Why is the differential ρ^2sinθ dρdθdφ? (the geometry)

We know the differential volume in the rectangular coordinate system is just dx dy dz but what is it in the spherical coordinate system? Many calculus have learned that dx dy dz = ρ^2sinθ dρ dθ dφ but why do we need the ρ^2sinθ ? This multivariable calculus tutorial provides a geometric approach to understanding the dV in the spherical coordinate system. Make sure you subscribe to ‪@bprpcalculusbasics‬ because we will do the dV by using the Jacobian soon!
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[@bprpcalculusbasics]

The Jacobian way: kzbin.info/www/bejne/q2Svd6Gcj8eeoZI

[@Anmol_Sinha]

Why is this video private :(

[@phill3986]

I like to think of it as a chunk of an orange peel

[@colmx8441]

Wake up babe, blackpenredpen just whipped out his Pokeball

[@Loots1]

is that a euphemism?

[@AlbertTheGamer-gk7sn]

Another way to interpret it is that in spherical coordinates, it involves a radius and a solid angle: dV = ρdρdΩ, and since dΩ = ρsin(φ)dφdθ, so dV = ρ^2sin(φ)dρdφdθ.

[@mitsosyt7853]

Can you make a video about multiple indefinite integrals (like the antiderivative of a function of multiple variables)? It seems that there are no theorems or even any mention of them online and it seem wierd how such a simple thought has no information online.

[@g4_61]

REALLY nice drawings; thank you! Please do the Jacobian method next; I would love to see you explain it.

[@MasayoshiTakanaka-w5k6c]

Are you dealing with measure and integration theory (especially Lebesgue integral) ?

[@programmer4047]

Please make a video on Jacobian

[@isharauditha4257]

throughout the entier last semester i was looking for this title and now you uploaded it after the exam

[@spyciarz170]

I HAVE AN EXAM FROM THESE THANK YOU BOSS

[@AsifAli-ym2cq]

That's 🍉

[@AlbertTheGamer-gk7sn]

Also, a similar approach is used in determining the surface area of a sphere (or anything in spherical coordinates) dS = = . Surface area of a unit sphere: ∬dS = ∬√((ρ^2*sin(φ))^2+(ρ*sin(φ)*ρ_φ)^2+(ρ*ρ_θ)^2)dφdθ where ρ_x is the partial derivative respect of x in the corresponding spherical ρ(φ, θ) function. In this case, ρ(φ, θ) = 1. Since the partial derivatives of the function = 0, then you will get this simple form ∬ρ^2*sin(φ)dφdθ, which is basically the differential of volume of a sphere without the dρ, due to a property that in an n-ball, the derivative of the volume respect to its radius is its surface area, so, dV = SdR, and solving this integral ∬{{0, 2π}, {0, π}}sin(φ)dφdθ = ∫{0, 2π}2R^2dθ = 4πR^2.

[@tobybartels8426]

Spherical coordinates are double polar coordinates; you can transform from rectangular coordinates to cylindrical coordinates and then from cylindrical coordinates to spherical coordinates. Each transformation puts an appropriate radius factor into the volume element, first r and then ρ. Then you write r using ρ and φ to get the final result. From the last video, we know that x = r cos θ and y = r sin θ give us dx dy = r dr dθ, so we also know that z = ρ cos φ and r = ρ sin φ give us dz dr = ρ dρ dφ in the same way. So dx dy dz = r dr dθ dz = r dz dr dθ = r ρ dρ dφ dθ. And since r = ρ sin φ, this is ρ² sin φ dρ dφ dθ.

[@jomariraphaellmangahas1991]

It's like you're explaining the perspective but in your own way I found this video interesting because it's one of the applications of camera lens

[@adrianavila6681]

Do the cylindrical coordinates differential please!

[@Kotuseid]

Your drawing skills are immaculate

[@TheMasterGreen]

what math is this?

[@Waller12]

Multi variable calculus / calc 3

[@jackrubin6303]

Thank you

[@r1ckthe]

Yo master can u help me solving this integral? ∫arcsin((x-2)/(x+2))

[@Mediterranean81]

Integration by parts

[@Mediterranean81]

Differentiate arcsin (x-2/x+2) Integrate 1

[@youceftalbi1077]

Like

[@Ntt903]

Yay!!!

[@iamtheone9242]

dear bprp, i searched for this video yesterday due to my final exam being today. However, you didnt upload the video untill today(AFTER MY FINAL EXAM) IT IS UNBELIEVABLE to do such a thing. REZALET...... NOT WATCHING YOU ANYMORE.

[@highviewbarbell]

lmaooo thats just tough luck man sorry to hear it xD