nlm.nih.gov/pubmed/23891110
The lunar cycle: effects on human and animal behavior and physiology. (Postępy Higieny i Medycyny Doświadczalnej, 2006)
https://www.ncbi.nlm.nih.gov/pubmed/16407788
Gout attacks and lunar cycle. (Medical Hypotheses, 2000)
https://www.ncbi.nlm.nih.gov/pubmed/11021320
CON
Do lunar phases influence menstruation? A year-long retrospective study. (Endocrine Regulations, 2013)
https://www.ncbi.nlm.nih.gov/pubmed/23889481
The influence of lunar cycle on frequency of birth, birth complications, neonatal outcome and the gender: a retrospective analysis. (Acta Obstetricia et Gynecolegica Scandanavica, 2008)
https://www.ncbi.nlm.nih.gov/pubmed/18607814
Birth rate and its correlation with the lunar cycle and specific atmospheric conditions. (American Journal of Obstetrics and Gynecology, 2005)
https://www.ncbi.nlm.nih.gov/pubmed/15970864
The effect of the lunar cycle on frequency of births and birth complications. (American Journal of Obstetrics and Gynecology, 2005)
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I am not sure why Phi posted, but Phi spurred me to get myself caught up again. On March 28th, Trump signed an executive order to roll back the Clean Power Plan. The EPA is holding hearings with the supposed intent of hearing out the public on what they want rolled back, but apparently the hearings are being announced with short notice, held at awkward times, and having more stakeholders and lobbyists than ordinary citizens.
According to Ars Technica's John Timmer, it will be quite the lengthy task to reverse the "greenhouse gas Endangerment Finding". Unless they can reverse it, they cannot repeal a plan unless they replace it with something else, and the replacement plans will probably involve reduced regulations on the grounds that the regulations are economically costly.
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That is very, very helpful! However there is still a lot unanswered with regards to the time derivative with the empty numerator, the changing integral interval units, and the mysterious use of square brackets.
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I am reading Schaum's Outlines, Electric Circuits Fifth Edition having taken both precalculus courses but not calculus.
Firstly, I am confused by the definition of power as the time derivative of watts (p = dw/dt) even though watts is a unit of power. More importantly, I am confused about how this derivative is reversed into an integral of power as below:
[math]w = \int_0^t p dt[/math]
I recognize that both have an undefined variable t, but I am baffled at why the below should be true regardless of the value of t:
[math](\int_0^t p dt) \div w = \frac{dw}{dt} \div p = 1[/math]
I get even more confused by some equations given to describe the behavior of an inductor, which stores energy in its magnetic field, charging when current (i) rises and discharging when current drops. I think I understand Ohm's law for an inductor, stated below with the constant of inductance L.
[math]v = L \frac{di}{dt}[/math]
[math]i = \frac{1}{L} \int_?^? v dt + k_1[/math]
I get confused at the inductor's equations for power and for the amount of energy stored in its magnetic field. The energy in its magnetic field is confusingly expressed in watts (w subscript L) rather than joules. Furthermore, I am confused by the change of interval from t1-t2 to i1-i2 followed by the change to a bracketed [i-squared-1 minus i-squared-2].
[math]p = vi = L \frac{di}{dt} i = \frac{d}{dt}[ \frac{1}{2} Li^2][/math]
[math]w_L = \int_t^t p dt = \int_i^i Li di = \frac{1}{2} L [i^2_2 - i^2_1][/math]
The book then concludes that "the energy stored in the magnetic field of an inductance is WL = (1/2)Li^2."
Obviously v is converted into the form of inductance times the time derivative of current. Then di/dt is multiplied by dt to get di, and this somehow changes the integral's interval (or the x-axis of the integral's function) from units of time to units of current. I am not sure exactly why this happens even though I see it happening a lot in the chapter. I presume that the function is now being plotted against the backdrop of changing current rather than changing time, but that is all I can say. Lastly, I have no clue what to make of the empty time derivative d/dt, nor the bracketed squares of i.
Thank you for any help you can give!
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As I ponder this, it seems as though narcissists lack integrity but still consider the consequences whereas the "secondary psychopath" acts without regard for the consequences. This could be why the secondary psychopath exhibits more deviant ("Social Deviance" subscale), or even criminal, behavior. This could be contrasted with impaired honesty-humility.
Grit or Honesty-Humility? New Insights into the Moderating Role of Personality between the Health Impairment Process and Counterproductive Work Behavior (Ceschi, startori, Dickert, Costantini, 2016)
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5147463/
Honesty-Humility measures the disposition to not take advantage of other individuals, even when there is no risk of unfavorable repercussion for such exploitation (Ashton and Lee, 2008). People low in Honesty-Humility are portrayed as egoistic, lying, haughty, fraudulent, unethical, hypocritical and cunning (Lee and Ashton, 2006).
Considering this description, it is not unexpected that low Honesty-Humility is linked with a certain range of questionable behaviors. Low Honesty-Humility individuals have an inclination to deceive, craft, and break rules, searching for the chance to take part in self-interested behaviors (Lee et al., 2005a). The HEXACO Honesty-Humility dimension has consistently predicted workplace deviance and CWBs [Counterproductive Work Behaviors].
This could also be framed in terms of group norms. We conform with the group to avoid being punished by the other group members, i.e. to avoid the consequences. Thus secondary psychopaths might be seen as violating group norms without regard for how the group will view their behavior. In some contexts, failure to conform with the group despite the consequences can actually indicate a high level of integrity, something narcissists appear to lack.
The prevalence and structure of obsessive-compulsive personality disorder in Hispanic psychiatric outpatients (Ansell et al, 2010)
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2862854/
Multi-factor models of OCPD were tested and the two factors - perfectionism and interpersonal rigidity - provided the best model fit. Interpersonal rigidity was associated with aggression and anger while perfectionism was associated with depression and suicidal thoughts.[...]
Perfectionism and detail-oriented criteria loaded strongly on the perfectionism factor, the workaholic criterion loaded moderately and packrat loaded weakly. Morality and reluctance to delegate loaded strongly on the interpersonal rigidity factor, stubbornness loaded moderately, and miserly loaded weakly.
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On the other hand, I'm not just jumping for joy at the idea of invoking the unknown. If I could encounter a presentation of consciousness emergence that actually seemed to have any rigorous traction whatsoever (even a hint), I'd reconsider - I agree with the general notion of not introducing extraneous unproven things into the model when the model already provides a plausible explanation. I like understanding things - I get a bigger kick out of that than I can put into words. But I've never seen anyone explain the how of emergence. Every presentation of it I've ever seen pretty much just says "It just does emerge." No explanation - just the claim.
To reiterate my original point, it is okay to say that mental phenomena arise from physical phenomena "just because" in the same way that it is okay to say that anything exists at all "just because". If a second moon suddenly materialized in orbit around Earth, the "why not" would be obvious: we have never seen moons materialize out of nothing; it is a violation of the law of conservation of mass. Each of us sees mental activity emerge from brain activity day after day, so it is not clear "why not."
No, that doesn't help me much. My core issue here is that all of the transistors in the computer are still just transistors. Each one still just has a charge distribution in it, and still just has two potential differences (Vgs and Vds) as inputs. They just have no "global" interconnectivity to fuel some mysterious emergence. If you can't look at one transistor and recognize a vehicle for awareness, having 100, or 1000, or 10^10^10 doesn't help.
On the other hand, we can see how algorithmic behavior emerges - we have excellent theoretical and engineering understanding of how to get arrays of transistors to do fancy, organized things. But in none of those cases does the computer "know" what it's doing - each transistor is still just doing its transistor thing.
You might have a point if all of the transistors had the same open/closed state and the same spatial orientation, but it is their configuration that causes something to emerge.
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My personal opinion is that there is no "hard problem" of consciousness, the problem which asks how mental states can arise from physical states. The problem of mental-from-physical is similar to the problem of something-from-nothing—people are asking "why?" rather than "why not?". Science looks at how things are composed, at emergence. Science breaks things down, or else it adds things up. It reduces causes into smaller causes, or else it adds causes together. What causes there to be causes may be an invalid question altogether, so why do we see problems with causes that connect physical phenomena to mental phenomena?
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These celestial cycles probably do affect our circadian rhythms, which would result in biochemical changes. Cortisol peaks in the morning during the "cortisol awakening response" and then cortisol gradually drops over the course of the day. Whereas cortisol has a diurnal cycle, testosterone has a semidiurnal cycle. The cortisol cycle takes a few days to adjust to a disrupted sleep schedule, but renin-aldosterone activity is intimately tied to sleep phases with aldosterone dropping during REM sleep and rising during non-REM sleep. The hypothalamic-pituitary-adrenal axis and renin-angiotensin-aldosterone axis are both related to mental disorders and perhaps bipolar disorder particularly. Don't forget that we probably aren't the only organisms with these internal cycles. We are immersed in and dependent upon the entire ecosystem.
Of course our modern world provides more ways to block out these natural cycles, but not everybody takes advantage of this. Those noisy rascles nextdoor might start waking you up sooner when the sun starts rising sooner. The seasonal cycles have more obvious effects that could explain the popularity of "sun-sign" astrology, which uses the month of your birth, but I wouldn't entirely discount the moon phases. People might prefer to stay out late at the beach when there is a giant night-light in the sky creating giant waves. On the other hand, they might prefer the new moon so that they will not be seen. It was thought that womens' menstrual cycles were tied to the moon phases, but this effect, if it were ever real at all, might be weakening as women utilize the power of birth control.
But I repeat. Astrology relies on much, much more than the seasons and moon phases, most of which could not be accounted for by modern astrophysics.
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The key here is science is involved in discovering mysteries and adventures. Social life is boring, predictable, and rarely an adventure.
I would think maniacs would be more interested in adventure than the depressed, but the two mood states may not be opposites but in fact weakly correlated. The mixed state may be the most dangerous mood state. I imagine some of the overlap may be due to overactivity of the stress response system, the hypothalamic-pituitary-adrenal (HPA) axis (so named because left-to-right each region or gland releases a hormone that stimulates the next region/gland).
Depressive and manic symptoms are not opposite poles in bipolar disorder. (Johnson, Morriss, Scott, Paykel, Kinderman, Kolamunnage-Dona, Bental, 2010)
https://www.ncbi.nlm.nih.gov/pubmed/20825373
I would think an overactive DMN would lead to a suppressed EAN and suppressed interest in math and science, but maybe focusing on math and science could be a healthy strategy for making the DMN "shut the @#$% up".
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Astoundingly, I forgot to include this.
Social and Mechanical Reasoning Inhibit Eachother (Scott Barry Kaufman of PsychologyToday)
In recent years, neuroscientists have identified two opposed brain networks: the default mode network (DMN) and the executive attention network (EAN). The standard view is that the DMN network is involved when we are engaged in internally focused tasks such as recalling deeply personal memories, daydreaming, sleeping, imagining the future, and trying to take the perspective of others. In contrast, the EAN network is thought to be involved when our attention is directed outward, and we must exert control over our limited attentional resources. For instance, the EAN is highly active when people are taking IQ tests (see here). When the DMN network is active the EAN network is quiet, and when the EAN network is active, the DMN network is quiet.For most people, our evolved brain networks automatically know when it's appropriate to come online and when it's best to lay dormant. For some people though, there isn't this automatic switching. For instance, those with autism have been shown to have an underactive DMN and an overactive EAN, which can lead to overthinking, a fascination with how objects work, and a lack of awareness of social subtly. In contrast, those with schizophrenia have been shown to have an overactive DMN and an underactive EAN, which can lead to overinferring people's intentions, disordered thought, and a breakdown of the boundary between self and other (see here for a review).I should add. Schizophrenics also have a lack of social interest, although this is a "negative symptom" and inhaled oxytocin is especially useful in treating their negative symptoms. However, autism is the condition associated with interest in math and science. Thus activation of the EAN in autism could be the key.
DMN activity is important to depression research. I don't know what the research says, but I do know that depression is often described as a "turning inward" that implies greater DMN activation. Here are a few links in case you want to know more.
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The only condition I know to be related to interest in science is the autism spectrum, which is still not described all that well and seems to have broader and narrower conceptualizations. Some argue that, in addition to Asperger's syndrome and autistic disorder, the spectrum should also include Pervasive Development Disorder (PDD) or even Childhood Disintegrative Disorder. It has been conceptualized as three-dimensional: social deficits, verbal ability deficits, and repetitive behaviors & restricted interests. I am not sure exactly which explains the interest in science. Being less interested in social behavior and less affected by peer pressure, perhaps they prefer concrete subject matters to the socially derived trivialities of small talk: fashion trends, fictional media, gossip, etc. They are more prone to depression, but I am not sure how this would be related, but both depression and autism appear to be connected with oxytocinergic abnormalities that impair the capacity to enjoy social interaction.
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This idea is essentially astrology. Astrology was paired with astronomy since the beginning of civilization, but modern astrology is hardly like the style of astrology that was predominant at least as far back as Ptolemy (100-170 AD). These astrologers thought they could predict events by how the planets were be aligned. They used ecliptic longitude because most of the planets in our solar system move along this plane, and they looked at the angle that was formed by two planets with Earth as the vertex. In this system, the full moon is an "opposition" (180 degrees) of the sun and the moon. At some point astrologers switched to personality readings to avoid the persecution associated with being a fortune teller. These full moon ideas are probably a remanent of this obselete science. Some celestial bodies do clearly influence our affairs. The sun determines our seasons as it traverses the plane of the ecliptic each year. The seasons influence sun exposure, food availability, the tides, etc. This doesn't mean the other 99% of astrology is accurate.
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Remembering the song "Paint It Black" and thinking the title is ironic.
Hair is cut now. Maybe some seasonal flowers will restore the neutrality. They're never quite the same after you borrow them and put them back.
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It isn't a superficial covering up.
The content is not
downloadedexecuted, which is why adblockers are considered a security feature.https://wiki.ubuntu.com/BasicSecurity#NoScript
NoScriptIf you unknowingly visit a compromised website, this will prevent scripts (i.e. programs) from running on your system.
Furthermore, the ad servers know when the ads have been blocked, which prevents the domain hosting those ads from collecting money.
Why Ad Blocking is devastating the to the sites you love (Ken FIsher, Arstechnica)
https://arstechnica.com/business/2010/03/why-ad-blocking-is-devastating-to-the-sites-you-love/
Finally, on a slow connection, you will notice that an ad blocker such as AdBlock Plus makes the pages load more quickly.
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Have you heard of volumetric alarms?
http://www.foxspyoutlet.com/security-equipment/smart-sensor-volumetric-alarm/
The knowledge might come in handy, but it is supposedly easy to wire an alarm switch into the door so that it will activate when the door opens. In contrast, a volumetric alarm probably has to be sitting in the open to work.
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I think you need a book on electronic engineering.
For some reason, electrical and electronic engineering are often combined at (pre-university) college level (at least in the UK). The electrical side skips over theoretical details beyond ohms law (other than those associated with complex impedance and Z-transforms) but the electronic side does explain how devices like diodes work.
Thank you. I figured any such book would include electronics (solid-state devices and circuit boards) in addition to electrical (wiring, power sources, digital multimeters).
Hello Mondie, it is a long time since we shared a thread.
Anyway I am somewhat confused as to what you are seeking.
Are you saying you understand the electrical/electronic engineering aspect from a point of view of connecting appropriate devices together, but not how the devices work and would like to understand how they work.
Or do you need help with auto electrics as well?
The main difference between auto electricity and domestic wiring is that domestic wiring is AC or alternating whilst autos run on DC or Direct current.
It is true that modern autos use an alternating generator (alternator) rather than a direct generator (dynamo) but this is converted in a module mounted on or within the alternator.
But yes the voltage is lower and the current higher for the same power requirement, but not otherwise.
Modern vehicles also have a plethora of sensors (transducers) which measure/monitor all sorts of activity in the vehicle its mechanics and surroundings.
Do you wish to know how these work, for example the lambda sensor in the cat converter?
This is the province of control engineering.
So tell us a bit more about your intended project so we can offer focused help and references.
Immediately, I want to start wiring devices into my car for security. They don't have to run off of the car battery, but I would prefer that they aren't easily removed. It's a long story that others find hard to believe. Someone is entering my car repeatedly, leaving behind mostly innocuous items, the worst being a baggy of piss snow under the seat that I was lucky to find. (You would think this wouldn't happen living across the street from the sheriff.) I want a basic alarm system, a light to keep my car lit at night, and an encrypted transponder for keyless entry if that wouldn't be too complicated.
I will have a more long-term interest as well, but I am extremely low on money. I seem to understand better or at least remember better when the theory is also explained. Furthermore, I find the theory interesting in its own right. I am becoming increasingly interested in Linux, networking, and all forms of security: device encryption, network security, home security, car security. Lastly, I have always enjoyed the natural sciences, but I know a lot more biology than chemistry.
Depends on how long you have. It will takes ages to learn I would recommend you follow this tutorial first https://www.allaboutcircuits.com/textbook/
Hmm!...
I think I will pass. I have to use the library Internet ever since my dad retired our home connection for some reason. I do have a laser printer, and I could print it at home. I would download the AAC PDFs to a flashdrive along with the printer driver. However, I will still be spending about $8.00 on printer paper for each volume—volume one is 560 pages as a PDF—, and a new cartridge will cost more than $50.00 if I run out of ink. In addition, the publication is probably more in-depth than I require at the moment.
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What books on electrical engineering go into depth on technique as well as theory? I was reading the electrical chapters of a highschool-level automechanics textbook only to find the descriptions shallow and confusing. The book's explanation of semiconductors was appallingly incomplete, and then my chemistry textbook filled the gap on band theory and doping. Unfortunately the chemistry textbook barely mentions diodes. I need an electrician's guide that assumes that I want to know the theory too and already do to an extent. Along those same lines, the book should be broadly applicable. Many of these books focus on household projects, but I will be doing automotive electricity. Compared to household wiring, car wiring has about ten-fold lower voltage and about ten-fold higher amperage, and it has to withstand constant vibration. Thus I probably want a bottom-up guide that goes from theory (bottom) to technique (top) and has wide-range applicability. At the same time, I don't want to spend too much time reading to prepare for my project.
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I had an interesting thought regarding dissociate fugue and hypnotic susceptibility. "Absorption" is related to hypnotic susceptibility and to "Openness to Experience". Perhaps this helps to explain why a person who has undergone dissociative fugue finds themself in an unfamiliar place after the alter wanders away from home and starts a new life. It's almost like extreme openness—the preference for novelty becomes so strong that the person just leaves and goes somewhere else.
Anyway, although there are probably many more, here is just one research paper that discusses empirically observable effects of hypnosis.
Oxytocin impedes the effect of word blindness post-hypnotic suggestion on Stroop task performance (Parris, Dienes, Bate, Gothard, 2014)
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4090953/
Raz et al. (2002) showed that the Stroop effect (Stroop, 1935), one of the most robust effects in cognitive psychological research, can be virtually eliminated following a post-hypnotic suggestion. The Stroop task requires participants to identify the colour of the font in which a word is presented, while ignoring the meaning of the word itself. When the written word is incongruent with the ink colour (red written in blue), the time it takes to identify the colour is increased relative to neutral stimuli (i.e. when the word is not colour-related, e.g. top written in red). When the word is congruent with the colour (red in red), colour identification time is reduced compared with neutral stimuli. The effect of word congruency on colour classification times is known as the Stroop effect. The Stroop effect has been referred to as the ‘gold standard’ of attentional processes and has long been held as strong evidence for the obligatory nature of word processing (MacLeod, 1992). However, a post-hypnotic suggestion describing the word dimension of the Stroop stimulus as being made up of ‘meaningless symbols’ and ‘characters of a foreign language’ (to be referred to as the word blindness suggestion) resulted in the virtual elimination of not only Stroop interference (Incongruent-Neutral Stroop trials) but also Stroop facilitation (Neutral-Congruent trials) in the reaction time (RT) data, and Stroop effects typically observed in error data. The authors argued that their results were inconsistent with the notion that the processes of visual word recognition are obligatory and that the post-hypnotic suggestion works via a top-down mechanism that modifies the processing of input words through a means not voluntarily available.1 -
You can make the argument that moral knowledge is actually knowledge of better versus worse. If the moral option is the better option, then any all-knowing being would recognize it as the better option and therefore choose it.
Does anybody's happiness matter more than anybody else's?
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watts and power, converting dw/dt to an integral, and changes in integral interval units
in Analysis and Calculus
Posted
I am interested by both, but my primary concern is electrical engineering. Regarding post #10 above, I do not know the Maxwell-Faraday equation. This equation is referred to by allaboutcircuits as "Ohm's Law for an inductor."
https://www.allaboutcircuits.com/textbook/direct-current/chpt-15/inductors-and-calculus/
The equation is also used in Schaum's to describe the behavior of an inductor. Perhaps it is over-simplified and I will see a more complex version down the road.
As I think about it more, I realize that you can graph a function or graph its integral. The interval after the brackets is useful when you are graphing the function—or its integral since they are equal and have the same graph—rather than generating an output from an input. That is to say:
[math]\int\limits_0^1 {xdx} = \left[ {\frac{{{x^2}}}{2}} \right][/math]
That would be incorrect since the integral is bound but the function is not. If I had to calculate the reverse, to calculate the integral from the function, I could not determine what interval should come after the integral except some undefined variables like t1 and t2. Thus the interval after the brackets:
[math]\int\limits_0^1 {xdx} = \left[ {\frac{{{x^2}}}{2}} \right]_0^1[/math]
Regardless, I still need to understand the final conversion in equations 5 and 6. In the case of 6, I still do not even know how to interpret the brackets in the last piece.