How Maxwell's Equations (and Quaternions) Led to Vector Analysis

[@lingarajpatnaik6514]

Kathy, a whole generation of physicists shall be grateful to you for your indepth research in to history of physics. Heavyside is one of my heros. And I learnt so many things from you today! Grateful. Very.

[@Kathy_Loves_Physics]

My goodness, thank you so much for your kind words.

[@ElectronFieldPulse]

​@@Kathy_Loves_Physics- Your videos are awesome, I love finding out about the history of science. Please keep doing this as long as you want!

[@powerjets3512]

A brilliant take. A school has lost a physics teacher, but the internet has gained someone inspiring us all with the deeds of those that have changed our world. Some particular favourites in this video.

[@Kathy_Loves_Physics]

Thanks, I am pretty proud of this one TBH (and I never expected to be interested in the history of math).

[@powerjets3512]

@Kathy Loves Physics & History You need to get to Edinburgh (Edwin's Borough. Nothing to do with Berg) and India Street.

[@Kathy_Loves_Physics]

I’ve been to Edinburgh but what is on India street?

[@lawrencedoliveiro9104]

One can mark milestones in science in terms of unification theories, where previously seemingly disparate phenomena are seen to be aspects of the same thing. The third great unification: Maxwell/Faraday, discovering that electricity and magnetism were aspects of the same thing. The second great unification: the realization that living and non-living matter obeyed the same laws (as when Wöhler was able to synthesize urea--a compound associated with living organisms--from ammonium cyanate--which was not). I guess Darwinian evolution is also a part of this, showing that there is no fundamental difference between humans and other living creatures. The first great unification: Newton’s gravitational theory, showing that the force that made things heavy (“gravid”) and fall on Earth was the same force that kept the planets and other celestial bodies in their orbits up in the heavens.

[@dougr.2398]

@@Kathy_Loves_Physics “James Clerk Maxwell was born in 1831 at 14 India Street.”

[@josephncraig]

Kathy, this old physicist really enjoys your videos, your spirit, your energy, and your delightful sense of humor! I am particularly drawn to your historical perspective, putting real people behind the names associated with equations. But, simply, your channel is a complete delight, both for its science as well as its humanity.

[@sphakamisozondi]

Wait, Heaviside did not have a college education? And he contribuated to vector calculus?! My level of respect for him went up by a large margin.

[@UKimpress]

Faraday was a book binder, Joule was a brewer.

[@Nakameguro97]

Not only that, Heaviside simplified Maxwell’s Equations to the compact 4 equations form that we study today. EM would look much nastier/more complicated without his contribution! That’s a huge point to these videos - to understand EM with enough clarity for engineering, we don’t need Quaternions. That was why engineers in the past chose to study Heaviside’s equations over Maxwell’s original ones.

[@suryahitam3588]

As an engineer who had physics as one of my courses my understanding of vector analysis and electromagnetism would have benefitted enormously from this video. You really do make physics gripping. If this video were a book I would not be able to put it down!

[@user-ij5lc1kw8r]

Words fail to express my joy of your lectures!

[@GoldenAgeMath]

What an excellent video! I learned more physics history in an hour than I did during my whole degree

[@FlaminTubbyToast]

As a student training to be a mathematician I find these videos to be truly inspired. I am particularly entertained by the restraint that Kathy shows regarding all important maths. Without a doubt any other professor would have flaunted the convergence and curl out at every turn. This is also quite the case in her early works in this series where i, j, and k would have called out as î, j hat, and k hat. It gives me much glee to recognize, know, and spot the precursors of the tools I use today. Without which, my work in physics would be particularly impossible. Kathy Joseph is a fantastic orator and has a fantastic taste in presentation. I have been keen to keep up with each upload.

[@Kathy_Loves_Physics]

Thank you 😍

[@ericephemetherson3964]

Have you noticed how mathematicians come up with thier own definitons of mathematics? For example: he took a triangle and set it on the side.......And suddenly we have a new symbol that nobody knows where it came from. As Bertrand Russel said: ''Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.''

[@ericephemetherson3964]

@@Kathy_Loves_Physics Have you noticed how mathematicians come up with thier own definitions of mathematics? For example: he took a triangle and set it on the side.......And suddenly we have a new symbol that nobody knows where it came from. As Bertrand Russel said: ''Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.'' I placed the whole Universe in a symbol ''U''. Then I divided it into smaller bits. So, U/number of atoms in the Universe (Nu). Then I divided it again U/Nu and I came up with one half of the Universe. I did that process a few times and began coming up with particles smaller than a quark. So, can I say that if I repeat the process of U/Nu one thousand times I will get something smaller than a quark? But then again if I claim that I came up with something smaller than a quark, am I correct about their existence? And again Bertrand Russell helped me when he said: ''Do the truths of mathematics follow purely by definition?'' We can't know that either because of Kurt Godel. No mathematical expression can proove itself. But smaller than quark particles come up in simple mathematics. So is with the side turned triangle. Is it true?

[@miguelmouta5372]

@@ericephemetherson3964. Not surprising. Russell was a notorious leftist bourgeois. Without God, but a believer in logic. Until Gödel vanished with his faith.

[@GlennElert]

I have been involved in physics education and history for a long time now and I have never heard this story told before. I love when your videos teach me something new.

[@phitsf5475]

One of my favourite things about physics and maths is the fact it has such a rich history. There are so many characters and the story takes so many twists and turns. Who needs novels when this exists?

[@juanreza4500]

Thank you for this wonderful human-side story of one of humanity's grand achievement.

[@myronplichota7965]

I nominate this video for a best of 2023 award!

[@Kathy_Loves_Physics]

Thanks!

[@robertbachman9521]

Just wow. I have been fascinated about the 'vector wars' for a while now. You are an outstanding researcher for sorting through all of this and putting out a memorable presentation. I thought I knew the whole story, but you surprised me at the end. I never knew that Heaviside did not write the modern form of Maxwell's Equations. I look forward to the next video in this series. PS, I have your book, the Lightning Tamers and highly recommend it.

[@robertbachman9521]

I have recently been reading more about quaternions and this leads to Geometric Algebra (Clifford Algebra !?). It turns out that Maxwell's equations can be collapsed from four vector equations into one Geometric Algebra equation. From what I have read quantum mechanics operators like 'spinors' can be incorporated into Geometric Algebra in some form. If some day we get a unification of gravity with the standard model, my guess is that some algebraic system will be invented to form a single equation for everything.

[@olenaerhardt7725]

Wikipedia states: Later, Oliver Heaviside studied Maxwell's A Treatise on Electricity and Magnetism and employed vector calculus to synthesize Maxwell's over 20 equations into the 4 recognizable ones which modern physicists use.

[@chrisfuller1268]

Words cannot express how much I loved this video. What an amazing story and I can't wait to learn more about how relativity caused us to think Maxwell had only 4 equations.

[@johneverson2433]

It amazes me how those physicists could figure all of this out and I have a hard time balancing a check book

[@Kathy_Loves_Physics]

They probably had a hard time, balancing a checkbook, too. I know I do!

[@dougr.2398]

@@Kathy_Loves_Physics I developed my own method of using positive and negative numbers for withdrawals, deposits (or credits and debits) above and below the « balance line » (the date of the bank statement, usually). Given enough time, I could convert it into a simple computer program, which would take more work than actually striking a balance (which I no longer need to do)

[@dougr.2398]

The fully relativistic (coordinate frame-independent) version of Maxwell’s equations, of course reduce the four Lorentz version (H. A. Lorentz, not Lorenz) to one equation

[@renegadedalek5528]

working out how to describe phenomena of the natural world in mathematical terms is much more satisfying than keeping the bank happy.

[@skybot9998]

Lol! I know it just flows out if them.

[@ericphantri96734]

This is more important than anything discovered before not only it's scientific significance but also the divinity hidden in natural laws

[@redknight344]

Thanks for talking about this and also talking about Heaviside!!! a great forgotten genius!

[@Kathy_Loves_Physics]

I’m not sure Heaviside is forgotten anymore. But he was definitely a genius.

[@keylanoslokj1806]

Forgotten only by those who have no clue of electrodynamics

[@redknight344]

@@Kathy_Loves_Physics believe me, many on the communications and electronics engineering fields dont know who Heaviside was!!!

[@mntlblok]

@@Kathy_Loves_Physics I hold electrical engineers in the highest esteem, but every time I meet one - with maybe one exception - I ask about Heaviside and his reduction of Maxwell's 20ish equations to four and they've never heard of either him nor the greater number of equations. And now you tell me it was Lorentz. 🙂 Oh, well. . .

[@kendebusk2540]

Kathy, I love your ability to take complicated subjects and break them down into manageable "bites" which nearly all can understand. Please keep up the good work!

[@Kathy_Loves_Physics]

Thanks ken

[@rogergreenwood1536]

Great stories about people most will never hear of, thanks. At least Heaviside got part of the ionosphere named after him, again something few people will have heard of.

[@siobhangraham7280]

I learned all about the maths of these subjects in university, but now to hear about the history and people behind them is fascinating - especially in such a detailed and passionate telling of it! Thank you!

[@Nakameguro97]

Wish I had these videos when I studied electrical engineering decades ago - the historical context of how mathematical physics came to be is fascinating. Subscribed!

[@ExistenceUniversity]

Kathy loves Physics and History, and I love Kathy! Thank you so much for all the work you do!!

[@AkamiChannel]

I love how you make so much use of primary sources. It makes me want to read more and dig in deeper. If you really want to understand a subject, you need to understand the history of that subject.

[@CliveBagley]

Finished it all with a few rewinds to be sure I followed properly. Absolutely masterful. Jolly well done!

[@Darthvanger]

Now it makes so much sense! I was so puzzled reading Maxwell... :)

[@068LAICEPS]

I have been waiting for 17 years this video (still waiting for a movie) about this war (since 2006 when I started learning Physics in the University). Thank you.

[@meniscusmedical2471]

This is just simply brilliant, like most many will have been trying to find out the history of how these equations began since their childhoods and you've made their lives so including mine so thank you!

[@debraj10]

Such a beautiful video. I should have had you as my teacher for Electromagnetism, but it is 30 years too late. It would also be great to have more episodes on the contributions of Grassmann and Clifford to multidimensional algebra and calculus, which were not appreciated by physicists at that time, but have become very relevant to physics today (differential forms, Clifford's geometric algebra etc.)

[@AndreaCalaon73]

I was going to write pretty the same. 😅

[@kantanlabs3859]

I was certain I had a good knowledge on the whole EM story. Then you came, and you bring so many elements I was missing on the various contributors as for instance Grassmann, the quaternion war, Wilson late contributions. Now I fill like I was only aware of the tittles of some chapters of the EM and vector analysis history book (Volta, Faraday, Maxwell, Heavyside, Lorentz), whithout any idea on the contents of the chapters themselves. Many thanks for these enlightning moments ! Maybe you may tell us one day the interesting story of Alessandro Volta and the discovery of the capacitor and the displacement currents. These were ignored by Maxwell and reintroduced by himself for mathematical coherence instead of empirical bases. I think it was the biggest mistake in Maxwell work and it now affects the way we perceive the whole EM domain. This late and etherical reintroduction of displacement currents, led according to me, to a poor understanding of the attached domain, to the inversion of the magneto-electric effect discovered and appropriatly named by Faraday into electro-magnetic induction, then to confusions between magnetic displacement and electric displacement, to a large ignorance of the electric influence domain and the large variety of influence machines that were built in Europe. More generally it led to a prety poor understanding of the electric near-field domain, that is often ignored in EM books (except in Melcher book). This large ignorance due in part to Maxwell disregard of Volta contributions is particularly pregnant in the states.

[@michaeldamolsen]

Thank you Kathy!

[@yutubl]

Wow! Thanks for this very exciting tour through the history of the development of mathematical operators from quaternions to vecor analysis used by Maxwell equations.

[@jerrymelcher6431]

I have watched all of Kathy's videos multiple times. And because of this one thus one I tripped over Geometric Algebra - the unifying mathematics for all physics by Grassman and Clifford resurrected by David Hestenes. Thanks Kathy. No wonder I never kept my vector calculus book. Too obtuse!

[@sholaibukun1352]

I first came across maxwell equation in 200 level and I have been wondering how he must have reason that through but this video has helped me cross every barrier of reasoning through. This is an epic clip. Thanks for taking such a time to do a great job ma

[@HitAndMissLab]

phenomenal and so well explained. Sometimes its easier to learn electromagnetism from Kathy's videos then from textbooks 🙂

[@Kathy_Loves_Physics]

Thanks. I am planning on writing a college level advanced EM book, but it is going to take me years and years as I have so much I want to do first and I am easily distracted.

[@brucesekulic5443]

Isn’t it a marvellous example of emergent understanding?

[@aravindbanoth5791]

I would have married you if I were close to your age so that i can listen to ur talk all the time. I just saw you video for the first time today while looking for story of Ludwig Boltzmann but now, I got hooked up to every video on your channel. When I don't understand something, i never look up for resources online because previously i just got some equations as explanations which even the explainer don't completely understand I feel. I used to read reseach papers or literature on those but they too sometimes overload me . Understanding something at roots really makes us motivated to learn something (atleast for some people). it makes nosense to keep things in our brain when we don't know where did it come from. Thank a lot. I'm you fan starting today.

[@PaulSpades]

As a programmer, I always though quaternions were related to matrix math, since I first encountered them in 3d graphics, not in my math education. Arguably, matrix operations, vectors, functions and quaternions in computer science are somewhat bastardized versions of their pure math versions. But this whole series set me straight, and cleared my mind regarding these concepts and how they came to be used. Fantastic video!

[@paulpiacentini]

I wish a man could do what you do so incredibly well Kathy, but I've yet to meet him. I'm 61 and I teach physics at William Thomsons old campus in Glasgow. Electrons allow you to teach me about my hood more than I can be bothered to listen to any voice around here. Hence that well deserved silver plaque behind you hen. You are braw, as we say in these parts ❤

[@supermikeb]

Man or woman I doubt you'll meet anyone that has done this depth of research as well as showing all of her scripts with citations (on her website).

[@cyclonasaurusrex1525]

That was amazing. My math IQ is pretty much limited to calculating tips (a fact I regret more deeply the older I get), but the characters, drama, and pacing had me captivated the entire time. What a gift!

[@alexandertownsend3291]

Its never too late to learn math. There are lots of free ebooks and tutorials. Let me know if you want any resource recommendations.

[@cyclonasaurusrex1525]

@@alexandertownsend3291I’d love that. One of my goals in (potential) retirement is to work my way from algebra to calculus. You know, the stuff I should have learned in HS and college.

[@alexandertownsend3291]

@@cyclonasaurusrex1525 Okay I would start by going to the youtube channel The Math Sorcerer. He gives good book recommendations for learning math at different levels. If you are more of a tutorial person there are tutorials on KZbin, Khan Academy, and Udemy. Udemy has both paid and free courses. If you have the budget for it, the paid ones are generally better, but the free ones can still be good. If it takes you a while to learn this stuff, that's fine. Take your time, take good notes, do lots ofvproblems, and enjoy the learning journey.

[@lopezb]

What a wonderful series of lectures! I have taught Vector Calc many times and never learned about quaternions before this. It's really a scandal. They were always mentioned as a curiosity. I will need a new chapter in my lecture notes!

[@cornfall]

8:40 Here and overall you’ve struck gold! Brilliant work!

[@AT-27182]

This history is thrilling and inspiring. It is also enlightening. Thank you so much for this work and for generously sharing it with us.

[@otiebrown9999]

Katy, An incredible history. Thank you!

[@zenken8534]

Thank You; for bringing us the fruits of your investigations and presenting animated insights into the lives and reasoning of great scientists.

[@justincheng3922]

I Second the Wow! Thankyou for this education. I have been self studying physics and so vectors and this provides much needed conceptual clarification.

[@garyarnold3141]

Kathy, I'm working my way through your videos. You're the best teacher / explainer of physics I've seen. I studied physics/chemistry up to about ordinary degree level so I'm familiar with some of the topics. I've found your videos on Tesla the most interesting, nobody else seems to tackle him. I've just ordered your book.

[@izaret]

This is a fantastic video connecting the dots on quaternions, Grassman Algebra, Maxwell equations and vector analysis. Amazing to see these powerful concepts are coming from a very small group of people who knew each other.

[@aunraza206]

I love your channel so much. I’m a biochemistry and physics student, I’ve been having one of the hardest times of my life recently and watching your videos is one of the things that make me genuinely happy. You’ve been helping me to keep going. Thank you for these fascinating videos! Keep being you!

[@LaurenceRonayne]

Great stuff Kathy, your weave of science with history is the missing bit that powers understanding.

[@ericfielding668]

I love applied quaternions. Thanks for the history. (Aside: Every time I hear "Maxwell," I think of his silver hammer.)

[@raymondfrye5017]

You love Quaternions?...and think of Maxwell's Silver Hammer? Must be from an influence coming "thru the bathroom window'.

[@mpolier]

Wow! Fascinating! You are also a great storyteller, I am a social worker and musician, and I’ve grown fascinated by quantum physics- as I listen to lectures, these names keep popping up. To hear the competitive humanistic history of mathematicians is very similar to the pursuit of the arts.

[@marktwain5232]

This presentation was absolutely fabulous! Just wonderful!

[@bertiewalker5140]

You are so good. Thanks for all the effort you put into these.

[@swamihuman9395]

- FANTASTIC/FASCINATING! (as always :)) - I don't know which I love more: math, or the history of math (sometimes along w/ physics) - and, you do a great job of elucidating everything. - Keep up the great content (I'll keep watching, w/ rapt enjoyment)...

[@Stelios.Posantzis]

You're a gifted story teller. I never thought I could tolerate 55 minutes of accounting (so I thought) of quaternions and Maxwell ( I only began watching in the belief this was going to be an account of quaternions' ) but it was a delightful breeze.

[@michaelsanchez7798]

This is the first video of yours I have seen. I love your enthusiasm for what most people would consider a terrible bore. The history of math and science is very interesting to me though. I will subscribe immediately. Click, done.

[@Nikolas_Davis]

I never knew the origin of the term nabla, and I use it all the time! 😲 Really fascinating!

[@pythagorasaurusrex9853]

Excellent! Love your videos. Especially this one is highly interesting about the development and history of modern notation what we take for granted. Instead it was developed over decades with fights and conflicts. I am honest... this is the first time in my life (I teach math for more than 20 years), that someone explained to me, where div and rot actually came from... quaternions! In all the text books I read, all lectures I attended, it was NEVER explained where divergence and curl actually came from. All books about vector analysis started with : "We define divergence / curl as ...". I am 54 years old and I learned something new. Life is beautiful! Thank you Kathy!

[@Kathy_Loves_Physics]

"Life is beautiful" - what a lovely response to learning something new. Thank you and I am so glad you liked it.

[@davidwilkie9551]

A most Educational reiteration of relevance, thank you.

[@Kathy_Loves_Physics]

You are welcome

[@ilanpi]

When I was young and ignorant, I was teaching linear algebra at Stanford. A student asked me if you could divide a vector by another vector. I said that you couldn't. It took 30 years to realize that you can. The result of dividing a vector by another vector is a quaternion...

[@TalaashDotCom]

From now on, you are gonna be my lifetime teacher. Thank you!

[@factChecker01]

The advantage of Geometric Algebra (aka real Clifford Algebra and Grassmann Algebra) is that it starts with a very simple geometric concept of an oriented n-dimensional volume (as opposed to the three mysterious complex components, I, j, k ) and takes that as far as it can go. For instance, it goes all the way to turning four (to eight) Maxwell equations (all interrelated) into one simple equation. It shows how much of the mystery of physics is based on the simple geometry of oriented volumes. The disadvantage of Geometric Algebra is that it involves a lot of tedious book-keeping in the mathematical manipulations and that there is a significant learning curve. Also, interpreting a simple (in Geometric Algebra) equation like Maxwell's equation has to be done one vector component at a time and that basically splits it into the four (to eight) separate Maxwell's equations. But I highly recommend it to anyone who wants to get a feel for what is really going on in Maxwell's equations.

[@owaiselectrical]

Kathy what are you? A physicist, a Historian, a teacher??? Amazing allrounder

[@Kathy_Loves_Physics]

Aw thanks. I guess I’m all 3.

[@magellan500]

This is one of the greatest presentations on math and science history I’ve ever seen. Many thanks for creating this.

[@Kathy_Loves_Physics]

Thank you ❤️

[@christopherjackson2575]

Kathy, I think your work is so valuable and well done. Brava!❤

[@robertschlesinger1342]

Very interesting, informative and worthwhile video.

[@skippy6086]

Vector calculus was my favorite area of math studies. Div, Grad, and Curl (the book and the functions) were my friends for years.

[@user-wc7em8kf9d]

Thank you so much. What a talent!

[@macbookpro1232]

Wow...... Simply Breathtaking ! As powerful an effect as I had while reading Einstein's biographies or watching Carl Sagan in my school going years. Thank You so much Kathy. You make me fall in love yet again. Brava!

[@taylorlevaur]

Thank you for making this video. I have studied ME's for a decade and seeing Tait's comment about Maxwell speaking to the next generation through the lips of the those he inspired spoke me. Words can not express how true that is.

[@Daniel-ih4zh]

This channel is a real gem.

[@deanrubine2955]

Awesome as usual, and definitely the first video I've seen in a while that included a "Part 8".

[@Kathy_Loves_Physics]

I was told that the best thing for my channel is to make shorter videos more often and um… I don’t follow directions. 🤷🏻‍♀️

[@dr.rahulgupta7573]

Excellent presentation 👌

[@kristhompson8112]

always love the backstory Girl thats whats brings it to life Thank You from myself Kris

[@tyggeln]

Thanks for taking a stand!! I really enjoy your videos very much!

[@douginorlando6260]

Standardizing nomenclature is underrated. And concise names help effective communicating as well as understanding. It seems a haphazard journey of the etymology of math included arrogance, bullying, and cliquish alliances we aren’t taught to expect from the scientific community.

[@mntlblok]

*Most* insightful.

[@clivedavis6859]

Thanks. Very interesting. We did not cover much of the history during my electrical engineering degree. We used the vector analysis. I only came across quaternions in my 3d animation hobby.

[@monoamiga]

...and Charles Steinmetz said a century ago that learning history is just as important as learning engineering!

[@redknight344]

part of this vector war is presented in the book of Paul Nahin about the life and works of Oliver Heaviside, so this video is a great complementary material for that chapter!!!

[@youreale]

I cannot put into words how this channel is good!!!

[@supermikeb]

Teaching science through its history was done a lot a long time ago. In about the 1800s as the concepts became more complicated they seemed to have not taken time to include the history. Kathy is bringing that back, and making learning fun and easier.

[@lkey1777]

WONDERFUL EXPRESSION, GOD BLESS...

[@caesarespinosa3131]

Kathy you are as magnificent as your work, your enthusiasm is contagious, thank you very much

[@mrfranksan]

You've done an absolutely marvelous job.

[@frankk6416]

Thanks for this. I just discovered your channel and this was a well spent 55 minutes :) Subbed.

[@KipIngram]

Kathy, you should look into Clifford Algebras, also known as geometric algebra. It mops all of this stuff - complex numbers, quaternions, vectors, etc. - into a single unified framework. I honestly think it's the path we should use to introduce kids to this stuff when they first start learning it.

[@johnvonleibniz]

Really excellent, engaging content.

[@Artyom178]

Astonishing! Thanks so much for this beautiful video!

[@jakublizon6375]

I love this lady. She's like a sweet elementary school teacher, with the slight wakiness found in a lot of college lecture halls.

[@wyattcanady8175]

My life would have been infinitely easier had I had this information when I started my studies in physics, in the 1970s!

[@clay.tennis]

Lots of learning such amazing depth in ALL of your videos!

[@lxoxrxexnx]

This is all such wonderful information. Thank you!

[@stevelang3171]

thanks Kathy! you're great. i hate inconvenient minus signs too! Lol someday explain Eigen vectors? thanks

[@jorgealzate4124]

It is funny how Heaviside talks about almost having forgotten algebra and trigonometry, because often the first time his name is heard is with the integration method of partial fractions that carries his name. Also, Heaviside is the example of how an Engineer must think: the practical, concrete, and unambiguous methods always triumph over the "perfect" ones, the latter are better suited to mathematicians 😊

[@MarkKrebs]

What fun to find you out here on the internet. Nice work!

[@johnfitzgerald8879]

Thank you, Kathy.

[@magellan500]

Just watched this again for the second time. You are truly a great teacher. Having discovered your channel I’m going to make sure to watch your other videos.

[@leonardbertaux6897]

Simply brilliant!

[@AnilYadav-nt9mf]

What an incredibly informative video... Amazing work.... 🎉

[@donallen7830]

I am a fan of the Civil War and Maxwell and to find Maxwell was creating his equations at the same time the North and South were at war is a fascinating two parallel time line events for me.

[@matemaicon]

What a amazing video! Greetings from Brazil.Your work is terrific. I just became a fan

[@drancerd]

Señora! No se imagina cuanto tiempo busqué un video este! De verdad la amo.