New way of Fourier series using the 4D complex numbers.

Warning: This post contains stuff that is not correct! Yet I decided to post it anyway so you can see that crafting math is also just keep on working until you have it right. The reason it does not work in the post below is that the basis functions I use are not all perpendicular to each other. And if you want to reconstruct a function or a signal s(t) with the basis functions as mentioned below, you will get weird overlap and the end result will not be correct.

Ok, that was a bit frustrating. But all in all I don’t have much reputation damage because more or less instantly I found another way of crafting the 4D Fourier series and that seems to work perfectly. So compared to the professional math professors who at one point in time accepted the quaternions and together with that stupid theorem of Frobenius concluded that 3D or 4D complex numbers are not possible, at least I don’t look that retarded. Sometimes I can be stupid too but at least it does not last for over one century. And may be that is also the reason that professional math professors absolutely do not want to talk about my work on 3D & higher dimensional number systems; admitting that you have been stupid for over one century is of course not an easy thing to do. And given the fact I am now unemployed for 17 or 18 years, rather likely the professional professors would rather be eating dog shit compared to speaking out my name… Once more we observe that in this world there is never a shortage of idiots.

In this post I use the coordinate functions of the exponential curve in the 4D complex numbers but I changed the period to 2 pi instead of a period of 8 that comes along with taking the log of the first imaginary unit. I also would like to mention that I use the so called modified Dirichlet kernel and because that kernel originates from Fourier analysis you must not get confused by the name ‘Dirichlet kernel’. The modified kernel is important (anyway for me) because it spits out all those coordinate functions for making exponential circles and curves in all dimensions possible. While if memory serves, the use of the Dirichlet kernel inside Fourier analysis is for using it in proofs of convergence. But may be I remember that completely wrong, after all it was about 30 years ago that I studied the Fourier stuff for the last time. The last two weeks were pleasant from the mathematical point of view, all that old Fourier stuff that somewhere still lingers around in my brain. But so much is gone, what is that Gibbs overshoot? Is that when a male math professor has his yearly orgasm? And what was the Parcival identiy? I don’t have a clue whatsoever.
This post is 7 pictures long, four are of 550×775 pixels and I had to enlarge the other three to a size of 550×850 pixels. So it is not a mess like the previous post where I just enlarged the pictures on the fly until all that text was there. Here it is:

Again, this way of recontruction does not work!

Likely all those basis functions have this problem, if you take the inner product of an arbitrary basis function against the same basis function with three times the speed, it is not zero. And as such it is not perpendicular…

For people who have never heard of inner product spaces done with functions I found a cute pdf where a lot of the basics are explained.

Inner product spaces.

I would like to be the 4D Fourier stuff done in a correct manner in the next post but sifting through what I wrote on 4D complex numbers I realized I never wrote about a de Moivre identiy for the 4D exponential curve. May be I will publish that in a separate post, may be not.
Anyway, have a good time and see you in the next update.

Cute search engine results found & intro 4D Fourier series.

About a week ago I started investigating how you could craft a Fourier series using the coordinate functions of the 4D exponential curve. The usual way the series of Joseph Fournier are done is with the sine and cosine that are also the building blocks of the exponential circle on the complex plane. So I needed to look up my own work on the 4D complex numbers because in the beginning of 2019 I stopped writing posts about them and after such a long time not every detail is fresh in your brain of course.
Anyway I did a Google search on 4D complex numbers and to my surprise this website popped up above where the quaternions were ranked. I was ranked number one. That was a great victory of course, it means that people are actually reading this stuff… In one of the screen shots below you see the quaternions once more topping my 4D complex numbers but from day to day it seems that Google is shuffeling the top results a bit so the search results look a bit more dynamic on a day to day basis.

This year I didn’t look at the search engine stuff at all, we still have that corona stuff going on and beside that why look at such boring stuff if I can do math instead? But I could not resist and went to the Microsoft Bing search engine. For years they never ranked this website on page 1 if you searched for ‘3D complex numbers’. But all those years if you looked into the picture search of the Bing search engine a giant fraction of the pictures was from this website. That was very strange, how can you return so much pictures from my website while never mentioning me at any significant position in the rankings of the html files? Ok ok most people say Bing is an inferior search engine compared to the Google search engine and as such not many people use the bing thing.

So once more and for the first time in this year 2020 I searched for ‘3D complex numbers’ on the Microsoft search engine. To my surprise instead of being burried down deep on say page 10, at Microsoft they had seen the light. Here is a screen shot:

Not bad, Reinko is in pole position & where is the competition?

In the next screen shot you see the html listing of Google when you search for 4D complex numbers. Today when I made the screen shot I was not ranked at no 1 but for some strange reason that did not make me cry like a baby in distress.

All in all there are 20 posts for 4D complex numbers.

And the last screen shot is about the Google thing for pictures when you search for four dimensional complex numbers. Luckily there is no competition but does that mean the rest of humanity is stupid as hell?

Of course not, it only means no one is interested in crafting 4D complex numbers for themselves. Professional math professors don’t want to talk about 4D complex numbers in public, so why are my internet search engine rankings that high? It might be that it is read by non math professors and that more or less explains the high rankings…

Google picture search on 4D complex numbers.

Ok, it is now 22 June and I finally wrote down what the new way of Fourier series is using the 4D exponential curve. Writing of the next post is almost finsihed and I think I am going to do it just like Joseph Fourier did. That is without any proof at al for the most important things…

Anyway, in the nexgt picture you see the Fourier series in a 4D style:

It could ber handy to look at the end of an old post from 01 Nov 2019, there I show you how you can use the modified Dirichlet kernels for finding parametrizations of the exponential circles & curves in 2, 3 and 4 dimensions. If it is possible to craft a 4D Fourier series (again this is only postulated so there is no proof at this date) you surely must try to understand the 4D modified Diriclet kernel…
Here is the link:

End of this post, till updates my dear reader.

A norm based on the eigenvalues of 3D complex and circular numbers.

Ah, finally it is finished. This work grew longer than expected but with a bit of hindsight that is also logical: for example I spell out in detail once more how to find the eigenvalue functions for a arbitrary number X. After all that is an important detail so it is worth repeating. But I skipped the proces of diagonalization because we do not need it in this post.
Yet if you teach math and the time has come to do the complex number stuff, you could show the students how to diagonalize the complex multiplication for numbers from the complex plane. Most of the time students only diagonalize just one matrix with some numbers in it and that’s all, they never diagonalize an entire family of matrices. So that is why that would be useful, on the other hand the eigenvalues for a number z from the complex plane is z itself and it’s conjugate… And say for yourself: diagonalizing a number z so that later you must multiply the eigenvalues (also z) is very useless, as a matter of fact it is hard to find anything that is more useless… And once you have explained that diag stuff is usefull and utterly un-usefull at the same time, you can point to the live of the average math professor: also utterly useless…

Of course in higher dimensions the proces of diagonalization is very handy because it gives you for example a way of calculating the logarithm of higher dimensional numbers. And that way can be used in any dimension while all other methods for finding a logarithm get more and more difficult (as far as I know).

In this post I also worked out in detail what the eigenvalues of non-invertible numbers are; the non zero numbers with a determinant of zero have at least one eigenvalue being zero. I calcualted the eigenvalues for the numbers tau and alpha for both the complex and circular 3D multiplication.

This is post number 150 so all in all on average I write just about 30 posts a year. That is a cost of about 2ÔéČ per post… ­čśë Luckily this hobby of 3D complex numbers is a rather cheap hobby while at the same time it keeps the mind sharp. A disadvantage is that if it takes me just 5 or 10 minutes to do some calculations with pencil and paper, it often takes 5 to 10 hours before it is turned into a post that is more or less readable for other people… And that is something I value highly; so often you come across sloppy explanations that are not carfully thought through. I don’t like that.

Originally I prepared 10 pictures to write the post on but I had so much text that I started expanding those pictures and in the end I made an 11-th picture to get it all on. So I just expanded those pictures to make the text fit more or less precise of most of them have weird sizes. May be it is better to just stick to the size of 550×775 pixels and just make more of them if needed and not this chaotic expansion on the fly.
Ok, here we go:

I expect that when you made it this far, you already know what the Cauchy/Schwarz inequality is. But in case you never heard of it, please try to understand that beautiful but very simple inequality. Here is a wiki: Cauchy-Schwarz Inequality. Link used:

Ok, this is more or less what I had to say on the subject of crafting a norm from eigenvalues. Don┬┤t forget in the complex plane the square of the norm is also the product of the eigenvalues of a complex number z. So for centuries the math professors are already doing this although I do not think they are aware that they use a product of eigenvalues. For them likely it is just some stuff that is ┬┤Just like Pythagoras┬┤.

End of this post.

TU Delft guy claiming the electron pair is in a super position…

I am working in the kitchen cutting the vegetables, cleaning them etc etc. It is a beautiful Spring day. In the living room the smart television stands on Youtube and it jumps to the next video on auto play. And oh no, it is that Delft weirdo again and he thinks that all kinds of things can be in a super position without offfering the tiniest experimental evidence. And why not, he always comes away with it. His name is Leo Kouwenhoven and he is a physics professor at the Delft university.
A tiny piece of my freshly cut vegetables falls to the floor, is that a sign of God? What to do my dear God? Select another video or listen to that crap again? I decide to listen to that crap again and why not make a new post of it? After all the way I view electron spin is just so different from what the Leo’s of this world make of it. In my view the electron pair in chemistry (and super conductivity) exists because electrons are magnetic monopoles and that is why they like to pair up. People like Leo think electrons pair up because they are in a super position.
So as a reader you have something to choose; it just cannot be more different as this…

Let me write a parody on this super position nonsense, here we go:

Atomic hydrogen consists of two particles that, when measured, have an electric charge. Here I have an apparatus that can measure the electric charge of one of those particles that make up atomic hydrogen. Fifty percent of the time it measures a positive electric charge and fifty percent of the time on average it says the measurement is a negative electric charge. So the probability of measuring a positive or negative electric charge is 50%. According to the laws of quantum mechanics, before a measurement is done those two particles are always in a super position. Only when you measure one of them, the electric charge of the other becomes instantly clear. If I separate the two particles in atomic hydrogen and bring one particle to another galaxy and I measure the particle that was left behind, say it is negative, in that case the other particle instantly becomes positive. That is quantum teleportation.

So far for this simple parody. Do you think the electron and the proton are in a super position or are it the so called Coulomb forces that held them together? Anyway, below you will find the video that right now is over four years old. Of course at present day in 2020 the Delft guys still have nothing to show when it comes to quantum computing and in my view that is not much of a miracle…

Leo is also known as the man of 40 million because Microsoft has invested 40 million US$ into the Delft way of making quantum computers (that is with Majorana fermions, these fermions are made of electrons and holes and supposedly they are their own anti-particle). I don’t think it will ever work but later it will be a good joke: Remeber the time Microsoft invested 40 million US$ in particles that are their own anti-particle?

So far for this kind of nonsense, in another development I am still working on the next math post upon a norm based on the eigenvalues that 3D complex and circular numbers have. Next week it should be ready to post it. In case you are interested, try to look for those so called eigenvalue functions in previous posts. In 3D (complex or circular number space) you have three of them and if you take an arbitrary number X, with these easy functions you can calculate the eigenvalues with two fingers in your nose. Below you see already what the basic idea is:

Ok, that was it for this small post upon magnetism. Thanks for your attention and till next week or so.

The RI has a new video on magnetic monopoles.

Yesterday all of a sudden there was a new video upon magnetic monopoles; naive & dumb as I was I only thought ‘Great may be I can learn something new!’ and I started watching.

The video from the Royal Institution is entertaining and as such not boring to watch. But for me there was nothing new to learn, so I started thinking about why this guy Felex Flicker behaves the way he does. After all he is a scientist and given the fact that physics is a so called ‘hard science’ all claims made should be backed up by experiments. Yet this Felix guy when he claims that magnetic domains in metals and electrons are magnetic dipoles, there is once more zero mentioning of any experimental evidence.

Compare that for example to how at CERN they study anti matter. From positrons and anti-protons they managed to make a bit of anti hydrogen. And they do as much experiments with it as possible and try to find out it ther properties of anti hydrogen are such as expected. And that is the way it should be, that is what I view as standard behaviour for a hard science. But for electrons they never ever even tried it. Over the years I have made a long list of troubles with the electron as a magnetic dipole. I can’t name them all here of course so let me pick up just one detail:

If electrons are magnetic dipoles, why do we only observe electron pairs (and unpaired electrons) but never larger structures?

Here you see the new Brexit style in UK clothing, it looks great:

Take for example atomic and molecular hydrogen, there is only stuff with an unpaired electron (atomic hydrogen) and stuff with an electron pair (the molecular version of hydrogen) and nothing else. That kind of behavior is not what one should expect if the electron was a magnetic dipole… Electrons never behave like the bar magnets in the next picture:

May be I should have formulated this a bit less rude. It is not personel or so.

My dear RI folks, it is in so many ways not logical that electrons are magnetic dipoles. So I more or less only wonder that psychological stuff: why do the professors behave like they do? Ok, most of the time it is bad for your carreer to go against the insights as shared in the group, but this electron stuff you tell is just not logical. And, in my view, more logic is found when you think of electrons as having a magnetic charge.

Enough of my preaching, here is the video:

This guy hangs together from electron pair bindings,
why only electron pairs?

Let me leave it with that. Likely in the next post I will show a new way of taking a norm in the 3D complex and circular numbers. It is all based on eigen values, for the 3D numbers you can make a norm out of the eigen values while for general matrices you can’t.

Hurray! Nuclear electric resonance found.

Always when physics people explain stuff like nuclear magnetic resonance and it’s cousin electron resonance, it is always explained in terms of alignment of the particle spin with the applied external magnetic field. In my view that is a bizarre explanation because that would cause hardly any acceleration of the nuclei and electrons, so how can that give some measureable em radiation?

Yet in medical applications like MRI there is plenty of em radiation to make an image from. Where does that come from? In my view where particles like electrons and protons carry magnetic charge and as such are all magnetic monopoles, the resonance works because there is actually something resonating… It must look a lot like harmonic resonance or like a mass on a spring if you want. Basically it should not make much of a difference if you use oscillating magnetic fields or an oscillating electric field. Ok, in practice like medical MRI scanning I don’t think you can use electric fields because most atoms and molecules in your body are not ions, that is they are neutral under electric fields oscillating or not.

To my surprise in a video about a so called ‘Breakthrough in quantum computing’ all of a sudden the concept of nuclear electric resonance came along. Ok, it was on the Youtube channel named Seeker, so often it is not carefully thought through, but anyway. it might be Seeker but the concept of nuclear electric resonance should have large similarities with nuclear magnetic resonance if my idea’s upon magnetic charge are correct…

Let us take the time and look at a few screen shots I made from that Seeker video:

Wow man, NER instead of NMR?

At some points in time the video will get highly confusing, after all it is the Seeker channel combined with the insights of that Australian team trying to make quantum computer with qbits made from magnetic spins. Of course that is not going to work because if permanent magnetism is a charge you just cannot make a super position of it. So if I am right, all those kind of quantum computer will never work. Let’s go to the next screen shot:

This is the confusing part: Electricity makes the magnetic moment wiggle.

Of course this fantastic part of the video is inspired by how the university people explain magnetic resonance. If you view the video below, please remark there likely is no arrow of a magnetic dipole anyway.

It has to be remarked however that atomic nuclei can have many protons and neutrons and as such all kinds of magnetic configurations should be possible. Next screen shot:

These people are experts in understanding the electron pair.

The guy on the left, I don’t know his name, explains the electron pair as next: These two electrons are in a superposition of spin up and spin down. It is just like man and wife, there are two persons but you do not know if it is the man or the wife. Only when you make a measurement on one of the electrons, you instantly know the spin state of the other electron…

Don’t forget those people from blah blah land have zero experimental evidence for the electron being a magnetic dipole. After having said that, why not go to the next screen shot?

I never ever heard of this guy, but he was Dutch so shame on me.

You should not feel much pity for Mr. Bloembergen. After all he got a Nobel prize so he died while still having plenty of money. You are looking only at an old photograph of just one more perfumed prince. Also, Nobel prize or not, it’s just another perfumed human being not understanding it is impossible for the electrons to have two magnetic poles.

After so many screenshots, enjoy the deep thinking as in the next Youtube video:

Every year we have quantum breakthroughs but never a real computer.

Before we split I want to link to a few experiments that I posted on the other website on 11 May. One of those experiments is completely undoable, the second requires a lot of work because there a beam of electrons should get split in half in a cyclotron. The third experiment is showing that magnetic domains always have surplusses of either north pole or south pole electrons. That is stuff I cannot do myself in my kitchen, garden or living room. The likelihood that someone else will pick that up in the next 10 years is relatively low, it is a wild guess but at best it will be something like 1% to at most 4 or 5%.
As you see my expectations are not very high. Say for yourself: how likely is it that an article about an experiment that validates the magnetic monopole character of electrons passes the peer review process?
That is not very high… Ok, end of this post; live well and think well.

Calculation of the circular exponential circle via ‘first principles’.

Oh oh, this is one of those posts where I only calculate in the 3D circular numbers while I classify it as 3D complex numbers. In the past when I made those categories on this website I did not want to have too many categories so that is why I only have 3D complex numbers as a category.

All in all this post (number 146 already) is not extremely important because over the years I have given many proofs that the parametrization for the exponential circle indeed fulfills all those equations like the sphere-cone equation of the fact the determinant is always one. On the other hand, if you have an important mathematical object like the exponential circles, it is always good to have as many proofs as possible. Just like there are many proofs for the theorem of Pythagoras, it would be strange if we only had one proof and nobody cares about more proofs to that theorem that more or less the central to a giant mountain of math.

What do I mean with ‘first principles’? Very simple: that is the summation formula for the exponent of a linear operator or the matrix exponential if you want. In this post I use a somehow slightly different number tau; I use a number tau that gives a period of 2 pi for the exponential circle. The reason is simple: that makes the long calculation much more readable.

Another thing I want to mention is that the long calculation is nine lines long. For myself when I read the works of other people I do not like it if calculations go on and on and on. I always try to avoid too long calculations or I just don’t write posts about them. Almost nobody reads the stuff it it’s too long and gets too complicated so most of the time I simply skip that. Beside that there is always 0% feedback from the mathematical community, so although I always year in year out try to keep it so simple that even math professors can understand it, nothing happens. Just nothing, so after all those years it is not much of a miracle I don’t want to engage with these overpaid weirdo’s at all. Likely if you are born stupid you will die stupid & I have nothing to do with that. Mathematics is not a science that is capable of cleaning itself up, the weirdo’s keep on hanging to their fantastic quaternions and their retarded ideas of what numbers & complex numbers are. Too much money and too much academic titles have not lead to a situation where the science of math is capable of cleaning itself when needed.

Enough of the blah blah blah, after all the physics professors have the same with their electron spin: where is your experimental proof that the electron is a magnetic dipole? For over five years nothing happens except a lot of weird stuff like quantum computers based on electron spin…

This post is five pictures long, for me it was cute to see how those three cosine functions slowly rise from the start of the long calculation. Also of importance is to notice that I had to use the simple formula for cos(a + b) = cos(a)cos(b) – sin(a)sin(b) that comes from the exponential circle in the complex plane. Just once more showing that 3D complex & circular numbers are indeed emerging from the 2D complex plane. Not that the math professional will react, but anyway…

Let’s go to the five pictures:

I think you must calculate them for yourself, grab a pencil and some paper and use the
fact that the circular multiplication uses j to the third power is 1.

Again, this is not a ┬┤very important┬┤ post. Given all those results and proofs from the past it is logical such a long calculation has to exist. It┬┤s relevance lies in the fact you simply cannot have enough proofs for the calculation of parametrizations of the 3D exponential circle.

Let me leave it with that. See you in the next post.

Three video’s for killing the time if needed.

This time a somewhat different post, just 3 video’s I thought are interesting to share for their own reasons. In the first video the American television physics professor Brian Greene goes beserk on the beauty of the exponential circle in the complex plane… Brian, like so many others, do not know what they are missing. So many spaces have exponential circles and curves and indeed they are beautiful.

The second video is about a question that is often asked: Is math invented or is it a discovery? I think this is a false way of looking at math, if you replace the word ‘math’ by ‘food’ you already understand this is a weird question: Is food invented or is it discovered? In my view that often goes hand in hand but opinions vary wildly on this subject. The video is an interview with the UK math professor Roger Penrose. I included this video because back in the 80-ties of the previous century Roger had written some books on the things known as spinors. A lot of so called scientists think that spinors have something to do with electron spin, there are even weirdo’s that think after the electron has encircled the nucleus once it’s spin state is altered so that after two rounds the electron has it’s original spin back… Oh oh for people like Roger and those others it will be a long way in understanding the electron cannot be a magnetic dipole. In all ways possible that is not logical. For example the unpaired electron is not magnetically neutral while the electron pair is. And there are a whole lot more examples to be given showing electrons simply can’t be magnetic dipoles. And you only have to use the thing called logic for that; no weird quantum mechanical stuff but just a magnetic charge on the electron gives much better results if you use the thing called logic.

The third video is about a weird line of reasoning that I have observed in many video’s. It is about explaining how those jets form that emerge from black holes and their accredion disks. The reasoning is that the plasma in the accretion disk goes around the black hole and if a charge goes round it produces a magnetic field & that is all explanation given always. That is nonsense of course, even spinning metals like when you are drilling a hole with your drill machine never produces a magnetic field because for every electron that goes round on average also a proton goes round and all in all there is no overall magnetic field created. But if the electrons are magnetic monopoles, they will have much more acceleration compared to the far more heavy protons and as such an accretion disk around a black hole should be positively charged all of the time and that explains why the magnetic fields are so strong over there.

Ok, I crafted 8 pictures from the stuff. For example I made a 4D generalization of the 3D outer product while explaining such math is an invention and not a discovery. After the 8 pictures I will post the three video’s that aroused my attention for one reason or another. Have fun reading it.

The link to Reason 82 as why electrons cannot be magnetic dipoles is
08 Feb 2020: Reason 82: More on solar flares.

And here are the three Youtubers to kill the time.

Ok, let┬┤s try to upload this bunch of stuff and see what happens.

Integration on the complex and circular 3D number spaces.

A lot of math professionals rather likely still think that 3D complex numbers do not exist, may be for reasons like there are non-invertible numbers or whatever what other reason they have. This post more or less proved such views are nonsense; for example a lot of math on the 2D complex plane does not rely on the fact it is a field (and as such only division by zero is forbidden).

But on the 3D complex and circular number spaces indeed it brings some complications if you have non-invertible numbers in the function you want to integrate over a particular curve. And I have to say that problem could be solved by using the special properties that those numbers have. In this post I only show some examples with the non-invertible number alpha (alpha is the midpoint of the 3D exponential circles and all multiples of alpha are also non-invertible so the line through 0 and alpha are all not invertible).

For me writing this was a good distraction away from all that negative news we have day in day, all those countries reporting daily death toll can make you a bit depressed… So when I am through with the daily news I always do some other stuff like calculating a few of such integrals. That is a very good antidote against all that bad news. After all there is not much gained if you constantly think about things you cannot change at all.

This post is relatively long; at first I crafted 12 pictures but it soon turned out that was not enough. So while filling the 12 pictures with the math and the text I expanded some of the pictures so they could contain more math & text. That was not enough and in the end I had to craft two more background pictures. All in all it is 14 pictures long, that is a record length for this website.

If in your own mathematical life you have performed contour integration in the complex plane, you must be able to understand how this works in the 3D spaces. And for those who have done the thing known as u-substitution on the real line: it is just like that but now this u thing is the parametrization of a path. All that stuff below with gamma in it is either the path or the parametrization of that path. Please remark that you must use the complex or the circular multiplication on 3D, just like integrating over a contour in the complex plane uses the 2D complex multiplication.

In case if you are not familiar with the number alpha that is found at the center of the exponential circle, use the search function of this website and for example look up ‘seven properties of the number alpha’.

I hope I have removed all faults, typo’s etc so that later I do not have to repair the math because that is always cumbersome. Here we go: 14 pictures long so this is hard to grasp in detail in just a few hours. But it is beautiful math & that is why I do this. For me math is a lovely hobby.

Enough of the blah blah blah, here we go:

Ok, let┬┤s first hit the button ┬┤Publish┬┤ and see what will happen…
It looks all right but a day after first publication I realized there was some missing text. It turned out I had to rename picture number 2 and now every thing was like it was planned.

Later I will flea through the rest of the text, if needed I will post more addenda. For the time being that was it so till addendums or till the next post.

Integration on the circular and complex 3D number spaces.

Ok, the math text is finally written. It took a long time but all in all I am very satisfied with the result. It will be a long post, I estimate about 12 pictures long and that is more or less a record length on this website. I have finished only two pictures and I will take my time to make the other ones because my mouse does not work properly. When I click with the computer mouse, very often that acts as a double click and that makes making pictures a laborsome task because of all the errors that double clicking gives. And when I have to repeat a series of clicks three or four times before it is ok, it will take some time. May be I should buy a new mouse,

Anyway to make a long story short: For years I stayed more or less away from crafting math about integration because it is hard to find a definition that would work always. My favorite way of using Riemann sums could not work always because of the existence of non-invertible numbers in the 3D spaces. And that gave some mathematical fear in my small human mind because path independence came with that way of Riemann summation. All in all it is beautiful math to think about: For example if in 3D you use a primitive to integrate over a closed loop, is it always zero?

So only the first two pictures are posted and I have no idea when all other pictures are finished. Here we go:

Oh oh, only later I observed a double click problem in this picture…

That was it, till updates.