Learn Integral

[Narcisa Flierl]

ButI always feel a lack of thorough understanding of the mathematical logic behind these calculations. So when I do such calculations, it's more like follow my habit than logical reasoning.

I think this is because I don't have a complete picture of the background math knowledge. So I want to spend some time (about one year) to make up for it. Otherwise it'll be a pity for my life.

During my searching, I found some articles/books useful to me.I will keep adding links to them below. Maybe they are just remotely related to this question. But they do make me aware of something new.

Laplace transforms were derived in a very strange way by Oliver Heaviside, who is considered by many to be the Father of modern Electrical Engineering. He created 'operator' methods for solving ordinary differential equations. (The 'D' operator was Heaviside's notation, and the algebraic method was his, including using partial fractions and his 'cover up' method for decomposing into partial fractions.) Most of what he did was not very rigorous, but it was brilliant, it worked, and he always checked his answers. The reason you have trouble tracing back to the source is because Heaviside was so arrogant and nasty to people at the time, that they vindictively set out to keep his name out of everything. Honestly. He used to openly and viciously insult Lord Kelvin. Heaviside was banned from publishing several times throughout his life for his open attacks through Journal articles.

Heavside deliberately set out to turn differentiation into multiplication, and he came up with expressions that morphed into something similar to what is now called the Laplace transform. But it didn't start off as something called the Laplace transform; when people found integral expressions similar to what Heaviside was using that could be named after someone else, they jumped at the chance to write Heaviside's name out of it. Heaviside noticed that time evolution operators for time-invariant systems (such as circuits) would have an exponential property. That is, if the solution operator acted on a state $x$ at time $0$, then the state $S(t)x$ at a time t seconds later when evolved again by $t'$ seconds should be the same as the state obtained by evolving the original state by $t+t'$ seconds. In other words, the solution operator would satisfy $S(t')S(t)x=S(t'+t)x$. Very abstract, very general for such systems, and obviously leading to something exponential. That's where the exponential in the Laplace transform comes from, and that's the level Heaviside worked at during the late 1800's! His operator methods allowed him to solve problems nobody else at the time could; otherwise people at the time would have gladly ignored Heaviside.

Overview of Heaviside's work, along with links to his publications: Heaviside Operator Calculus.

I highly recommend this person's web page; it's entertaining, informative, and has excellent references.

Integral Transform is a huge subject. In my opinion, you should also have a strong background in Ordinary Differential Equation, Partial Differential Equation, and Real/Complex Analysis. Linear Algebra and Calculus is a "must" known subjects if you want to know where does Integral Transform come from. On the other hand, I found that Google and even Wikipedia don't have much information about integral transform in general, they just talk about some specifics topic such as Laplace, Fourier Transform, etc.There is a new book by K. Wolf from Springer and you should check it out, it's basic but deep enough in theory though. There is another book about ODE, but it has a section about Laplace Transform, and it's really details though not just a Transform table and showing you how to do it. I think it's a book by William A. Adskin.

As far as I know, there is no unified subject that specifically deals with integral transforms in general. Different integral transforms come up in different contexts. You are probably better off asking yourself, "What sort of engineering/mathematics do I want to study?" Depending on your answer, this may lead you to the study of some particular integral transforms.

Similarly, there are probably no books that deals with integral transforms generally - there are just too many to discuss. But you will find plenty of excellent books that discuss just a few transforms at a time, possibly, even focusing on just one. Some (maybe all) of the transforms you have listed have books entirely devoted to their study, and the Fourier transform essentially has an entire subfield of mathematics devoted to it. These should just be a Google search away.

As for the requisite knowledge, it depends on the particular transform you study, and in what depth. A solid background in calculus and linear algebra is definitely a must. If you go deep into the theory of such transforms, you will probably begin to encounter more sophisticated tools from real, complex, and functional analysis.

Take a cinematic journey through the major stages of human development, using a series of 22 carefully-curated film clips (and more than 30 video games) to illustrate some of the most important qualities of each stage.

Note: The films and characters below are typically far too complex to be described by a single stage, which is why the qualities below are only being applied to the specific film clips, and not to the rest of the film, character arc, or filmmaker's perspective. We are not saying Lord of the Rings is a "magenta movie", for example, or that Gandalf a "magenta character", because both would include multiple combinations of views, values, and ideas from multiple different stages. We are focusing on specific moments here that distill the essence of each of these stages, and not making judgments on the rest of the films they are excerpted from.

The Crimson altitude (alternatively known as "infrared", or "beige" in Spiral Dynamics) signifies a degree of development that is in many ways imbedded in nature, body, and the gross realm in general. Crimson altitude exhibits an archaic worldview, basic physiological needs (food, water, shelter, etc.), a self-sense that is minimally differentiated from its environment, and is in nearly all ways oriented towards physical survival. Although present in infants, Crimson is rarely seen in adults except in cases of famine, natural disasters, or other catastrophic events. Crimson is also used as a kind of catch-all term for many earlier evolutionary stages and drives.

The human story begins with the Crimson stage, a period when we begin to use our new, comically oversized brains in order to find new ways to live, new ways to think, new ways to communicate, and new ways to manipulate our environment. The dawn of the individuated self.

First-person shooters are often associated with the Red stage, where the primary goal is to dominate anything and everything on the screen. Watch as Ryan and Corey look at several examples of Red video games, including Fortnite, Wolfenstein, the Grand Theft Auto series, Assassin's Creed, and the game that kicked off the genre, Doom.

This famous "dueling of the anthems" scene demonstrates healthy Amber nationalism as a Nazi song is drowned out by a tearful cast (which included real-life refugees from recent Nazi invasions) singing "La Marseillaise".

Denzel Washington shows what Amber leadership can look like within a family environment. Amber hierarchies are typically known as "dominator hierarchies" with a very rigid chain of command, where respect and discipline is expected, and where power is only exercised from the top down.

The Amber stage is often the home of team-based multiplayer games such as Battlefield 5 and Destiny 2, as well as some historic-based games such as the "feudalism simulator" Crusader Kings 3. The Amber stage is also associated with concrete-operational thinking, embodied by the classic game Tetris, as Ryan and Corey discuss here.

Captain Picard challenges the Amber chain of command to make an impassioned defense for universal humanitarian rights, and explains why men, women, and androids of Orange conscience cannot blindly follow orders.

Interestingly, unlike most of the earlier stages, there do not seem to be any specific game genres that are strongly associated with the Green altitude. However, Green content and themes are very common in many of today's games. Watch as Corey and Ryan discuss some of these games: Animal Crossing, BioShock, Disco Elysium, and what was perhaps the prototypical postmodern game, Metal Gear Solid 2.

Integral films often excel at weaving transpersonal themes, multiple layers of symbolism, and spiritual states of consciousness together into something the public actually wants to watch. In this keystone scene from the third Matrix movie, Keanu Reeves discovers the transcendent golden "whoa" behind all things.

Hugh Jackman gets enlightened. Again we find the familiar idea of a love that goes beyond death, reinforced by a cinematic integration of microcosm and macrocosm (another important polarity at this stage) as close-up shots of fluid dynamics and chemical reactions are used to represent the transpersonal clouds and currents of the subtle realm.

Corey W. deVos is editor and producer of Integral Life. He has worked for Integral Institute/Integal Life since Spring of 2003, and has been a student of integral theory and practice since 1996. Corey is also a professional woodworker, and many of his artworks can be found in his VisionLogix art gallery.

Integral Life is a member-driven digital media community that supports the growth, education and application of Integral Philosophy and integrative metatheory to complex issues in the 21st century. Integral Life offers perspectives, practices, analysis and community to help people grow into the full capacities of integral consciousness in order to thrive in a rapidly-evolving world.

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