Surface Area of a Sphere (equation derived with calculus)

The equation for the area of a sphere is derived by summing up small ring elements of area along its perimeter.

[@haydenhattenbach8223]

Calculus is so satisfying because everything that you were ever told just “to remember” without the reason now has an explanation

[@Batman_94]

Amen

[@magicianwizard4294]

most people who really appreciate math hate when they're forced to memorize something without any explanation. That's why I like calculus. I'm a few years ahead of the curriculum, give me about 3 years and I'll come back to this comment saying I hate calculus.

[@san_d_ptel]

Please find a good teacher then..😑

[@mert446]

@@magicianwizard4294 when you get to analysis you will

[@THEMATT222]

I agree!!! Reasons behind mathematical formulas is one of the main reason I love math!!!!!!!!!! Without it, I wouldn't like/enjoy the subject. That's why one of my favourite topics in math is the proofs.

[@hellboy_____2019]

The way you hold your pen is just amazing

[@saulremihernandez314]

Thanks! :)

[@peter_roth_8812]

Calculus is almost everything.

[@jibran8410]

almost *everywhere*

[@AmitKumarPrajapati]

Yeah you are right...

[@MusicMidas]

None of my calculus instructors explained why a particular method works (disk, shell, Green's theorem, etc.). Instead they threw formulas at us that we were expected to memorize. As a consequence, I effectively learned only to follow directions and not to truly understand calculus. I appreciate that you have broken this down into simpler steps to show WHY and HOW the actual calculus is applied in this derivation. Thanks!

[@nilaksh007]

Forget everything, did anyone noticed the perfect circle ?

[@MarkMcDaniel]

He had to have used a marker compass to get it that good.

[@odio_stationofficial3420]

whytf u askin' us to *'forget everything'* ?? JK 🤣

[@rodrigo-vl7bi]

fantastic, this is exactly the kind of video i was looking for, a purely mathematical explanation

[@aparnabiswas8312]

You explanations are absolutely fantastic. Got a great help. Thanks

[@ceo1OO]

I agree... very clear... the confusion is probably in the circumference element ds and understanding where it came about... its analogous to the surface area of the round part of a cylinder..2(pi)rh.. but instead of h, we use the differential for the smallest change in circumference. Other confusion may come from the need to express the integral in terms of theta instead of one of the co-ordinate variables, x,y,z, etc.. knowing all the basic formulas for circles will make things much easier..

[@martinzone8153]

The step aside at the end was a nice touch.

[@saulremihernandez314]

+Martin Zone Thanks!

[@TSthegamer2000]

Your way of making Integral symbol is very unique 😊

[@matrixate]

Absolutely brilliantly done. Thanks so much!!!!!

[@ptyptypty3]

Please, Step aside so I can see how Much Trouble I really am IN!! lol.. just kidding, that was a GREAT efficient Explanation!! you are very GOOD... I feel the same about your Video on the Area of a Circle!!!.... and to think.. I"m only NOW discovering your videos.. FIVE years after you posted them!!.. thank you!!

[@Geo877]

This is fantastic! It was well explained, thank you

[@georgegiani8587]

Very well done. From this problem solving exercise I can see he is a very good math teacher. Clear, concise lesson!

[@saulremihernandez314]

Thanks!

[@leif1075]

@@saulremihernandez314 Wait shouldnt it be from 0 to 2pi since it's the full half circle not jjst a quarter of it?? You didn't factor in the whole thing?

[@saulremihernandez314]

@@leif1075 not the way the integral is set up. I multiplied by two to take both halves into account. If you were to do it solely by an integration, you would integrate from -pi/2 to pi/2. Hope this helps.

[@leif1075]

@@saulremihernandez314 Ok yea sorry I was looking at it wrong. I did ot the other way where you go all the way around and integrate from zero to pi. And the reason in that method or in this one thst you only go from zero to pi and not from zero to 2pi is that you are creating and integrating tje full circles spanning the sphere and not just in the top semicircle as it might initially look? Thanks

[@fuahuahuatime5196]

This is a nice approach. You could also just use a double integral and solve for the surface area of a sphere.

[@yi-chenhu7794]

EXTREMELY HELPFUL! HIGHLY RECOMMEND!

[@aryensujjan]

please give this man an oscar

[@deadxfall7377]

lol

[@agrikagupta8095]

could we do this for volume too using pi*r^2*ds?

[@JoshLaurence_ETTO]

How is your board so perfectly white?!

[@CDChester]

SAME!

[@jackyleecs]

Elegant and graceful. Thanks. :)

[@Kram_Nebula]

ad for some blood sucking website feeding off the pain of a younger generation mocking us for being poor, " Still watching math tutorial videos from 2006? *scoff* ". 2011 actually and still incredibly helpful, saved me $40 and didn't have to tear me down to teach me. People like the man in this video are what our legacy as a whole is built on, not power hungry fear mongers clamoring for status. Sometime soon a child living in a fly over state in Nowhere America, that doesn't have access to advanced STEM programs or facilities will be watching this very same video and through hard work and perseverance will bring something revolutionary into this world that makes us all better. Be the change you want to see and leave each day hoping today was your day to contribute to someone else's future.

[@hazembenromdhane1124]

Thank you for the video it's very informative !

[@LockOn0705]

Thanks for saving my day

[@MrNoway20]

thank you . you are very useful. i subbed and liked! but please tell me if you do home tutors ? and if you do , do you live in united kingdom , Wales , Cardiff? it is unlikely but i do try :)

[@swarnendupachal2579]

Wowwwww, that drawing ❤️❤️, Sir thank you very much ❤️❤️

[@asriel4013]

For anyone wondering why s = R*Theta, it is the formula for the Arch length. Totally forgor 💀and got confused when decided to refresh some of calculus, and I am willing to bet someone is in the same predicament as I.

[@saulremihernandez314]

Why is the formula (s=r*theta) true? I’ve never found an satisfactory way to explain this. It has to do with the way theta is defined. In my view anyway. Perspectives welcome here.

[@asriel4013]

@@saulremihernandez314my bad for not going in depth. So basically the arch length is (s = r*theta) when radians are used. Using degrees, the formula would be s = (theta)*(pi/180*)*r or s = 2*pi*r*(theta/360*). If we then convert 180 and 360 into radians, it all simplifies to s = (theta*r) again. I don't know if this is what you were wondering about but anyway.

[@noraelazoui9689]

Thank you so much for your explanation

[@LGFZ136]

i got it clear but the way you explained was more clear and good

[@gautamsaroay1085]

Thank You Sir❤ I was finding this way of proof of area of sphere using calculus.

[@Zurenio]

Graande Saul ... Muy bueno...!!!

[@abhijeetkumar-hv6qh]

Can anyone tell me how s =rtheta

[@diegoalgo1296]

That whiteboard is soooo pleasing

[@andrzejpawowski8293]

Why S is equal to R theta and what is S anyway?

[@SWAGGALaGGAZ]

Billy Mays has been resuscitated

[@JonStoneable]

Thanks, and a question: isn't 0 to pi/2 a quarter of the sphere? Why don't we multiply by four?

[@saulremihernandez314]

because we're summing the circumference area elements due to the circle at that elevation. At theta=0, that area element is related to the circle all the way around. Then we sum from the equator to the top of the sphere and the circles get smaller and smaller as we move up. By summing in this way, we get the area of the top hemisphere by summing from zero to 90 degrees (pi/2). Hope this helps.

[@boltez6507]

Hey just had a question ...why can we consider it to be made up of small rings of radius r and integrade it from 0 to r and then double it for the other side

[@saulremihernandez314]

bc of the limits of integration. please specify little r vs cap R in your question.

[@boltez6507]

@@saulremihernandez314 hey thanks for replying on such an old video I thought that we could consider the sphere to be made up of large of stripes/rings of length 2πx and of width dx and Then itegrate it from 0 to R(the radius of the sphere)twice to get the SA of sphere 2*(0 to R)∫2πx dx However now i think that the width of the ring/stripe would not be dx maybe??

[@singh2702]

@@boltez6507 What about for every (rcosx; rsinx) you attach a corresponding perpendicular (r-rcosx) over the interval 0-pi/2 , this will result in all the heights and lengths contained in an area r^2. Then multiply that area by 2pi for semi sphere and multiply by 2 to get the surface area of full sphere.

[@nicholasmartin6353]

Great video man, but my OCD drove me nuts during this video. SURFACE area of a sphere.

[@Wjzj29q0]

For finding area why you multiply 2×p×r×R×theta? Is it rule or not?

[@lalitdebnath]

Thanks a lot, it was great

[@judicatoraldaris7604]

I wonder what is the theta here And why would the circumference of a slice equals the theta that’s not horizontal to it times the radius of it

[@singh2702]

Take two coordinates (rcosx , rsinx) , people don't realize that if sinx is increased by r times then that new point , rsinx , is increased by r in the x direction. So before integrating all points on the circumference , rsinx must be multiplied by r again giving us r^2sinx. Then it's easy to integrate to get surface area take the definite integral of 2pir^2sinx over o-pi.

[@ravipatnaik9936]

very good explanation. Thank you very much.

[@saulremihernandez314]

Thanks!

[@NotLegato]

can you tell me why the following method doesn't work? radius is 1, x^2 + y^2 = 1, so y = +/-sqrt(1-x^2). take the area of the circle going around the sphere: 2pir * dx, so the integral becomes integral from -1 to 1 of 2pi(sqrt(1-x^2)), from which you can substitute cos(theta) for sqrt(1-x^2) and cos(theta) for dx as well, so the integral just becomes 2pi*cos^2(theta), which is easy enough to solve. but still, the answer is wrong.

[@singh2702]

It works by R being squared. Reason for this is for a point to stay on a radius , no matter its length, it must be multiplied by r in the vertical direction and then again by r in the horizontal direction. Now you've chosen r to be 1 so your coordinates are just sinx and cosx and 1x1 is 1. This will leave you with a function 2pi*sinx , integrate that with interval 0-pi. You will have correct answer👍

[@quinnwang4660]

Why was dø ignored in the end? Since you are multiplying it instead of stating it as "in respect to ø".

[@Thecowthatcould55]

When you integrate the d theta is always removed. Your taking the antiderivative of the function, so any derivatives are removed, that's how I see it.

[@mohithkankanala8336]

Calculus is everything. It is my life. Thanks!!!!!!!!!!!

[@saulremihernandez314]

Mohith Kankanala I agree it is important! Thanks for your kind words. Tell your friends brother.

[@LVCID777]

What is the inverse of a sphere ?

[@rawanfouda2291]

beautiful

[@MusicMidas]

You are correct: 2pi radians is 360 degrees. I think your friend's reasoning is that the circumference of a circle is 2pi times the radius. A sphere is a circle rotated 2pi radians about an axis (imagine a coin spinning on its end). Thus 2pi rotated through 2pi is 4pi... however, this logic only appears to work here because it is a sphere. The video is more or less explaining why the calculus involved in the derivation works- obviously it would be easier to just go look up the formula!

[@bionickchief]

what if we take the infinitesimal surface of π*R*dθ and integrate it from 0 to 2π? can any1 do it for me?

[@1dan1609]

nick bio you have nothing to integrate now. You need an expression in terms of theta.

[@pashagaranin9571]

What is d? as in what does it represent?

[@LarthVolos]

So you took circle's area and turn it around by the length of full arc.

[@saulremihernandez314]

Not sure if this is a question?

[@undistractedmindset]

Tysm!

[@tushar9247]

Finaly i find the soln tnx a lot

[@vahaha1175]

Great! How about finding surface area of spherical cap

[@saulremihernandez314]

Should be half.

[@RunstarHomer]

just change the bounds. letting theta range from -pi/2 to pi/2 gives you the whole sphere, but you can integrate over any subset of that range that you want

[@aliourangiasr8066]

isnt anti derivative of cosine theta = -sine theta?

[@monkeycarz]

No, it is not. The anti-derivative of cosine theta is sine theta. As a test, try taking the derivative of the anti-derivative, and you see it will give you cosine theta, your original function.

[@skeletoncrusader]

The derivative of cosine theta is -sine theta while the antiderivative of sine theta is -cosine theta.

[@charlesrockafellor4200]

The term "antiderivative" is simply another word for "integral".

[@diyakomoradi8473]

How do you get your whiteboard to look so white without sacrificing the quality of anything else?

[@saulremihernandez314]

Great question. I produced these videos in a studio that I built in Denton, Texas at the University of North Texas. Since then I’ve tried creating the same effects without success. One thing you have to do is increase the aperture of the camera. Take a look at my videography playlist for more details.

[@gateway8833]

Sir, you have superb videos and I have learned a great deal watching them so please do not take this question as criticism, why would someone need to go thru all these steps instead of using 4 pi r squared? Is it so one can derive the area of less than a full Sphere? Thank you for your time.

[@agrikagupta8095]

this is the proof

[@aldovanderberg2519]

Thanks so much all the other idiots out there use shapes and pre-school object to show the formula and finally my hero arrives with calculus. Thanks so much.

[@raghavarvoltore6517]

Who's your here??

[@destinymyles4531]

I learned it to where the circumference is pi x the diameter and that the area was pi x the radius squared

[@robato1997]

i love the channel name

[@andyjones2457]

Surely integrating from zero to pi by 2 only gives a quarter sphere, you would have to multiply by four to find the area of the entire sphere. Of course that wouldn't give the accepted formula though. Could someone explain please? Any help would be much appreciated.

[@bzztbzztboy]

from another reply by the video's creator: "because we're summing the circumference area elements due to the circle at that elevation. At theta=0, that area element is related to the circle all the way around. Then we sum from the equator to the top of the sphere and the circles get smaller and smaller as we move up. By summing in this way, we get the area of the top hemisphere by summing from zero to 90 degrees (pi/2). Hope this helps. "

[@andyjones2457]

Milton Lee Thanks for pointing that out! I had assumed he was summing area segments of the quater hemisphere. Rather than summing area elements of the entire circumference. A bit silly, seeing as he clearly constructs the area element as a small circumference. I’ll chalk that one up to my mid revision cram brain

[@poorniahmohanajayanthrao3945]

why is s=r*theta??????

[@tommybaird3880]

A little late, but it represents the arc length. Formula for the length of an arc: (arc length) = (radius) x (angle in radians).

[@jtflypegasus]

Thanks man.

[@happyfrezar2813]

Thanks man

[@SaiKiran38252]

Perfect and great explaination

[@saulremihernandez314]

Thanks! Tell your friends.

[@user-zq1jg8dy8h]

I had seen many videos. But this is the one that satisfy me👌👌👌👌 (And this is the 100th comment)

[@Abishek2314]

nannu tamil........

[@Subhanii24]

I completly understood..🙂👍👍👍 thanks..

[@saulremihernandez314]

Excellent!!!

[@vratislavgoldie7386]

why did we take the limit for hemisphere from 0 to pie/2 rather than 0 to pie to cove whole hemisphere

[@saulremihernandez314]

bc each angle accounts for the area of a circle that goes all the way around.

[@vratislavgoldie7386]

@@saulremihernandez314 got it sir thanks

[@channaiahchakravarthi3352]

You are goat(greatest of all time)🎉

[@adventureboy444]

I tried (2 pi y dx) with y = √(r^2 - x^2) and wonder why it didn't working

[@lalitdebnath]

Calculus, the king of mathematics

[@SkillslliK]

I need good math teachers, *screaming intensifies*

[@ngyufeng6205]

Great video. Calculus is the best.

[@TriptoPal]

Nice & Cool.

[@dedly13]

this video is cool, i like this video

[@patrickfiset1694]

Why this technique does not work for the volume of a sphere ? Using that, I'm getting (2*pi^2*R^3) / 4. So, a 2 instead of 3, and a pi squared !

[@saulremihernandez314]

I'd have to see your work. You can email me. Go to the "About Us" page for my email.

[@AkashRaj-pk4kx]

Did you got your answer please tell me

[@mugdha1942]

Whoa!

[@gautamkumar5578]

Why can't we find volume of sphere by same method.integrating πr^2.R dtheta

[@saulremihernandez314]

That would work. This one was for the surface area but I did the volume in another video. There are two ways to do the volume. The way you described or by summing spherical shells.

[@gautamkumar5578]

Yes by integrating spherical shell I got my answer but by doing this I didn't get the answer.

[@gautamkumar5578]

If I integrate πr^2.dy and use the equation r^2+y^2=R^2(circle equation)then the answer came correct but I have to know why integrating wrt to theta I get wrong one

[@saulremihernandez314]

@@gautamkumar5578 I'm not exactly sure how to answer without seeing your work but here's my take on the limits. If you're doing it the way you described, you can 1. integrate from 0 to pi/2 and then multiply by 2. or... 2. integrate from -pi/2 to pi/2 and the answer should come out directly that way. For the first way here, you would multiply by 2 because the integration would only give half the volume. Hope this helps.

[@Hoakshila]

Thank you to the Billy Mays of Calculus

[@saulremihernandez314]

Haha thanks!

[@yr4137]

Fantastic, thank you to share your knowledge, quietly. Above all, I learn math words in English :-)

[@mohitmadnani7742]

im using this for my maths ia

[@greedyinterntainment5973]

Why can't we do the same when calculating the volume? But just use pi*r² instead of 2pi*r

[@saulremihernandez314]

You can. Just needs to be 4*pi*r^2. See this clip: kzbin.info/www/bejne/jXecYYNurNarp7M

[@saulremihernandez314]

Oh I see, yes that should work also. Try it. Let me know if that works.

[@greedyinterntainment5973]

@@saulremihernandez314 It works only when integrating with respect to cosine of theta timed with R When integrating with respect to θ I got the result of (pi^2)(r^3)/2

[@AkashRaj-pk4kx]

​@@greedyinterntainment5973please tell me can we use this method

[@jen-hungfang5437]

what is S?

[@saulremihernandez314]

S=arc length.

[@RohanKumar-xd9mr]

Nice

[@saulremihernandez314]

Thanks

[@realityversusfiction9960]

The "Laws of Mathematics and Differential Geometry" are Universal in nature; and are universally creative of all realities in nature, because they ""unlike theoretical mathematicians"" obey the "Universal Laws Of Mathematics"; And they " DO NOT APPROXIMATE"! A Circles Length Diameter Line 120 cm 120 cm's x 3 = 360 cm A circle has 360 degrees Each degree is 1 cm long 3 X r2 120 cm x 120 cm Square = 14. 400 sq cm 1 right angle = 120 cm 120 cm ÷ 2 = 60 cm radius 60 cm x 60 cm = 3,600 sq cm 3,600 sq cm x 3 = 10, 800 sq cm to the area of the Circle Which is 3/4 of the area to the 14, 400 sq cm square Or simpler method Diameter 120 cm x 12 = 14, 400 sq cm to the square Diameter 120 cm x 9 = 10, 800 sq cm to the circle

[@skepticmoderate5790]

Lol what. That's not true at all.

[@yttean98]

Neat.

[@realityversusfiction9960]

From The Circle to the Sphere = Simple Empirical Arithmetic Pi = 22/7 as an improper/vulgar fraction, which translates to being 3 whole units of 7, and 1/7th of 1 whole unit remaining Or 3 diameter lengths consisting of 7 measurement units, with one measurement unit (1/7th of one diameter) remaining PROOF Finding The Length Of One Degree Of A Circle 1. Use a line measuring 120 centimeters as a Diameter Line 2. Multiply the 120 cm Diameter Line by 3 3. The Circle is 360 cm long 4. The Circle has 360 degrees 5. Each Degree is 1 cm long Finding The Area Of A Circle 3 X r2 (Three times radius squared) Given a Square measuring 120 x 120 cm; the Square will have an area of 14. 400 square centimeters. 1. Use one right angle of the 120 x 120 cm square as a Diameter Line 2. Divide the 120 cm Diameter Line by 2, to give a Radius of 60 cm 3. Square the 60 cm radius: 60 cm x 60 cm = 3, 600 square centimeters 4. Multiply the 3, 600 square centimeters by 3 = = 10, 800 square centimeters Therefore the Area of the Circle is 3/4 that of the 120 x 120 cm 14, 400 square centimeter square CONCURRENCE The Ancient Sumerian masters of geometry and mathematics defined this empirical reality, more than 2000 years before the Alexandrian Museum Plagiarizing Greeks. Reference: “Estimating The Wealth” Encyclopedia Britannica. A Babylonian cuneiform tablet written some 3,000 years ago treats problems about dams, wells, water clocks, and excavations. It also has an exercise in circular enclosures with an implied value of π pi = 3. The contractor for King Solomon's swimming pool, who made a pond 10 cubits across and 30 cubits around (1 Kings 7:23) used the same value. Sumerian Method 1. Diameter 120 cm's 2. Circumference 360 cm's 3. Circumference squared 129, 600 sq cm's 4. Divide 129, 600 by 12 = 10, 800 sq cm's to the area of the circle However one question for me does remain begging, did they also manage to achieve this? Twelve Steps From The Cube, To The Sphere Calculating the surface area and volume of a 6 centimeter diameter sphere, obtained from a 6 centimeter cube. 1. Measure the (a) cubes height to obtain its Diameter Line, which in this case is 6 centimeter’. 2. Multiply 6 cm x 6 cm to obtain the square area of one face of the cube; and also add them together to obtain the length of perimeter to the square face = Length 24 cm, Square area 36 sq cm. 3. Multiply the square area, by the length of diameter line to obtain the cubic capacity = 216 cubic cm. 4. Divide the cubic capacity by 4, to obtain one quarter of the cubic capacity of the cube = 54 cubic cm. 5. Multiply the one quarter cubic capacity by 3. to obtain the cubic capacity of the Cylinder = 162 cubic cm. 6. Multiply the area of one face of the cube by 6, to obtain the cubes surface area = 216 square cm. 7. Divide the cubes surface area by 4, to obtain one quarter of the cubes surface area = 54 square cm. 8. Multiply the one quarter surface area of the cube by 3, to obtain the three quarter surface area of the Cylinder = 162 square cm. CYLINDER TO SPHERE 9. Divide the Cylinders cubic capacity by 4, to obtain one quarter of the cubic capacity of the Cylinder = 40 & a half cubic cm. 10. Multiply the one quarter cubic capacity by 3, to obtain the three quarter cubic capacity of the Sphere = 121 & a half cubic cm, to the volume of the Sphere. 11. Divide the Cylinders surface are by 4, to obtain one quarter of the surface area of the Cylinder = 40 & a half square cm. 12. Multiply the one quarter surface area by 3 to obtain the three quarter surface area of the Sphere = 121 & a half square cm, to the surface area of the Sphere Confirmation by Weight Given that the 6 Centimeter Diameter Line Sphere was obtained from a Wooden Cube weighing 160 grams, prior to it being turned on a wood lathe into the shape of a sphere The Cylinder of the Cube would weigh 120 grams The waste wood shavings would weigh 40 grams Given that the Cylinder weighed 120 grams The waste wood shavings would weigh 30 grams. Note: And ironically you can also obtain this same result by volume, using Archimedes Principle. www.fromthecircletothesphere.net

[@danieldollopac6803]

hey billy mays

[@dhimansaha3248]

It's quite difficult

[@ibaryabaq9197]

wait, what is S? he just threw S up there.

[@saulremihernandez314]

the arc length.

[@espn168]

0 to pi/2 is 90 degree. isn't that just 25% of the sphere? so shouldn't you multiply it by 4?

[@saulremihernandez314]

you multiply by 2 because each angle corresponds to a circle that goes all the way around the sphere. so the integral we calculated take care of the top half. thus multiply by 2 to get the bottom half also. hope this helps.

[@espn168]

@@saulremihernandez314 still trying to picture it...surely the video is correct 'cause it derived the correct answer..

[@saulremihernandez314]

@@espn168 go to 5:10 and pause the video. see how at the angle drawn, the area element goes all the way around?

[@espn168]

@@saulremihernandez314 Thanks Saul, I'll sink in soon I hope. Just having some contradicting thoughts for now. 'cause a hemisphere is 0 to pi and and full sphere is 0 to 2pi.

[@saulremihernandez314]

@@espn168 not with this coordinate system. a hemisphere is from 0 to pi/2. a sphere is from -pi/2 to +pi/2 with this coordinate system. go to 5:10 again and pause. see it? I integrated from 0 to pi/2 and then multiplied by 2 but you should get the same answer by integrating the same expression from -pi/2 to +pi/2 and then you wouldn't have to multiply by anything.

[@qualquan]

not bad

[@SornRarm-yk7qr]

didnt know Drake know about calculus

[@zaplinaki9999]

quite so.

[@Wise_videos_]

He is a lefty

[@SWAGGALaGGAZ]

sorry if its mean... :/

[@jrjr18181]

lol you did not the system. will you not show how to ge the formula without intergrating straightaway???????????

[@tgmaths9021]

Ha

[@airajoycastro170]

it's so easy

[@johnbatchler8551]

not what I'm looking for

[@aldovanderberg2519]

How

[@norbertocriado7718]

Very poor explanation. Only useful for anybody who already have all the math, and geometric elements, to fill the gaps. So is the science so difficult for the masses...

[@saulremihernandez314]

this was a derivation. those typically aren't easy. calculus was a pre-requisite for this derivation. no way around that.