Integration in Spherical Coordinates

[@menoima9501]

underrated,underrated,underrated,underrated,underrated,underrated

[@sulfur9915]

I’ve been having severe test anxiety all year in Calc 3, and these videos are the first time in months I’ve been able to study, understand, and feel good about the material. Just wanted to say thank you for doing these! Nothing else stuck until now, and I’m so very appreciative!

[@levishell9280]

Me this year

[@shitvanddiewe1345]

@@levishell9280same 🥲

[@internetroamer8063]

Wow only after I finished the video I realised it is not 700k views but 700... The video is great! Visuals make it very easy to follow and the length of the video makes it very easy to grasp. Hope your channel will grow a lot in the near future! Thanks a lot for your teaching.

[@intsciencewing6870]

Visuals + facial expressions + hand moments, all are working in harmony with each other.

[@gnaved]

This is the first youtube channel I found that has in-depth explanations with great visuals of all the main topics in Calculus. Just wanted to say I always view your videos before every calculus lecture so I can actually visualize the content. Amazing work!!

[@DrTrefor]

So glad they are helping!

[@user-amanfeildconvecture]

​@@DrTreforyou deserves

[@raisaraminnadia7642]

I was searching for spherical trigonometry for a very long time as the couldn't figure it out by my own,,,I also lost my olympiad marks for not having a agony grasp on integration and spherical trig!!! I just earned all the basic concepts from this video !!!!no stumble whole solving problems ...thank u sir !!!u must be a professor in princeton University!!!!

[@killerwithoutcanary]

I stumbled into my lecture 5 minutes late and was introduced to these 3 variables and was so lost where they came from when I was used to r and theta, thank you so much this makes a lot of sense. Also, the jacobian too makes sense a bit where my professor brought it in from.

[@Bill0102]

This is sheer brilliance. I found something with a similar message, and it was beyond words. "The Art of Meaningful Relationships in the 21st Century" by Leo Flint

[@observever7808]

The illustration was brilliant, solved my questions at once.

[@SoumikRoyBAI]

Literally watched every video in a day and im feeling extremely confident about multivariable calculus now. Amazing explanations😍

[@DrTrefor]

Great to hear!

[@punyaputtprakaivichien4727]

I was surrenderring on advanced calculus, however, your teaching makes me able to stands up and fight against it again. Thank you Fr.Trefor

[@comfychairs795]

the visuals are super helpful in comprehending what each variable corresponds to :)

[@DrTrefor]

Happy to hear that!

[@bethj7106]

Thank you! Seeing where the conversion from cartesian to spherical coordinates comes from was extremely helpful!!

[@tomatrix7525]

If you want to understand the ‘scaling factor’ it”s just an extension of the 2d jacobian. The 2d jacobian translates a narly parallelogram into a square, with a scaling factor to account for the lost/gained area. In 3d, we convert a narly ‘parallelogram cube’ (squished cube made up of four ugly parallelogram sides) into a nice cube (area is simply (w)(h)(l) but dealing in infinitely small differential components as the deltas tend to 0, as with all calculus...but again with a scaling factor required.) The area of the parallelogram cube can be described as the area of a parallelogram side (as calculated by the 2d jacobian, which comes from the cross product, times the 3rd dimensional component, which the third dimensional component, say w, can also be converted into a nice straight component in our new system, but with a scaling factor, say it is called z in our nice straight system, w can be described at (dw/dz) dz . So now we have to take the determinant of a 3d matrix.

[@saleplains]

This is so helpful in this time of unexpected transition to online learning. I feel bad for my teacher because he is an older guy and was doing great at teaching calc 3 in person but its difficult to learn from him online this video had excellent graphics and really helped.

[@l.w915]

Honestly i can't believe i didn't see this channel until recently... your channel's absolutely amazing! Very well orgnized , easy to find subjects and very understandable way of presentation. Keep up the great work!❤

[@sazokuotsutsuki8848]

Today i missed the math lecture talking about spherical coordinate system in triple integrals and this video saved me fr

[@kshitizpoudel-jp7cc]

I just discovered the channel today. I really love your content and the way you explain everything.

[@DrTrefor]

Glad you enjoy it!

[@marianaasko6604]

This is amazing. I was so confused and I didn't even understand what was happening. But now I know :) Pretty sure this is going to save my GPA.

[@masteroftheuniverse7418]

These vidoes have been doing a great job at explaining these topics. Great job!

[@DrTrefor]

Glad you like them!

[@unsabhatti3639]

This is exactly what I was looking for. Thank you.

[@amatris]

7 min can teach you better than 2 hours. Thank you a lot.

[@TheCypher93]

I really wish you would solve these integrals in this video and others. I know it's trivial to you but a lot of struggling students can mostly conceptualize what you're saying but it's the actual calculus that makes the problems tricky. Probably below your paygrade but me and others would really appreciate it. This is still an amazing video though thank you.

[@torfoes]

best explanation on youtube!!

[@SciHeartJourney]

This was excellent! Thank you very much Dr. Bazett.

[@rakoonberry7879]

This video is so good. Love the illustrations!

[@cesarpascualramos7796]

Wow loved it. Your way of explaining it was really straightforward. Of course you need previous knowledge in calculus to comprehend it. Thanks, it think I get spherical coordinates now!

[@KRITISHPALRECHABRS]

You make such difficult concepts so easy..i love the way you teach. Thanks :-)

[@DrTrefor]

You're very welcome!

[@shishirallu3439]

Thanks for the explanation. The 3d model helped a lot, dr. trevor.

[@skcr3220]

this guy saved my engineering program

[@abdulraheemtahir6269]

The dude atleast deserves a million followers! A T L E A S T !!!

[@ogunsadebenjaminadeiyin2729]

Super useful, excellent visuals

[@krittikasarkar6989]

SIMPLY AMAAAAZINGG!!!💯💯❤️❤️❤️❤️❤️❤️

[@bryan-9742]

great video! I use theta on my zed so I was like, "oh no I did this all wrong!" before I realized my variables are just switched.

[@hubenbu]

unveil the beauty of Calculus

[@baqerghezi1342]

Wow you really make math easy good job professor. Thank you

[@DrTrefor]

Thank you so much!!

[@aymenazhar8946]

Helpful video 😍😍 Love from Pakistan❤❤❤

[@kaweaakuna7325]

Azimuthal angle👍 thanks bro💪

[@caleb5688]

I'd love to see a video breaking down the geometry behind the formula!

[@hannarula]

Thank you for an explanation, great video!

[@iyadindia862]

You know how to teach well ❤️

[@thomasclark210]

I often see maximum limits on theta and phi. Many volume integrals can be done over the range 0

[@ruzzcraze1862]

I hear my brother talk about how I definitely want to get trefor for math at my local university, I didn't realise that its the same trefors videos I watch, as the one that taught my brother.

[@DrTrefor]

That is amazing:D

[@ApunHandles]

Love this video ,it made abstract things to concrete

[@DrTrefor]

Glad you enjoyed!

[@nevis4567]

Thanks man! Big help before my calc lecture

[@DrTrefor]

Glad it helped!

[@syabongajele6431]

well explained SIR much appreciated

[@emtnan1733]

You are the best 👏🏻

[@kaokaoylan4510]

Thank you, its been a great help!

[@ice_the_kicker]

So very very very very well done!

[@DrTrefor]

Thank you!!

[@pasataizkonzuma]

Can you explain dV breakdown? The geometry behind it? or maybe point to a resource?

[@ethandrost6974]

I'm sure it's time of year where you see the view count on your videos jump lol. Great job on this one! I feel like I have a much better grasp on spherical integration now. Have a good one and keep up the good work.

[@hassanconteh6654]

Thanks doc u are such a wonderful explainer

[@DrTrefor]

Thank you so much!

[@newaymamushet7956]

brilliant teacher

[@rizqiadriyanto1090]

thanks you Dr. Trefor

[@eexxo0]

Nice video. I'd like to add that in physics the annotations of Θ and 𝜙 are inverted as per ISO/IEC 80000

[@muhammadumarsotvoldiev8768]

thank u. Amazing video

[@cedricpulmano9234]

THE GOAT! HE IS THE GOAT!

[@kylewong2067]

Thank you, good sir!

[@abdulmalek1118]

Thanks a lot Can I know what program do you use for constructing these brilliant illustrations

[@ohsnapigetit4631]

Dig the model, very helpful

[@smokey7146]

Would've been really helpful if you'd actually show how we derived the complex dV in spherical coordinates cuz I expected you'd be one of the very few ppl who'd explain it on youtube and I'd fully understand😢 Nonetheless great video and can't thank you enough !!!

[@cavelinguam6444]

Awesome Channel

[@xXEpicxManXx]

Beautiful video!

[@DrTrefor]

Thank you very much!

[@saurabhjoshi3507]

Plz give some examples for more understanding

[@josephhajj1570]

But why we multiply by the jacobian is there any proof

[@cesarpascualramos7796]

Yeah there is! From what I got from my class, it is a way to "adjust" for deformation. This deformation being caused by the change of variables from Cartesian to Spherical, or really to any other coordinate system as long as I know. Change of variables is based on Linear Maps, which can retain shapes and geometry but cannot retain Areas. Thats why we adjust, we adjust for that change in the area of the region of integration. I am just a student so I dont fully comprehend this, but maybe you can look further into it yourself

[@Han-ve8uh]

6:45 Why is the function just 1? How about 2, 3, ... ? Is the answer to this the same reasoning as why the function is 1 for the previous triple integrals in cartesian coords too? (I also don't understand why the function is 1 there)

[@coleadams567]

so useful. Thanks!

[@danielbornstein3701]

Im confused as to why the theta bounds are from 0 to 3pi/2 and not pi/2 to 2pi given the projection of this specific shape onto the xy plane. If we look at it like the normal unit circle with a horizontal x axis wouldnt the bounds be pi/2 to 2pi?

[@irwinedbert2436]

what if for example i have a function sqrt(x^2/y) and then i find the integral of this function, that is bounded by x^2+y^2+z^2=1 ( Spherical Coordinat System). Do i get the intersectional volume between these two function?

[@AsmitSou]

hello how you made us imagine why Jacobian came in role for converting normal form to parametric form or polar form in case of double integral similarly can make video on triple integral for rohsqr* sinphi.... thank you

[@michaelgomez3364]

This is excellent

[@mlop2484]

thnkx for helping out the science;!)

[@calebmyatt1021]

Thank you, I was confused about the (phi?) values

[@abeekuampah2419]

Very helpful video ! Do you have a video on how to get the limits of the integration in a more general way?

[@Mysoi123]

Thank you so much Sir!

[@sectumsempre]

AMAZING!

[@petelok9969]

Trefor, if the phi angle was not constant but a some function of z would we then proceed using the f(z) as our function instead of 1, would that work? Peter

[@jacksoncrocker1400]

Why does the rotation of Theta start at the Y-axis. I would have written the limits of integration as pi/2 to 2pi. Don't we usually start on the x-axis for the rotation of Theta?

[@GuohaoChen-jx9to]

I'm a little bit lost why f(rou, fi, theta) = 1 will give me the volume. Could anyone explains this to me? Thanks!

[@benpct5555]

What about for an oblate spheroid of equation (x^2/a^2 + y^2/a^2 + z^2/b^2 = 1) such as the Earth (where a and b are the equatorial radii and polar radii respectively)?

[@carultch]

Start with the equation you provided: x^2/a^2 + y^2/a^2 + z^2/b^2 = 1 Replace each of the Cartesian variables, with their equivalent expressions in spherical coordinates. I'll use theta for the complement of the latitude, and phi for longitude. x = r*sin(theta)*cos(phi) y = r*sin(theta)*sin(phi) z = r*cos(theta) where r is the general radial coordinate form the origin (r*sin(theta)*cos(phi))^2/a^2 + (r*sin(theta)*sin(phi))^2/a^2 + (r*cos(theta))^2/b^2 = 1 Factor out r^2: r^2*[(sin(theta)*cos(phi))^2/a^2 + (sin(theta)*sin(phi))^2/a^2 + (cos(theta))^2/b^2] = 1 Factor the first two terms inside the brackets, to gather what they have in common: r^2*[sin(theta)/a^2 *(cos(phi))^2 + sin(phi))^2) + (cos(theta))^2/b^2] = 1 We see the fundamental pyth trig identity, and can replace it with 1, eliminating all dependence on p: r^2*[sin(theta)/a^2 + (cos(theta))^2/b^2] = 1 Solve for r: r = 1/sqrt((sin(theta))^2/a^2 + (cos(theta))^2/b^2) And we have our spherical coordinate equation r(theta, phi) for an oblate spheroid in 3-dimensions, which only depends on theta and the two parameters a and b.

[@benpct5555]

@@carultch Just realised I never thanked you for this, appreciate it!

[@meletaitumelo9550]

Hey… I’m a little confused on how you found the interval of theta to be exactly between 0 and pi/6

[@recsoncanton9965]

May i ask what will happen to the function inside the integral if the distance of rho is not equal to 1?

[@abdullahe3674]

yes Prof plz resume and complet the playlist :) kind regards

[@jagsproduction]

Please make a video on surface, volume differential in spherical coordinate system!

[@adamsusel3801]

Any recommendations for a video that does get into the geometry? My brain doesn’t like blind formulas lol

[@DrTrefor]

Try it yourself! Earlier in the playlist I did polar nice and carefully, it is the same idea for spherical just have to be careful with messier trig.

[@shachidevmahato]

thak you so much sir. i want to know how you represent this and how you make this type 3d graph.

[@mohamedasaad7883]

أنا أريد أن أعرف أيضا

[@mohamedasaad7883]

لكن قد يكون برنامج الجيواجبرا او الماتلاب

[@mohamedasaad7883]

لكن انا اريد ان أعرف كيف يقوم بتركيب هذه الرسمات علي الفديو

[@tsalt]

Hello. How is dV calculated in ur video?

[@scottboyer7774]

watching this the night before my calc 3 final!

[@DrTrefor]

Good luck!

[@DrTrefor]

Good luck!

[@BraveGisgo]

amazing!!

[@sangeetachauhan7498]

sir i try to calculate the value of dv and get rho ^2.d theta . d rho . dfi i did not get the sin fi

[@ahmedrohaim3366]

I am trying to understand the physical meaning for spherical harmonics by understanding the shapes of cones (orbitals) in polar coordinates as shown here. I still struggle, I cannot quite imagine it yet. Could you make a video for that?

[@kavehsiah4021]

Damn, I was hoping the geometry of the end of the integral would be explained. I've seen the rho^2 term be derived from a Jacobian before, and I was wondering why we were able to do that.

[@jasonfrimpong7208]

Thank you

[@justinw8370]

Hi, the equation of this is a sphere, so isn't f(p,phi,theta)= p , so we have to integrate over p^3sin(phi)? Why is this not the case?

[@josephmonte1470]

Hi, you have great videos! I was wondering what software you use to create your images in the 3d coordinate system?

[@nurulirwanie4703]

sir, how to find the upper limit and lower limit for theta if given x ≥0 and y ≥ 0?

[@ankitghanshala7382]

0 to π maybe 🤔

[@s.tahsin.r2288]

thanks a bunch

[@mrfrk23]

thanks. help a lot

[@divyenduroychoudhuri7267]

Sir it was a very good explanation. But I have one doubt. Why we took the limit of theta from 0 to (3 pi / 2). Wouldn't it be 0 to (2pi) ? Please clarify my doubt. Thank you sir

[@cesarpascualramos7796]

If you take a look, there is a sector missing from the cone. The missing sector is 1/4 of a full revolution, which turns out to be (3/2)π expressed in radians. Hope you understand!

[@anandapatmanabhansu]

Thankyou sir

[@larich370]

thank u soooo much

[@headmasterschool5844]

Tanks teacher