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Integration in Spherical Coordinates

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Dr. Trefor Bazett

Dr. Trefor Bazett

Күн бұрын

Spherical Coordinates is a new type of coordinate system to express points in three dimensions. It consists of a distance rho from the origin to the point, an azimuthal angle phi between this line and the z axis, and a radial angle theta which is analogous to the angle in polar coordinates when you project onto the xy plane. In this video we will talk about integrating regions in spherical coordinates. Much like polar the conversion introduces a factor of rho squares sin phi in the integrand. The choice of which coordinate system to use depends on the symmetries of the region involved and the types of terms in the integrand, but in many cases spherical coordinates are easier to integrate than trying to deal with the same region in cartesian or polar coordinates.
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Пікірлер: 163
@menoima9501
@menoima9501 3 жыл бұрын
underrated,underrated,underrated,underrated,underrated,underrated
@sulfur9915
@sulfur9915 Жыл бұрын
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!
@internetroamer8063
@internetroamer8063 4 жыл бұрын
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
@intsciencewing6870 3 жыл бұрын
Visuals + facial expressions + hand moments, all are working in harmony with each other.
@gnaved
@gnaved 3 жыл бұрын
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
@DrTrefor 3 жыл бұрын
So glad they are helping!
@user-amanfeildconvecture
@user-amanfeildconvecture Жыл бұрын
​@@DrTreforyou deserves
@raisaraminnadia7642
@raisaraminnadia7642 3 жыл бұрын
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!!!!
@observever7808
@observever7808 3 жыл бұрын
The illustration was brilliant, solved my questions at once.
@SoumikRoyBAI
@SoumikRoyBAI 3 жыл бұрын
Literally watched every video in a day and im feeling extremely confident about multivariable calculus now. Amazing explanations😍
@DrTrefor
@DrTrefor 3 жыл бұрын
Great to hear!
@tomatrix7525
@tomatrix7525 3 жыл бұрын
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.
@Bill0102
@Bill0102 7 ай бұрын
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
@comfychairs795
@comfychairs795 3 жыл бұрын
the visuals are super helpful in comprehending what each variable corresponds to :)
@DrTrefor
@DrTrefor 3 жыл бұрын
Happy to hear that!
@punyaputtprakaivichien4727
@punyaputtprakaivichien4727 2 жыл бұрын
I was surrenderring on advanced calculus, however, your teaching makes me able to stands up and fight against it again. Thank you Fr.Trefor
@bethj7106
@bethj7106 Ай бұрын
Thank you! Seeing where the conversion from cartesian to spherical coordinates comes from was extremely helpful!!
@kshitizpoudel-jp7cc
@kshitizpoudel-jp7cc 3 жыл бұрын
I just discovered the channel today. I really love your content and the way you explain everything.
@DrTrefor
@DrTrefor 3 жыл бұрын
Glad you enjoy it!
@saleplains
@saleplains 4 жыл бұрын
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
@l.w915 3 жыл бұрын
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!❤
@marianaasko6604
@marianaasko6604 4 жыл бұрын
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.
@sazokuotsutsuki8848
@sazokuotsutsuki8848 9 ай бұрын
Today i missed the math lecture talking about spherical coordinate system in triple integrals and this video saved me fr
@TheCypher93
@TheCypher93 Жыл бұрын
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.
@masteroftheuniverse7418
@masteroftheuniverse7418 3 жыл бұрын
These vidoes have been doing a great job at explaining these topics. Great job!
@DrTrefor
@DrTrefor 3 жыл бұрын
Glad you like them!
@unsabhatti3639
@unsabhatti3639 3 жыл бұрын
This is exactly what I was looking for. Thank you.
@KRITISHPALRECHABRS
@KRITISHPALRECHABRS 3 жыл бұрын
You make such difficult concepts so easy..i love the way you teach. Thanks :-)
@DrTrefor
@DrTrefor 3 жыл бұрын
You're very welcome!
@cesarpascualramos7796
@cesarpascualramos7796 2 жыл бұрын
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!
@baqerghezi1342
@baqerghezi1342 2 жыл бұрын
Wow you really make math easy good job professor. Thank you
@DrTrefor
@DrTrefor 2 жыл бұрын
Thank you so much!!
@amatris
@amatris 3 жыл бұрын
7 min can teach you better than 2 hours. Thank you a lot.
@hubenbu
@hubenbu 2 жыл бұрын
unveil the beauty of Calculus
@bryan-9742
@bryan-9742 4 жыл бұрын
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.
@krittikasarkar6989
@krittikasarkar6989 2 жыл бұрын
SIMPLY AMAAAAZINGG!!!💯💯❤️❤️❤️❤️❤️❤️
@smokey7146
@smokey7146 9 ай бұрын
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 !!!
@shishirallu3439
@shishirallu3439 2 жыл бұрын
Thanks for the explanation. The 3d model helped a lot, dr. trevor.
@SciHeartJourney
@SciHeartJourney 2 жыл бұрын
This was excellent! Thank you very much Dr. Bazett.
@caleb5688
@caleb5688 2 жыл бұрын
I'd love to see a video breaking down the geometry behind the formula!
@torfoes
@torfoes 4 жыл бұрын
best explanation on youtube!!
@rakoonberry7879
@rakoonberry7879 3 жыл бұрын
This video is so good. Love the illustrations!
@ruzzcraze1862
@ruzzcraze1862 2 жыл бұрын
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
@DrTrefor 2 жыл бұрын
That is amazing:D
@emtnan1733
@emtnan1733 Жыл бұрын
You are the best 👏🏻
@abdulraheemtahir6269
@abdulraheemtahir6269 3 жыл бұрын
The dude atleast deserves a million followers! A T L E A S T !!!
@newaymamushet7956
@newaymamushet7956 2 жыл бұрын
brilliant teacher
@skcr3220
@skcr3220 4 ай бұрын
this guy saved my engineering program
@thomasclark210
@thomasclark210 9 ай бұрын
I often see maximum limits on theta and phi. Many volume integrals can be done over the range 0
@pasataizkonzuma
@pasataizkonzuma 2 жыл бұрын
Can you explain dV breakdown? The geometry behind it? or maybe point to a resource?
@eexxo0
@eexxo0 4 жыл бұрын
Nice video. I'd like to add that in physics the annotations of Θ and 𝜙 are inverted as per ISO/IEC 80000
@kaweaakuna7325
@kaweaakuna7325 2 жыл бұрын
Azimuthal angle👍 thanks bro💪
@aymenazhar8946
@aymenazhar8946 2 жыл бұрын
Helpful video 😍😍 Love from Pakistan❤❤❤
@ogunsadebenjaminadeiyin2729
@ogunsadebenjaminadeiyin2729 3 жыл бұрын
Super useful, excellent visuals
@iyadindia862
@iyadindia862 2 жыл бұрын
You know how to teach well ❤️
@ethandrost6974
@ethandrost6974 Жыл бұрын
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.
@cedricpulmano9234
@cedricpulmano9234 Жыл бұрын
THE GOAT! HE IS THE GOAT!
@muhammadumarsotvoldiev8768
@muhammadumarsotvoldiev8768 2 жыл бұрын
thank u. Amazing video
@ApunHandles
@ApunHandles 3 жыл бұрын
Love this video ,it made abstract things to concrete
@DrTrefor
@DrTrefor 3 жыл бұрын
Glad you enjoyed!
@nevis4567
@nevis4567 3 жыл бұрын
Thanks man! Big help before my calc lecture
@DrTrefor
@DrTrefor 3 жыл бұрын
Glad it helped!
@xXEpicxManXx
@xXEpicxManXx 3 жыл бұрын
Beautiful video!
@DrTrefor
@DrTrefor 3 жыл бұрын
Thank you very much!
@saurabhjoshi3507
@saurabhjoshi3507 3 жыл бұрын
Plz give some examples for more understanding
@hannarula
@hannarula 4 жыл бұрын
Thank you for an explanation, great video!
@syabongajele6431
@syabongajele6431 3 жыл бұрын
well explained SIR much appreciated
@cavelinguam6444
@cavelinguam6444 Жыл бұрын
Awesome Channel
@ice_the_kicker
@ice_the_kicker 3 жыл бұрын
So very very very very well done!
@DrTrefor
@DrTrefor 3 жыл бұрын
Thank you!!
@sectumsempre
@sectumsempre 2 жыл бұрын
AMAZING!
@hassanconteh6654
@hassanconteh6654 3 жыл бұрын
Thanks doc u are such a wonderful explainer
@DrTrefor
@DrTrefor 3 жыл бұрын
Thank you so much!
@kaokaoylan4510
@kaokaoylan4510 2 жыл бұрын
Thank you, its been a great help!
@kavehsiah4021
@kavehsiah4021 9 ай бұрын
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.
@josephhajj1570
@josephhajj1570 4 жыл бұрын
But why we multiply by the jacobian is there any proof
@cesarpascualramos7796
@cesarpascualramos7796 2 жыл бұрын
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
@calebmyatt1021
@calebmyatt1021 2 жыл бұрын
Thank you, I was confused about the (phi?) values
@mlop2484
@mlop2484 11 ай бұрын
thnkx for helping out the science;!)
@abdulmalek1118
@abdulmalek1118 2 жыл бұрын
Thanks a lot Can I know what program do you use for constructing these brilliant illustrations
@jasonfrimpong7208
@jasonfrimpong7208 2 жыл бұрын
Thank you
@Mysoi123
@Mysoi123 2 жыл бұрын
Thank you so much Sir!
@danielbornstein3701
@danielbornstein3701 9 ай бұрын
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?
@ohsnapigetit4631
@ohsnapigetit4631 3 жыл бұрын
Dig the model, very helpful
@kylewong2067
@kylewong2067 3 жыл бұрын
Thank you, good sir!
@abeekuampah2419
@abeekuampah2419 11 ай бұрын
Very helpful video ! Do you have a video on how to get the limits of the integration in a more general way?
@larich370
@larich370 3 жыл бұрын
thank u soooo much
@anandapatmanabhansu
@anandapatmanabhansu 2 жыл бұрын
Thankyou sir
@nurulirwanie4703
@nurulirwanie4703 2 ай бұрын
sir, how to find the upper limit and lower limit for theta if given x ≥0 and y ≥ 0?
@coleadams567
@coleadams567 2 жыл бұрын
so useful. Thanks!
@rizqiadriyanto1090
@rizqiadriyanto1090 3 жыл бұрын
thanks you Dr. Trefor
@abdullahe3674
@abdullahe3674 3 жыл бұрын
yes Prof plz resume and complet the playlist :) kind regards
@michaelgomez3364
@michaelgomez3364 3 жыл бұрын
This is excellent
@jacksoncrocker1400
@jacksoncrocker1400 2 жыл бұрын
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?
@meletaitumelo9550
@meletaitumelo9550 Жыл бұрын
Hey… I’m a little confused on how you found the interval of theta to be exactly between 0 and pi/6
@continnum_radhe-radhe
@continnum_radhe-radhe 2 жыл бұрын
🔥🔥🔥
@user-ov7ik4sb9t
@user-ov7ik4sb9t Жыл бұрын
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
@jagsproduction
@jagsproduction 3 жыл бұрын
Please make a video on surface, volume differential in spherical coordinate system!
@shachidevmahato
@shachidevmahato 3 жыл бұрын
thak you so much sir. i want to know how you represent this and how you make this type 3d graph.
@mohamedasaad7883
@mohamedasaad7883 Жыл бұрын
أنا أريد أن أعرف أيضا
@mohamedasaad7883
@mohamedasaad7883 Жыл бұрын
لكن قد يكون برنامج الجيواجبرا او الماتلاب
@mohamedasaad7883
@mohamedasaad7883 Жыл бұрын
لكن انا اريد ان أعرف كيف يقوم بتركيب هذه الرسمات علي الفديو
@user-iz2et4kj9y
@user-iz2et4kj9y Ай бұрын
Thanks!
@GuohaoChen-jx9to
@GuohaoChen-jx9to Жыл бұрын
I'm a little bit lost why f(rou, fi, theta) = 1 will give me the volume. Could anyone explains this to me? Thanks!
@bengper8044
@bengper8044 3 жыл бұрын
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
@carultch Жыл бұрын
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.
@Han-ve8uh
@Han-ve8uh 3 жыл бұрын
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)
@mrfrk23
@mrfrk23 4 жыл бұрын
thanks. help a lot
@ethan2077
@ethan2077 4 ай бұрын
How do you know phi is between 0 and pi/6??
@adamsusel3801
@adamsusel3801 3 жыл бұрын
Any recommendations for a video that does get into the geometry? My brain doesn’t like blind formulas lol
@DrTrefor
@DrTrefor 3 жыл бұрын
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.
@irwinedbert2436
@irwinedbert2436 3 жыл бұрын
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?
@schizoframia4874
@schizoframia4874 Жыл бұрын
Why is there an R under the triple integral
@spehropefhany
@spehropefhany 2 ай бұрын
R is for Region
@ahmedrohaim3366
@ahmedrohaim3366 2 жыл бұрын
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?
@recsoncanton9965
@recsoncanton9965 2 жыл бұрын
May i ask what will happen to the function inside the integral if the distance of rho is not equal to 1?
@justinw8370
@justinw8370 3 жыл бұрын
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?
@maanmoa
@maanmoa Жыл бұрын
your a legend
@iluvyunie
@iluvyunie Жыл бұрын
can always count on trefor
@tsalt
@tsalt 2 жыл бұрын
Hello. How is dV calculated in ur video?
@s.tahsin.r2288
@s.tahsin.r2288 4 жыл бұрын
thanks a bunch
@sebjornsprauten1406
@sebjornsprauten1406 3 жыл бұрын
amazing!!
@petelok9969
@petelok9969 3 жыл бұрын
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
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