What is the vector field expressed in spherical coordinates?

[barnflakes]

Express the following vector field in spherical coordinates. (The
answer should be in a form that uses the unit vectors of the curvilinear coordi-
nate system and coefficient functions that are written in terms of the curvilinear
coordinates.)


$\underline{F} = -y \underline{i} + x \underline{j} + (x^2 + y^2)\underline{k}$

OK, so I've obtained the equation:

$\underline{F} = rsin\theta(-sin\phi\mathbf{i} + cos\phi\underline{j} +rsin\theta\underline{k})$ simply by substituting $x = rsin\theta cos\phi$ etc. into the above equations. Now how do I express this in terms of the unit vectors $\mathbf{e}_r,\mathbf{e}\phi, \mathbf{e}_\theta$ ??

 

[cristo]

Well, what are the unit vectors in spherical polars in terms of Cartesian unit vectors?

 

[barnflakes]

$\underline{e}_r = sin\theta(cos\phi \underline{i} + sin\phi{j}) + cos\theta\underline{k}$

$\underline{e}_{\theta} = cos\theta(cos\phi \underline{i} + sin\phi{j}) - sin\theta\underline{k}$

$\underline{e}_{\phi} = -sin\phi \underline{i} + cos\phi{j}$

I can't see how to write the above equation in terms of these unit vectors...

 

[gabbagabbahey]

$\underline{e}_r = sin\theta(cos\phi \underline{i} + sin\phi{j}) + cos\theta\underline{k}$

$\underline{e}_{\theta} = cos\theta(cos\phi \underline{i} + sin\phi{j}) - sin\theta\underline{k}$

$\underline{e}_{\phi} = -sin\phi \underline{i} + cos\phi{j}$

I can't see how to write the above equation in terms of these unit vectors...

You'll need to solve these 3 equations for i, j, and k. Then substitute the solutions into the equation from your previous post.

 

[barnflakes]

I mean really, I don't mean to sound ungrateful or anything, but how stupid do you think I am? I know what I have to do, I just don't know how to do it. In any event, I've solved it by myself. Note for the future: your method is slightly long winded. Thanks anyway!