Converting cartesian unit vectors to spherical unit vectors

[kde2520]

Homework Statement

Well, it's all in the title. I just need to show that Gauss's theorem applies to this fluid flow and have converted all my (x,y,z) components to their respective (r,theta,phi) versions, but I can't remember the spherical counterparts of $$\hat{x}$$,$$\hat{y}$$,$$\hat{z}$$.

 

[Dick]

Wouldn't it actually be easier to just look them up rather than to post here? http://mathworld.wolfram.com/SphericalCoordinates.html

 

[kde2520]

Wouldn't it have been easier to look at the page you linked to and see that there is no direct conversion there than to assume that I didn't already do that and reply to my post?... maybe not. Thanks though.

 

[Dick]

I thought you wanted $$\hat{r}, \hat{\theta}, \hat{\phi}$$. I would call those the "spherical counterparts" of the cartesian basis vectors. What do you want?

 

[kde2520]

You're right, I didn't word it very well. Here's what I was trying to compute:

$$\vec{u} \cdot d\vec{A}$$

where

$$\vec{u} = (2xy^{2}+2xz^{2})\hat{x} + (x^{2}y)\hat{y} + (x^{2}z)\hat{z}$$

and

$$d\vec{A} = r^{2}sin\theta d\theta d\phi \hat{r}$$.

I first converted the magnitudes of $$\vec{u}$$ to spherical coordinates, and what I intended to do was convert $$\hat{x}, \hat{y}, \hat{z}$$ to spherical as well. Does that make sense?

Anyway, I ended up using $$\hat{r} = sin\theta cos\phi \hat{x} + sin\theta sin\phi \hat{y} + cos\theta \hat{z}$$, leaving the unit vecors in the cartesian basis.

But then when I tried to compute the scalar product I ended up with a virtually unintegratable expression. The left side of the equation (remember I was trying to show that Gauss' theorem holds over a spherical region $$a^{2} = x^{2} + y^{2} + z^{2}$$) came out to $$\frac{8}{3} \pi a^{5}$$.

Do you think I was setting it up right? I was thinking that I probably skrewed up some algebra while computing the dot product.

 

[Dick]

If you are integrating over a sphere of radius a, then the normal vector is (x*xhat+y*yhat+z*zhat)/a. Does that suggest any simplifications? Sorry, it's kind of late here so I haven't really thought this through.

 

[Dick]

Another tip. Once you have your x,y,z expression for the integrand. You can choose the axis of the spherical coordinate system to point along any cartesian axis.

 

[ber70]

When I converted Laplacian from cartesian to spherical I use these unit vectors. I think my weblog will be usefull for you.
http://buyanik.wordpress.com/2009/05/02/laplacian-in-spherical-coordinates/"