Integration in Spherical Coordinates

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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[@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!

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

[@unsabhatti3639]

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

[@DrTrefor]

Happy to hear that!

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This video is so good. Love the illustrations!

[@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.

[@SoumikRoyBAI]

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

[@DrTrefor]

Great to hear!

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

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Glad you enjoy it!

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[@DrTrefor]

Glad you like them!

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Thank you for an explanation, great video!

[@amatris]

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

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best explanation on youtube!!

[@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.

[@ogunsadebenjaminadeiyin2729]

Super useful, excellent visuals

[@shishirallu3439]

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

[@caleb5688]

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

[@KRITISHPALRECHABRS]

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

[@DrTrefor]

You're very welcome!

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[@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.

[@kaokaoylan4510]

Thank you, its been a great help!

[@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.

[@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 !!!

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[@DrTrefor]

Thank you so much!!

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Glad it helped!

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

[@syabongajele6431]

well explained SIR much appreciated

[@thomasclark210]

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

[@kaweaakuna7325]

Azimuthal angle👍 thanks bro💪

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unveil the beauty of Calculus

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Love this video ,it made abstract things to concrete

[@DrTrefor]

Glad you enjoyed!

[@abeekuampah2419]

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

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Thank you!!

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Dig the model, very helpful

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[@josephmonte1470]

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

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Beautiful video!

[@DrTrefor]

Thank you very much!

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Plz give some examples for more understanding

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This is excellent

[@pasataizkonzuma]

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

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thnkx for helping out the science;!)

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[@mrfrk23]

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[@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?

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

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[@calebmyatt1021]

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

[@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

[@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.

[@user-ov7ik4sb9t]

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

[@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?

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🔥🔥🔥

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thank you sir.....

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Great!!!!!

[@bengper8044]

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.

[@recsoncanton9965]

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

[@jasonfrimpong7208]

Thank you

[@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.

[@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?

[@headmasterschool5844]

Tanks teacher

[@scottboyer7774]

watching this the night before my calc 3 final!

[@DrTrefor]

Good luck!

[@DrTrefor]

Good luck!

[@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?

[@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)

[@anandapatmanabhansu]

Thankyou sir

[@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

[@meletaitumelo9550]

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

[@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

[@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?

[@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!