Cutting straight to the chase, I need help with the derivation best described in this video here @ 7:48: https://youtu.be/AjDSLq-Pzcs?t=7m48s

By this point in the video, Adam Beatty has already established:

x=r*cos(Φ)*sin(θ) , y=r*sin(Φ)*sin(θ) , z=r*cos(θ)

It also makes sense to describe a general cartesian vector as:

__A__= A

_{x}

__i__+ A

_{y}

__j__+A

_{z }

__k__

And to then transfer the same logic to spherical and say:

__A__= A

_{r}

__r__+ A

_{θ }

__θ__+ A

_{Φ}

__Φ__

This makes the somewhat logical progression that we can just sub-in our x,y,z values into this general spherical vector formula to get

__A__= r*cos(Φ)*sin(θ)

__i__+ r*sin(Φ)*sin(θ)

__j__+ r*cos(θ)

__k__

He describes getting A

_{r}, A

_{θ}, and A

_{Φ}as simply just doing the partial derivative of our

__A__function by the respective variable (r, θ , Φ)

And this is what gives us the "End Result" that I can only ever seem to find.

A

_{r}= cos(Φ)*sin(θ)

__i__+ sin(Φ)*sin(θ)

__j__+ cos(θ)

__k__

A

_{θ }= r*cos(Φ)*cos(θ)

__i__+ r*sin(Φ)*cosθ)

__j__- r*sin(θ)

__k__

A

_{Φ}= -r*sin(Φ)

__i__+ r*cos(Φ)

__j__

My issue with this is: "How"?

Multiple comments point out that there appear to be the missing [sin(θ)]'s for the A

_{Φ }term which one would expect to still be there, but he never quite responded. In other words, it should look like this, no?:

A

_{Φ}= -r*sin(Φ)*sin(θ)

__i__+ r*cos(Φ)*sin(θ)

__j__

______________________________________________________

All other tutorials on the matter kind of just go into the Jacobian and spit out the ultimate final answer (saying that you need r

^{2}sin(θ)drdθdΦ when doing the change of coordinates)

But still, I mange to consistently see those terms [A

_{r}, A

_{θ}, and A

_{Φ}] in that layout. I never see such tutorials explain where those answers come from; this YouTube video was the closest it got to explaining (save the otherwise seemingly erroneous typo)

An additional resource which appears to kind of just conjure up the answer via Jacobian may be found here: https://en.wikipedia.org/wiki/Vector_fields_in_cylindrical_and_spherical_coordinates#Spherical_coordinate_system

But I'm not the best when it comes to following abstract math when it's just delivered with such little context. And yet again, it still fails to have that sin(θ) which is my cause for all the confusion in the first place.

______________________________________________________________

Anyway, thank you in advance for whatever help you may provide.

-Joshua

I suppose I should also mention that the way Adam works with the spherical coordinates is that

r = Length of vector

Φ = Angle within the xy-plane (Azimuthal angle)

θ = Angle off of the Z-axis

(Also, terribly sorry if formatting is a bit choppy, evidently these Theta and Phi symbols throw off the text in some kind of way.)