How Can We Divide Lamé Coefficients When Some Are Zero?

[LagrangeEuler]

Sometimes in calculations authors uses
$$\frac{1}{h_1h_2}=\frac{h_3}{h_1h_2h_3}$$
where $h_i, i=1,2,3$ are Lame coefficients. For instance in spherical coordinates $h_r=1$, $h_{\theta}=r$, $h_{\varphi}=r\sin \theta$. I am not sure how we can divide so easily Lame coefficients when some on them obviously can be zero for certain values of parameters. Can someone give me some explanation? Thanks a lot in advance.

 

[anuttarasammyak]

I have no background of the physics there but the formula seems multiplying the same number to denominator and numerator, so obviously right except the number is zero.

 

[Mark44]

Changed problem level from A to B. The underlying concept of Lame coefficients might be advanced, but in the posted problem all that was done was to multiply a fraction by 1 in the form of $h_3$ over itself.

 

[pasmith]

I am not sure how we can divide so easily Lame coefficients when some on them obviously can be zero for certain values of parameters. Can someone give me some explanation? Thanks a lot in advance.

You can divide a function by another function provided that the denominator is not identically zero; this reduces the domain by excluding points where the denominator is zero. In the context of Lame coefficients these are points where the coordinate system breaks down, with a single point being referred to by multiple distinct coordinate tuples. If you need to analyze something at these points, the answer is to use a different coordinate system.