Phy,chem and math discussion thread

[James Brown]

Anyone knows what is the relationship between definite integration and the sum of rectangles?

 

[James Brown]

The heat has to come from somewhere.

Let's say, a cooling tower

 

[Horus Lupercal]

Let's say, a cooling tower

Ok.

a cooling tower

 

[Blazer Ayanami]

LMAO!

 

[Altaïr]

Anyone knows what is the relationship between definite integration and the sum of rectangles?

Yes.

 

[Horus Lupercal]

Anyone knows what is the relationship between definite integration and the sum of rectangles?

Yes.

Aren't they getting married once lockdown restrictions are lifted later in the year?

 

[Altaïr]

Aren't they getting married once lockdown restrictions are lifted later in the year?

I have different projects in mind...

 

[James Brown]

When will "sucking back" occur?

 

[Mars Pathfinder]

When will"sucking back" occur?

Wen it's finished?

 

[Horus Lupercal]

When will "sucking back" occur?

Oh the innuendo...

 

[James Brown]

Anyone knows the reason behind the chain rule of differentiation?

 

[James Brown]

Also, why radian is used in calculus not angles?

 

[Pink]

Radians are based on a fundamental property of circles. They're the same anywhere in the universe.

Degrees are arbitrary, the number of degrees in a circle could have been be whatever, 360° was picked because it's "nice".

Radians4life!

Also radians are simply simpler for some things.

 

[Altaïr]

Anyone knows the reason behind the chain rule of differentiation?

Yes: maths
There's a demonstration behind that that justifies it.

Also, why radian is used in calculus not angles?

That's an unit that appears to be more natural when doing maths.

The simplest example is when you want to calculate the length of a circular arc subtended by an angle θ. When θ is expressed in degrees, the formula is:
L = 2π × R × θ/360

If θ is expressed in radians, you simply have:
L = R × θ

If you only look at that you could have the feeling that this unit has been chosen just to make that formula simpler. But there's way more!

If you take some fundamental relations like this one:
Lim sin(x)/x = 1
x -> 0

Or the derivatives of the trigonometrical functions:
(sin(x))' = cos(x)
(cos(x))' = -sin(x)
And so on...

Or even the Taylor series of the trigonometrical functions:
sin(x) = x - x^3/6 + x^5/120 ...

If you have studied complex numbers, you also probably know the Euler relation:
exp(i×θ) = cos(θ) + i×sin(θ)

ALL OF THOSE are not true anymore if you express x in something else than radians.
Remove radians, and maths will never look the same!

 

[8BitCosmonaut]

chemistry

Last time I did chemistry, I ended up on the central intelligence's most wanted terrorists list.

 

[8BitCosmonaut]

Also radians are simply simpler for some things.

Throughout my entire engineering course, radians were only useful at simplifying mathematical calculations and computer programming, when applied in technical illustrations for the machine shops, the radian becomes an absolute fucking nightmare to use.

I mean, i'd rather write 20 degrees than 0.349066 rad.

 

[8BitCosmonaut]

Yes: maths
There's a demonstration behind that that justifies it.


That's an unit that appears to be more natural when doing maths.

The simplest example is when you want to calculate the length of a circular arc subtended by an angle θ. When θ is expressed in degrees, the formula is:
L = 2π × R × θ/360

If θ is expressed in radians, you simply have:
L = R × θ

If you only look at that you could have the feeling that this unit has been chosen just to make that formula simpler. But there's way more!

If you take some fundamental relations like this one:
Lim sin(x)/x = 1
x -> 0

Or the derivatives of the trigonometrical functions:
(sin(x))' = cos(x)
(cos(x))' = -sin(x)
And so on...

Or even the Taylor series of the trigonometrical functions:
sin(x) = x - x^3/6 + x^5/120 ...

If you have studied complex numbers, you also probably know the Euler relation:
exp(i×θ) = cos(θ) + i×sin(θ)

ALL OF THOSE are not true anymore if you express x in something else than radians.
Remove radians, and maths will never look the same!

I might need your help in the near future, its been nearly a year since I've done any calculus, and I've forgotten many basics already, like integration, differentiation, laplace transform and all that.

At this point I'm no longer taking my military conscription service seriously anymore, its all just a waste of time and I wanna dedicate myself to re-sharpening my calculus before I enter university.

What I am most interested in is the real world implementation of calculus, so far everything I've learnt are theoretical, non practical.

 

[Altaïr]

I mean, i'd rather write 20 degrees than 0.349066 rad.

I'd rather write 1 radian than 57.295779 degrees

Joke apart, the degrees have their advantages of course.

What I am most interested in is the real world implementation of calculus, so far everything I've learnt are theoretical, non practical.

I can see if I can help, but you know I don't practice all those things a lot anymore myself. It will really depend. I can probably help with derivation/integration, but Laplace transform is too far away from me now, I would have to relearn it myself. Not that I couldn't, but I don't have the time for this to be honest.

 

[8BitCosmonaut]

360° was picked because it's "nice".

Wasn't because it was nice. 360 was the 'Babylonian' number of days it took for the earth to rotate around sun (although it was later proven to be 365) , that's why a full circle was divided into 360 degrees.

I'd rather write 1 radian than 57.295779 degrees

Joke apart, the degrees have their advantages of course.

Wait, I just noticed something, whenever degrees become a whole number, radian goes nuts. Whenever radian becomes a whole number, degrees goes nuts.

I guess degree is more favorable in the engineering world because it was more "straightforward"? While Radians are more favorable in mathematics because formulas regarding them are much simpler?

And by straightforward... have you seen a radian protractor? It looks like an absolute nightmare to use.

I can see if I can help, but you know I don't practice all those things a lot anymore myself. It will really depend. I can probably help with derivation/integration, but Laplace transform is too far away from me now, I would have to relearn it myself. Not that I couldn't, but I don't have the time for this to be honest.

Thanks man, I could always try annoying the folks on Quora if my question proves too difficult. Hey I've never asked you this, what do you work as? Looking at your interests on the forums, I guess you are a software programmer of some sorts?

 

[Altaïr]

Wasn't because it was nice. 360 was the 'Babylonian' number of days it took for the earth to rotate around sun (although it was later proven to be 365) , that's why a full circle was divided into 360 degrees.

I didn't know that, but additionally, 360 is a number that has a lot of dividers in practice. Which makes it practical to measure a third of a circle, a quarter, a tenth... In fact, 360 is divisible by all numbers from 1 to 10, with the exception of 7.

Thanks man, I could always try annoying the folks on Quora if my question proves too difficult. Hey I've never asked you this, what do you work as? Looking at your interests on the forums, I guess you are a software programmer of some sorts?

Yes, I'm a software programmer. However I usually don't do a lot of maths in my job. Not at that level at least. It's just that I was good at that, and I still remember a few things

 

[Horus Lupercal]

You're all talking degrees and rads, I'm a soldier so I work in mils. 6400 of them in a circle, and they work for easy calculation of accuracy of navigation and firing at distances.

A mil is a metre deviation at a kilometre.

 

[Pink]

Sounds like a baby Parsec.

This here jeep did the Nuremberg in 100 mils

 

[8BitCosmonaut]

A mil is a metre deviation at a kilometre.

Used for artillery fire missions?

 

[Horus Lupercal]

Used for artillery fire missions?

Amongst other things. It makes corrections for all kinds of things really easy as long as you know the rough distance to target.

If you have mortars firing at a target 500m away from you and they're 100 mils off target, you can work out a very simple correction (100 mils is 100 metres deviation per kilometre. At 500 metres, the rounds are landing 50 metres off) and send that information back pretty quickly 'come left 50' and the mortar controller will work out the rest for you for the next fire mission.

That's why military binoculars have those vertical lines on them. They're set at specific mils apart to facilitate this for you.

If you've not got anything calibrated like binos or a prismatic compass, even body parts can be useful. For example, a finger held at arms extension is roughly 30 mils.

 

[Horus Lupercal]

It works for navigation as well. A Silva compass has minimum gradients of 50 mils. So when you're using it over long distances, you can expect to be a distance out (a mininum of 50 metres for every kilometre walked). Which is why you keep your nav checkpoints close together (preferably every few hundred metres if you're doing dead reckoning over featureless terrain) to minimise those errors