Help understanding math problem

[bobhays]

I need help understanding this vector calculus problem. I have the solutions too but I still dont get it.

So the problem is this

 

 

and the solution is

 

The part I dont understand is in the 2nd half of the solution (the checking part), how are they getting dA. I thought dA was just the variables not being held constant. For example dA = dR*d(theta) if the other angle is constant. How are they getting dA?

 

EDIT: I kind of understand how the dA is found because of the conversion from cartesian to spherical coordinates. I still don't understand how in the first half when taking the gradients why the factors in front of the partial derivatives are there i.e (1/r^2) and (1/r*sin(theta)) etc. and then the inverse is multiplied inside the derivative like r^2 in the first coordinate.

[minecraftyeet1]

I saw the first image and im out get a staff

member to help pm them

[Miguel Batista]

Yeah, the solution is not clear on that. They are using something called variable replacement where they call da the area integral that you have to calculate on each of the surfaces. Its just that. It makes the integral easier to solve.

It is not very useful here but as a matter of principle you should get used to using it. Also, teta is not constant.

Edit: well, this is not really variable replacement. Its called da because it is an area element. I left previous paragraph in because message is the same. Use variable replacement as a principle.

 

The solution otherwise is pretty self explanatory.

[bobhays]

1 minute ago, Miguel Batista said:

Yeah, the solution is not clear on that. They are using something called variable replacement where they call da the area integral that you have to calculate on each of the surfaces. Its just that. It makes the integral easier to solve.

 

It is not very useful here but as a matter of principle you should get used to using it. Also, teta is not constant.

 

The solution otherwise is pretty self explanatory.

I understand the dA part, and theta is constant for the integral of the bottom surface. What I dont understand is in the top half. When they took the gradient they added terms and I have no idea where they came from.

[Miguel Batista]

4 minutes ago, bobhays said:

I understand the dA part, and theta is constant for the integral of the bottom surface. What I dont understand is in the top half. When they took the gradient they added terms and I have no idea where they came from.

They didn't add terms. It has to do with spherical coordinates divergence. The formula is 1/r2 d/Dr (r2dfr) + 1/r sin(teta) d/dteta (sin(teta) fteta) + 1/r sin(teta) dfphi/dphi

 

When you integrate the divergent you have to integrate over the three coordinates and what you "added" to the equation was r2sin(teta) (the terms inside the brackets on r and teta coordinates.

 

That is just it. Now I don't remember the deduction but that is the divergence formulae you need to know.

 

[bobhays]

Ok apparently it was just another one of the tricks like the dA trick for converting from cartesian to spherical. Teacher going too fast skipping sections of the book. Thanks for the help though!

[Miguel Batista]

Just now, bobhays said:

Ok apparently it was just another one of the tricks like the dA trick for converting from cartesian to spherical. Teacher going too fast skipping sections of the book. Thanks for the help though!

Its not a trick. Its the formula... You are not converting coordinates you are calculating the divergent in spherical coordinates.

[jenia1]

if i understand you correctly. when you switch to spherical coordinate form thus terms replace the X,Y,Z coordinates to r,theta,gamma. if i remember correctly they are the  Jacobian here is the full formula 

:

[KkEvIn_Heart]

0 isn't equal to 1.

 

Thank me later.