# What is your favorite equation?



## SilverNautilus (Apr 3, 2020)

If possible, show the equation, mention its name and indicate why you consider it special.


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## Skimt (May 24, 2020)

The simplicity behind oxidation and reduction. It's amusing how my old man taught me about refilling old lead batteries with water, and how I'm now teaching him about equation imbalance and electrolysis. At the top of the white board in green there's just some acidic behavior. 










I only have a "Chemistry 1" education at university, which is the minimum required for an electroengineer. 

Now that plain old algebra has become such a breeze I have a much healthier appreciation for it. 

I'm not a big fan of integration and differential equations.


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## SilverNautilus (Apr 3, 2020)

Hi @*Skimt !*
Thanks for your answer! 




> The simplicity behind oxidation and reduction. It's amusing how my old man taught me about refilling old lead batteries with water, and how I'm now teaching him about equation imbalance and electrolysis


Great, I've had this kind of conversation with my father too. 



> At the top of the white board in green there's just some acidic behavior.


I'm not very good with this. If at any time you have some chance to explain this to me, I would appreciate it, but don't feel compromised. 



> I'm not a big fan of integration and differential equations


Differential equations are really fascinating to me


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## SilverNautilus (Apr 3, 2020)

I will share the equation that I have chosen: _Snell's law_










It is honestly not my favorite, it is not the most spectacular, as you can see. I could say that I am fascinated by Maxwell's equations, but I have made a more personal choice. I like this equation for its derivation from the electric and magnetic fields corresponding to the incident electromagnetic wave. This derivation seems particularly beautiful to me since I saw it in an electromagnetic theory class.


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## Skimt (May 24, 2020)

I'm not a professor of chemistry so I might not have fully grasped it myself and might say something I shouldn't have. Take it with a pinch of phosphate  (You'll get it.)

There are weak and strong acids, and the only thing I can tell you about strong acids is that their chemical compound differs from weak acids. H3xO4 is considered a weak acid, where H3 and O4 are constant, and x is an atomic variable. The particular acid used on the board is P (phosphor), making it a "phosphoric acid" (H3PO4). In Norwegian the naming convention is "x-syre", while in English it's "x-ic acid". The naming convention for acids should be all the same whether they're weak or strong.

Whenever a Hydrogen splits from this acid (H3xO4), the Oxygen loses the electron that the Hydrogen provided it with. The H-atom then becomes positively charged and the O-atom becomes negatively charged (ionization). This new compound is not considered an acid anymore.

In Norway, we simply refer to H2PO4, HPO4, and PO4 as "phosphate" (or "fosfat"; "x-at"), and when we want to go into depth we call them "dihydrogen phosphate" and "hydrogen phosphate", respectively. The "di-" is greek for two, indicating that there are two Hydrogen atoms. These are all called salts, or "leftover salts".

The basic rule of thumb here is that they can go both ways (equations after all). So, these leftover salts can turn into acids again .

Chemical equations uses basic algebra, so once you know how electrons work and you've seen how it's done, you should be able to get the hang of it relatively quick.



SilverNautilus said:


> I will share the equation that I have chosen: _Snell's law_
> 
> View attachment 865453
> 
> ...


What happens when you derive them? Sinus turns into Cosinus, and Cosinus turns into negative Sinus (I don't have the formula in front of me), and then there's the product rule.


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## SilverNautilus (Apr 3, 2020)

> Take it with a pinch of phosphate  (You'll get it.)


Ok, i've got it! 



> There are weak and strong acids, and the only thing I can tell you about strong acids is that their chemical compound differs from weak acids. H3xO4 is considered a weak acid, where H3 and O4 are constant, and x is an atomic variable. The particular acid used on the board is P (phosphor), making it a "phosphoric acid" (H3PO4). In Norwegian the naming convention is "x-syre", while in English it's "x-ic acid". The naming convention for acids should be all the same whether they're weak or strong.
> 
> Whenever a Hydrogen splits from this acid (H3xO4), the Oxygen loses the electron that the Hydrogen provided it with. The H-atom then becomes positively charged and the O-atom becomes negatively charged (ionization). This new compound is not considered an acid anymore.
> 
> ...



I definitely need to study this 




> What happens when you derive them? Sinus turns into Cosinus, and Cosinus turns into negative Sinus (I don't have the formula in front of me), and then there's the product rule.


Well, by "derivation" I was referring rather to the _deduction_ of Snell's Law (Also known as "Law of Refraction")
This derivation is usually made from the _Fermat principle_, with a much simpler development.

Deriving (deducting) Snell's law from electrodynamics is more complex (and beautiful). I honestly do not remember the whole development, but I leave you a part of the approach
(sorry for the image quality)


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## EasyR01 (May 27, 2020)

Easy. Pythagoras thereom.


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## Skimt (May 24, 2020)

SilverNautilus said:


> ...


Have you by any chance heard of Gauss-Jordan elimination?


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## DoIHavetohaveaUserName (Nov 25, 2015)

Skimt said:


> Have you by any chance heard of Gauss-Jordan elimination?


yep.


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## Skimt (May 24, 2020)

I figured if they're interested in differential equations then the Gauss-Jordan elimination (among others) might interest them.


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## SilverNautilus (Apr 3, 2020)

Skimt said:


> Have you by any chance heard of Gauss-Jordan elimination?


Yes


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## 17041704 (May 28, 2020)

1 + 1 = 2
It's one plus one equals two.
Because you can do it with one hand and do it again with another hand. 
You can also do it with both hands.


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## HAL (May 10, 2014)

My favourite is the Navier Stokes Equation for a rotating body.










I like it because it's a pretty mean equation, and I used it extensively during the final year of my undergrad to study and derive a shitload of very interesting laws in geophysical fluid dynamics.

It's special because it sets the basis for the entire human understanding of how oceans and atmospheres flow around a planet.


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## SilverNautilus (Apr 3, 2020)

> I like it because it's a pretty mean equation


I agree with you, it's a fascinating equation!



> and I used it extensively during the final year of my undergrad to study and derive a shitload of very interesting laws in geophysical fluid dynamics.


Amazing!

Would you mind telling me a little more about your work? 
I have been working in the geophysics area for a long time, but never with fluid mechanics. 



> It's special because it sets the basis for the entire human understanding of how oceans and atmospheres flow around a planet.


Beautiful!
I don't consider myself good at fluid mechanics. Despite that I think it is one of the most fascinating areas of physics. 
I honestly haven't studied much about applications to geophysics. 
Are you a physicist or did you study a different career? 

thanks for your answer @HAL !


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## HAL (May 10, 2014)

SilverNautilus said:


> I agree with you, it's a fascinating equation!
> 
> 
> 
> ...


I'm not a physicist at all haha. I studied physics for undergrad and fluid dynamics was a favourite topic, and then in my final year geophysical fluid dynamics was the icing on the cake. _Super_ interesting! But, aside from that, I got pretty bored of academia and went into programming. 

I can tell you a bit about the geophysical fluid dynamics stuff I studied though... It was absolutely fascinating, it pretty much explained the movement of all major planetary flows in the ocean and atmosphere. 

First there's the obvious one, the coriolis force, which causes wind to deflect eastwards as it heads to the poles. But there's actually so much more that happens due to coriolis effects. For example, when wind is blowing over the ocean, it causes currents in the ocean (almost all ocean movement is due to wind), but the coriolis force causes the water to move perpendicular to the direction of the wind. This means that easterly winds either side of the equator push oceans in to the equator, leaving the water with nowhere to go. And THAT is why there's a jet of water (aka the gulf stream) racing up the western Atlantic. It's just escaping the pressure of all the water being driven into the equator!








The equation for the gulf stream looks like this: 








Sadly I can't even remember what half of the variables mean, other than to say they represent things like wind force, friction with the ocean floor, or the angle of latitude that you want to calculate for.

What else? Geostrophic balance. This one's amazing. Typically you would imagine wind to move from regions of high pressure to regions of low pressure. Common sense, right? Well, Coriolis effects cause flows to move _around _regions of differing pressure. We actually witness this every day in weather reports.

In the image below, black arrows are showing wind direction. Common sense says wind should move directly from H regions to L regions, but it doesn't. It goes around them!








The equation for geostrophic balance looks like this:









It's basically saying that the coriolis forces that want to send the flow sideways are equal to the inertial forces that want to move the flow by direct pushing/pulling, and so you end up with flows that simply go _around _the push/pull (pressure) regions.

What else?

Rossby waves. These are the thing causing the 'concertina' pattern in the jet stream. They're caused by a thing called 'conservation of vorticity'. It's similar to the the more widely known principles of 'conservation of momentum', or at a more fundamental level, conservation of energy. Wind on the equator is at a much wider radius of global rotation than winds on the poles. As the wind moves to the poles, it attempts to maintain the rotation, which manifests as a concertina pattern in the jet stream.









The equation for Rossby waves looks like this:









It's a fairly standard wave equation, where the wave speed,







, is defined primarily by the angle of latitude you're looking at.

I think that's enough for now 

I think what fascinated me most about this topic is that there are actual mathematical equations to describe such huge, complex and seemingly chaotic events at the planetary scale. And all those equations come from that one main rotational navier stokes equation that I posted earlier.


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## PathSeeker (Aug 3, 2020)

1*0=0. It's special because of the Interesting number paradox, but this time it's "Special equation". Though I suppose it has some properties that could be interesting... I feel that there are many beautiful equations out there, but this one just seems quite well-rounded.


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## Glittris (May 15, 2020)

1 + 1 = -1.72...

I am just joking with you, I was never good at math, but that equation has a special meaning to me... xD


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## BigApplePi (Dec 1, 2011)

Here are two:

2 + 2 = 2 * 2 = 4 because the elements are the same, the functions are different yet give the same answer.


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## BigApplePi (Dec 1, 2011)

This equation is slightly more difficult and so far no one has solved it though many have tried:

*All non-trivial zeros of the zeta function have the real part = 1/2.*

These are terrific equations but must be treated as a package. Don't ask me to explain them:









Riemann hypothesis - Wikipedia







en.wikipedia.org


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## attic (May 20, 2012)

I have forgotten most of what I learnt in maths... But I will have to be ordinary I think and say Phytagora's, as it is the only one I have actually found myself using now and then. I did actually look up snell's law a year or so ago too, but I never ended up doing something with it, I think I will sometime though, so need to learn to understand that stuff again. I would like to be able to understand easily where light will end up when putting windows and mirrors at various angles etc.


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## Electra (Oct 24, 2014)

attic said:


> I have forgotten most of what I learnt in maths... But I will have to be ordinary I think and say Phytagora's, as it is the only one I have actually found myself using now and then. I did actually look up snell's law a year or so ago too, but I never ended up doing something with it, I think I will sometime though, so need to learn to understand that stuff again. I would like to be able to understand easily where light will end up when putting windows and mirrors at various angles etc.


Same with me, I had to relearn allmost everything after highschool when doing college again these last years and I am _still_ trying to improve the math, even though I finished in march


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## 558663 (Aug 9, 2020)

I don't really have a favorite equation because I like pretty much all of them, but a notable one is Euler's identity:










It's often called the most beautiful equation because it links five fundamental constants in a concise form.

e - Euler's number, comes up a lot in exponentials and studies of change, base of the natural logarithm
i - the square root of -1, used in expressing complex numbers
π - pi, appears in a lot of areas of mathematics
1
0
One way to visualize it is using the complex plane, with the "x"-axis being the real axis and the "y"-axis being the imaginary axis. Any number can be represented as re^iφ where φ is the argument, i.e. the angle that you have to turn from the positive real axis to get to the arrow, and r is the modulus (length) of the arrow. If we take the end of an arrow of length 1 and rotate it, we get the unit circle as shown below. We need r = 1 to visualize this identity.










If you plug in φ = π, this would mean turning π radians counterclockwise. Since we established that we are turning in the unit circle, the arrow lies on the negative real axis and ends on -1. So we get the following equation:










Rearranging all terms to one side results in the identity. It's not exactly a proof but hopefully it gives a visualization of this identity.

Another equation that stands out is the functional equation of the Riemann zeta function:










It's probably been on my mind lately because I've been trying to understand Riemann's zeta function. I believe this equation implies that ζ(s) becomes zero at s = -2n, which are called trivial zeros. Some say that trivial zeros are called trivial because they're easy to find, I don't know how true this is. The famous Riemann hypothesis states that the non-trivial zeros lie in the line of Re(s) = 1/2, which is still unsolved to this day. The article that Riemann published has more background on this topic, but I don't know enough to understand it.

There are many other cool equations, but I think this reply is long enough!


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## SgtPepper (Nov 22, 2016)




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## Behnam Agahi (Oct 27, 2020)

*The Principle of Least Action*


























Untitled


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