Is there a mistake in this explanation of spherical coordinates?

[leinadle]

The explanation for this is probably very trivial, but I am not seeing it. I'm reading through Ramamurti's "Basic Training in Mathematics" and in the Multivariable Calculus section for spherical coordinates he says

Using the fact that sinθdθ = dcosθ = dz

and then shows an integral. Shouldn't there be a negative sign before the cosθ? And how is this equal to dz?

Thanks in advance

 

[Mark44]

The explanation for this is probably very trivial, but I am not seeing it. I'm reading through Ramamurti's "Basic Training in Mathematics" and in the Multivariable Calculus section for spherical coordinates he says



and then shows an integral. Shouldn't there be a negative sign before the cosθ? And how is this equal to dz?

Thanks in advance

Yes, there should be a minus sign.
Apparently things started with this:

z = cosθ
dz = d(cosθ) = -sinθ dθ

It looks to me like the author is doing an ordinary substitution (z = cosθ) before carrying out the integration.

 

[leinadle]

Yes, there should be a minus sign.
Apparently things started with this:

z = cosθ
dz = d(cosθ) = -sinθ dθ

It looks to me like the author is doing an ordinary substitution (z = cosθ) before carrying out the integration.

Ah, I see. Thanks for the quick response!