Vector field change of variables

[ballzac]

Homework Statement

I just need to be able to change a vector field from spherical to cartesian
The question is about verifying stokes theorem (curl theorem) for a given vector field within and on a given path. It says not to use spherical coordinates, but the vector field is given in spherical coordinates.

Homework Equations

The Attempt at a Solution

I tried using the equations that relate r, theta, and phi to x, y, and z. One thing is, that it gets very messy, and I don't think it's meant to be. Another thing is that I don't know what to do with the unit vectors, as I have components that are (now) functions of x, y, and z but are pointing in the r, theta, and phi directions. I gather if I fixed that up, the messyness would disappear, but I don't know how to do it.

I also considered finding the curl in spherical coordinates and then using a Jacobian determinant to evaluate the integral, but I thought this might be kind of cheating.

If I can figure out how to change the vector field from spherical coordinates to cartesian I will have no trouble doing the integrals. I have a feeling I would've learned this in first or second year calculus, but I can't remember :( Any help would be appreciated.

 

[lanedance]

hi ballzac

i think you're correct you need to change the spherical unit vectors into cartesian unit vectors, have a look at this
http://en.wikipedia.org/wiki/Unit_vector

once you substitute in for the unit vectors and the variables, you should be left with a vector in terms of x,y,z and the cartesian unit vectors

you can always check your answer with the spherical form of the curl operator

 

[ballzac]

hi ballzac

i think you're correct you need to change the spherical unit vectors into cartesian unit vectors, have a look at this
http://en.wikipedia.org/wiki/Unit_vector

once you substitute in for the unit vectors and the variables, you should be left with a vector in terms of x,y,z and the cartesian unit vectors

you can always check your answer with the spherical form of the curl operator

Thanks for that. I'm still having a lot of trouble because it gets so complicated when I start changing the variables. There must be an easier way. Anyway, I think maybe if I just change the unit vectors (thanks to the link you posted), and leave the variables in polar form, then I can still maybe work it out in cartesian coordinates. Haven't thought this one through, but it might work. Thanks again for the help. I'll keep working on this...

 

[ballzac]

I'm going back to trying to change all the variables. One problem is that I have terms that are like sin(arcos(z/r)) this is obviously a projection of r in the x,y plane, but I don't know how to describe it precisely in cartesian coords because I don't know what phi direction it points in, arggg. This is causing me problems, lol.EDIT: I've found a list of trigonometric identities on wiki that has trig functions of inverse trig functions, and this seems to be simplifying quite dramatically, so we'll see how it goes.

 

[lanedance]

can you post the original vector field?

 

[ballzac]

certainly,

$$\textbf{f(r)}=r cos^2\theta\textbf{\hat{r}}-r cos\theta sin\theta\textbf{\hat{\theta}}+3r\textbf{\hat{\varphi}}$$


WTF? I typed it in tex, and when previewing it it has come up as something completely different. In fact it looks like something I typed in a completely different thread once. ? Very strange. Anyway, in case it doesn't come up properly above (if it contains A and a it is not the right thing)... here it is

f(r)=rcos^2(theta)r_hat-rcos(theta)sin(theta)theta_hat+3rphi_hat

It is actually problem 1.56 in griffiths "Introduction to electrodynamics", which in the solution manual is done in spherical coordinates, hence why we're not allowed to use them.

 

[ballzac]

Here is the question
https://www.physicsforums.com/attachment.php?attachmentid=18133&stc=1&d=1237876170
After this it says not to use spherical coordinates, but I didn't scan that.I have tried to calculate what the field is in cartesian coordinates, and after substituting the equations for the variable and the unit vectors, I cam up with this (the bit circled in red is the field that I calculated...sorry if the rests so messy. I thought I'd leave it into a t least show a little bit of how I got to the final bit.)
https://www.physicsforums.com/attachment.php?attachmentid=18132&stc=1&d=1237876170
Underneath you can see that I started calculating the line integral for what I labeled as segment 1. Then (I didn't scan this bit) I tried to do segment 2, but x is zero along segment 2, and there are two integrals for the second part, one for the y direction and one for z. In the y part, you can see that I end up needing to divide by zero when I set x=0.

I'm sure that it's not meant to be this complicated. I can't imagine how difficult it would be to get the curl the way I'm doing it.

EDIT:Damn, the pics don't seem to be working. EDIT2: Oh I see, duh.

Oh, also, I left r in the equations where it still exists just for brevity.

Yet another edit: I am getting somewhere. I'm sure I can (and will need to) simplify it further, but now I have

$$-3y\sqrt{x^2+y^2+z^2/(x^2+y^2)}\hat{\textbf{x}}+3x\sqrt{x^2+y^2+z^2/(x^2+y^2)}\hat{\textbf{y}}+z\hat{\textbf{y}}$$And for the grand finale...I've got the left hand side done and I know it's right because it agrees with the version in the book done in spherical coordinates. I should be okay from here. Thanks again for the help.

 

[lanedance]

Hi Ballzac

I just had a go and after substitution and cancelling get to:
$$-3r.sin(\phi).\hat{x} + 3r.cos(\phi) .\hat{y} + r.cos(\theta).\hat{z}$$

edit: ok - you're all good, cool