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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This video was created by Dr. Trefor Bazett. I'm an Assistant Teaching Professor at the University of Victoria.

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[menoima]

underrated, underrated, underrated, underrated, underrated, underrated

[sulfur]

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!

[internetroamer]

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.

[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.

[Bill]

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[raisaraminnadia]

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!!!!

[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!!

[punyaputtprakaivichien]

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

[SoumikRoyBAI]

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

[observever]

The illustration was brilliant, solved my questions at once.

[bethj]

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

[comfychairs]

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

[TheCypher]

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.

[sazokuotsutsuki]

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

[tomatrix]

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.

[amatris]

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

[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.

[kshitizpoudel-jpcc]

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

[l.w]

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!❤

[ruzzcraze]

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.