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So I am working on my assignment, and have a question about converting coordinates. I dont know whether I should ask here or the math SE, so lets give it a try here.

The force in question is $$\vec{F} = -k\,r^{-n}\,\hat{r}$$

I know the conversion equations but I have no idea how to do it. I need to split the $\hat{r}$ into its corresponding i, j, k vectors. I wanna take the curl to make sure that its zero and hence the force is conservative.

Note: the question asks to be converted. I know how to already do it in spherical coordinates.

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1 Answer 1

up vote 3 down vote accepted

$$r^2 = x^2 + y^2 + z^2$$

What does that make $r^{-n}$?

As for converting $\hat{r}$, the position vector can be written

$$\vec{r} = r \hat{r}$$

in spherical coordinates, but it can also be written

$$\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$$

in rectangular coordinates. Therefore, the two are equal.

$$r \hat{r} = x\hat{i} + y\hat{j} + z\hat{k}$$

From there you can finish up finding a formula for $\hat{r}$ in rectangular coordinates.

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1  
What $r^{-n}$ be? Would it be $=x^{-n}+y^{-n}+z^{-n}$? –  masfenix Feb 18 '11 at 2:21
    
Nope, but if you think about it a little more I'm sure you'll see how to do it. How do you take a number to the power -n? What would happen if that number is r^2? –  Colin K Feb 18 '11 at 2:49
    
I don't think I am doing this right. –  masfenix Feb 18 '11 at 4:53
    
@masfenix, Ouch on the algebra! –  Carl Brannen Feb 18 '11 at 5:37

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