Question

write the equation in spherical coordinates

Answer 1

\[z = x^2+ y^2\]

Answer 2

\[x=rhosin(\phi)\cos(\theta)\]
\[y=rhosin(\phi)\sin(\theta)\]
\[z=rhocos(\phi)\]

Answer 3

yup now you just gotta plug these in. :)

Answer 4

so far I have \[rhocos(\phi) = (rhocos(\theta)\sin(\phi))^{2} + (rhosin(\theta)\sin(\phi))^{2}\]

Answer 5

The final answer was simplified to \[\csc(\phi)^{2}\]

Answer 6

I'm not sure how that was done

Answer 7

well you can factor stuff out of the right side like the rho and sin phi, actually you can divide out a rho from both sides as well, and don't forget this kind of rule of how the exponent distributes to help you out : \[(abc)^2=a^2b^2c^2\]

Answer 8

ok: \[\cos(\phi) = rhocos(\theta)^2\sin(\phi)^2 + rhosin(\theta)^2\sin(\phi)^2\]

Answer 9

So i cancelled out the row on the left side and distributed the square

Answer 10

*rho

Answer 11

awesome, ok now on the right side factor out as much as you can and see if you recognize anything.

Answer 12

ohhhhhh, well spotted

Answer 13

haha cool

Answer 14

ok so now I have \[\cos(\phi) = rhosin(\phi)^2\]

Answer 15

hmm that looks different than what you said slightly but earlier what you wrote wasn't really an equation so idk exactly. but since 1/sin^2 is gonna be csc^2 when you solve for rho maybe we missed getting rid of cos phi somehow by accident lets see hmmm

Answer 16

The solutions they have are in the attachment question 9. A.

Answer 17

I just don't understand the step after plugging in the values for x,y, and z

Answer 18

hmm it looks like they messed up and forgot the cosine in there honestly

Answer 19

ya I think you're right. do you understand the step after that though?

Answer 20

its like they cancelled out one rho and then divided both sides by sin(phi)^2

Answer 21

yeah that's basically what you ended up with, just gotta rearrange thisby dividing both sides by sin^2 which is the same as multiplying both sides by csc^2 \[\cos(\phi) = \rho \sin(\phi)^2\]

Answer 22

So you think they just messed up the answer, cause they don't even have rho in the final answer

Answer 23

oh yes they do, it's written as \[\rho = \frac{1}{\sin^2 \phi}=\csc^2 \phi\] but they messed up and it should be \[\rho = \frac{1}{\cos \phi\sin^2 \phi}=\csc^2 \phi \sec \phi \]

Answer 24

alright. Thanks for the help, I really appreciate it.